VOLATILITY MODELING
The Stock Price Assumption
Consider a stock whose price is S
In a short period of time of length ∆119905 the return on the stock is normally distributed
∆119878
119878sim 119873(120583∆119905 1205902∆119905)
where 120583 is expected return and 120590 is volatility
The Lognormal Property
It follows from this assumption that
Since the logarithm of 119878119879 is normal 119878119879 is lognormally distributed
2
or
2
22
0
22
0
TTSS
TTSS
T
T
lnln
lnln
The Black-Scholes-Merton Formulas for Options
119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873 1198892
119875 = 119870119890minus119903119879119873 minus1198892 minus 1198780119873 minus1198891
where
1198891 =ln
1198780119870
+ 119903+1205902
2119879
120590radic(119879)
1198892 =ln
1198780119870
+ 119903minus1205902
2119879
120590radic(119879)= 1198891 minus 120590 119879
The N(x) Function
N(x) is the CDF of a standard normal random variable that is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x
5
Inputs of BS
Inputs
Current price 1198780
Strike price 119870
Risk-free rate 119903
Time to expiration 119879
Volatility 120590
Output
Price of the option
The Volatility
Volatility is a measure of the dispersion of returns ie a simple measure of asset return volatility is its variance over time
More formally volatility is standard deviation of the rate of return in 1 year
The standard deviation of the return in a short time period time Dt is approximately tD
Example Google stock prices last one year
In the BS model what value should be chosen for 120590 when pricing an option
Variance does not capture volatility clustering (periods of turbulence periods of calm impacts of economics events etc)
ndash It is a measure of unconditional variance
ndash It is a single number of a given sample
ndash It does not incorporate the past information
Volatility
Implied Volatility
The implied volatility of an option is the volatility for which the Black-Scholes-Merton price equals the market price
There is a one-to-one correspondence between prices and implied volatilities
Traders and brokers often quote implied volatilities rather than dollar prices
11
Implied Volatility
Consider a call option
119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873(1198892)119891 120590 = 119862 minus (1198780119873 1198891 minus 119870119890minus119903119879119873(1198892))
Find the zeros of 119891(120590)
Example
Example
Implied Volatility (IV)-Task
Why is IV important
Stocks vs Options You can theoretically hold a stock forever-no timeline Whereas when dealing options you have to deal with time component T and the implied volatility (IV how far do traders expect the stock to move)
ATM How much would you willing to pay for an ATM option
OTM How much would you be willing to pay for an OTM option
Market participantsrsquo belief determines how far would the stock move
Implied Volatility (IV)-Task
When IV is high people are willing to pay more money for an option contract
When IV is low people are willing to not pay a lot of money for an option contract
Suppose a stock with an IV of 20 is trading at $100 The expected range is $80-$120
68 of the time the stock will trade between $80 and $120
IV How do we know if IV is high or low
Consider two stocks Google and FedEx with IVs equal to 15
How do we determine if 15 is a high or a low volatility for GoogleFedEx
Suppose from the historical data for Google the high was 40 and the low was 10 Whereas for FedEx the high was 25 and the low was 5 Which stock would you trade
Lookup the historical highs and lows for each stock and determine the relative ranking-percentile
Auto-correlated Heteroscedasticity
Heteroscedasticity or unequal variabilityvariance
Auto-correlated heteroscedasticity
hellip is called the ARCH effect
Estimating Volatility
Autoregressive conditional heteroscedasticity (ARCH)
Exponentially weight moving average (EWMA)
Generalized autoregressive conditional heteroscedasticity
(GARCH)
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
The Stock Price Assumption
Consider a stock whose price is S
In a short period of time of length ∆119905 the return on the stock is normally distributed
∆119878
119878sim 119873(120583∆119905 1205902∆119905)
where 120583 is expected return and 120590 is volatility
The Lognormal Property
It follows from this assumption that
Since the logarithm of 119878119879 is normal 119878119879 is lognormally distributed
2
or
2
22
0
22
0
TTSS
TTSS
T
T
lnln
lnln
The Black-Scholes-Merton Formulas for Options
119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873 1198892
119875 = 119870119890minus119903119879119873 minus1198892 minus 1198780119873 minus1198891
where
1198891 =ln
1198780119870
+ 119903+1205902
2119879
120590radic(119879)
1198892 =ln
1198780119870
+ 119903minus1205902
2119879
120590radic(119879)= 1198891 minus 120590 119879
The N(x) Function
N(x) is the CDF of a standard normal random variable that is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x
5
Inputs of BS
Inputs
Current price 1198780
Strike price 119870
Risk-free rate 119903
Time to expiration 119879
Volatility 120590
Output
Price of the option
The Volatility
Volatility is a measure of the dispersion of returns ie a simple measure of asset return volatility is its variance over time
More formally volatility is standard deviation of the rate of return in 1 year
The standard deviation of the return in a short time period time Dt is approximately tD
Example Google stock prices last one year
In the BS model what value should be chosen for 120590 when pricing an option
Variance does not capture volatility clustering (periods of turbulence periods of calm impacts of economics events etc)
ndash It is a measure of unconditional variance
ndash It is a single number of a given sample
ndash It does not incorporate the past information
Volatility
Implied Volatility
The implied volatility of an option is the volatility for which the Black-Scholes-Merton price equals the market price
There is a one-to-one correspondence between prices and implied volatilities
Traders and brokers often quote implied volatilities rather than dollar prices
11
Implied Volatility
Consider a call option
119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873(1198892)119891 120590 = 119862 minus (1198780119873 1198891 minus 119870119890minus119903119879119873(1198892))
Find the zeros of 119891(120590)
Example
Example
Implied Volatility (IV)-Task
Why is IV important
Stocks vs Options You can theoretically hold a stock forever-no timeline Whereas when dealing options you have to deal with time component T and the implied volatility (IV how far do traders expect the stock to move)
ATM How much would you willing to pay for an ATM option
OTM How much would you be willing to pay for an OTM option
Market participantsrsquo belief determines how far would the stock move
Implied Volatility (IV)-Task
When IV is high people are willing to pay more money for an option contract
When IV is low people are willing to not pay a lot of money for an option contract
Suppose a stock with an IV of 20 is trading at $100 The expected range is $80-$120
68 of the time the stock will trade between $80 and $120
IV How do we know if IV is high or low
Consider two stocks Google and FedEx with IVs equal to 15
How do we determine if 15 is a high or a low volatility for GoogleFedEx
Suppose from the historical data for Google the high was 40 and the low was 10 Whereas for FedEx the high was 25 and the low was 5 Which stock would you trade
Lookup the historical highs and lows for each stock and determine the relative ranking-percentile
Auto-correlated Heteroscedasticity
Heteroscedasticity or unequal variabilityvariance
Auto-correlated heteroscedasticity
hellip is called the ARCH effect
Estimating Volatility
Autoregressive conditional heteroscedasticity (ARCH)
Exponentially weight moving average (EWMA)
Generalized autoregressive conditional heteroscedasticity
(GARCH)
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
The Lognormal Property
It follows from this assumption that
Since the logarithm of 119878119879 is normal 119878119879 is lognormally distributed
2
or
2
22
0
22
0
TTSS
TTSS
T
T
lnln
lnln
The Black-Scholes-Merton Formulas for Options
119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873 1198892
119875 = 119870119890minus119903119879119873 minus1198892 minus 1198780119873 minus1198891
where
1198891 =ln
1198780119870
+ 119903+1205902
2119879
120590radic(119879)
1198892 =ln
1198780119870
+ 119903minus1205902
2119879
120590radic(119879)= 1198891 minus 120590 119879
The N(x) Function
N(x) is the CDF of a standard normal random variable that is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x
5
Inputs of BS
Inputs
Current price 1198780
Strike price 119870
Risk-free rate 119903
Time to expiration 119879
Volatility 120590
Output
Price of the option
The Volatility
Volatility is a measure of the dispersion of returns ie a simple measure of asset return volatility is its variance over time
More formally volatility is standard deviation of the rate of return in 1 year
The standard deviation of the return in a short time period time Dt is approximately tD
Example Google stock prices last one year
In the BS model what value should be chosen for 120590 when pricing an option
Variance does not capture volatility clustering (periods of turbulence periods of calm impacts of economics events etc)
ndash It is a measure of unconditional variance
ndash It is a single number of a given sample
ndash It does not incorporate the past information
Volatility
Implied Volatility
The implied volatility of an option is the volatility for which the Black-Scholes-Merton price equals the market price
There is a one-to-one correspondence between prices and implied volatilities
Traders and brokers often quote implied volatilities rather than dollar prices
11
Implied Volatility
Consider a call option
119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873(1198892)119891 120590 = 119862 minus (1198780119873 1198891 minus 119870119890minus119903119879119873(1198892))
Find the zeros of 119891(120590)
Example
Example
Implied Volatility (IV)-Task
Why is IV important
Stocks vs Options You can theoretically hold a stock forever-no timeline Whereas when dealing options you have to deal with time component T and the implied volatility (IV how far do traders expect the stock to move)
ATM How much would you willing to pay for an ATM option
OTM How much would you be willing to pay for an OTM option
Market participantsrsquo belief determines how far would the stock move
Implied Volatility (IV)-Task
When IV is high people are willing to pay more money for an option contract
When IV is low people are willing to not pay a lot of money for an option contract
Suppose a stock with an IV of 20 is trading at $100 The expected range is $80-$120
68 of the time the stock will trade between $80 and $120
IV How do we know if IV is high or low
Consider two stocks Google and FedEx with IVs equal to 15
How do we determine if 15 is a high or a low volatility for GoogleFedEx
Suppose from the historical data for Google the high was 40 and the low was 10 Whereas for FedEx the high was 25 and the low was 5 Which stock would you trade
Lookup the historical highs and lows for each stock and determine the relative ranking-percentile
Auto-correlated Heteroscedasticity
Heteroscedasticity or unequal variabilityvariance
Auto-correlated heteroscedasticity
hellip is called the ARCH effect
Estimating Volatility
Autoregressive conditional heteroscedasticity (ARCH)
Exponentially weight moving average (EWMA)
Generalized autoregressive conditional heteroscedasticity
(GARCH)
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
The Black-Scholes-Merton Formulas for Options
119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873 1198892
119875 = 119870119890minus119903119879119873 minus1198892 minus 1198780119873 minus1198891
where
1198891 =ln
1198780119870
+ 119903+1205902
2119879
120590radic(119879)
1198892 =ln
1198780119870
+ 119903minus1205902
2119879
120590radic(119879)= 1198891 minus 120590 119879
The N(x) Function
N(x) is the CDF of a standard normal random variable that is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x
5
Inputs of BS
Inputs
Current price 1198780
Strike price 119870
Risk-free rate 119903
Time to expiration 119879
Volatility 120590
Output
Price of the option
The Volatility
Volatility is a measure of the dispersion of returns ie a simple measure of asset return volatility is its variance over time
More formally volatility is standard deviation of the rate of return in 1 year
The standard deviation of the return in a short time period time Dt is approximately tD
Example Google stock prices last one year
In the BS model what value should be chosen for 120590 when pricing an option
Variance does not capture volatility clustering (periods of turbulence periods of calm impacts of economics events etc)
ndash It is a measure of unconditional variance
ndash It is a single number of a given sample
ndash It does not incorporate the past information
Volatility
Implied Volatility
The implied volatility of an option is the volatility for which the Black-Scholes-Merton price equals the market price
There is a one-to-one correspondence between prices and implied volatilities
Traders and brokers often quote implied volatilities rather than dollar prices
11
Implied Volatility
Consider a call option
119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873(1198892)119891 120590 = 119862 minus (1198780119873 1198891 minus 119870119890minus119903119879119873(1198892))
Find the zeros of 119891(120590)
Example
Example
Implied Volatility (IV)-Task
Why is IV important
Stocks vs Options You can theoretically hold a stock forever-no timeline Whereas when dealing options you have to deal with time component T and the implied volatility (IV how far do traders expect the stock to move)
ATM How much would you willing to pay for an ATM option
OTM How much would you be willing to pay for an OTM option
Market participantsrsquo belief determines how far would the stock move
Implied Volatility (IV)-Task
When IV is high people are willing to pay more money for an option contract
When IV is low people are willing to not pay a lot of money for an option contract
Suppose a stock with an IV of 20 is trading at $100 The expected range is $80-$120
68 of the time the stock will trade between $80 and $120
IV How do we know if IV is high or low
Consider two stocks Google and FedEx with IVs equal to 15
How do we determine if 15 is a high or a low volatility for GoogleFedEx
Suppose from the historical data for Google the high was 40 and the low was 10 Whereas for FedEx the high was 25 and the low was 5 Which stock would you trade
Lookup the historical highs and lows for each stock and determine the relative ranking-percentile
Auto-correlated Heteroscedasticity
Heteroscedasticity or unequal variabilityvariance
Auto-correlated heteroscedasticity
hellip is called the ARCH effect
Estimating Volatility
Autoregressive conditional heteroscedasticity (ARCH)
Exponentially weight moving average (EWMA)
Generalized autoregressive conditional heteroscedasticity
(GARCH)
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
The N(x) Function
N(x) is the CDF of a standard normal random variable that is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x
5
Inputs of BS
Inputs
Current price 1198780
Strike price 119870
Risk-free rate 119903
Time to expiration 119879
Volatility 120590
Output
Price of the option
The Volatility
Volatility is a measure of the dispersion of returns ie a simple measure of asset return volatility is its variance over time
More formally volatility is standard deviation of the rate of return in 1 year
The standard deviation of the return in a short time period time Dt is approximately tD
Example Google stock prices last one year
In the BS model what value should be chosen for 120590 when pricing an option
Variance does not capture volatility clustering (periods of turbulence periods of calm impacts of economics events etc)
ndash It is a measure of unconditional variance
ndash It is a single number of a given sample
ndash It does not incorporate the past information
Volatility
Implied Volatility
The implied volatility of an option is the volatility for which the Black-Scholes-Merton price equals the market price
There is a one-to-one correspondence between prices and implied volatilities
Traders and brokers often quote implied volatilities rather than dollar prices
11
Implied Volatility
Consider a call option
119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873(1198892)119891 120590 = 119862 minus (1198780119873 1198891 minus 119870119890minus119903119879119873(1198892))
Find the zeros of 119891(120590)
Example
Example
Implied Volatility (IV)-Task
Why is IV important
Stocks vs Options You can theoretically hold a stock forever-no timeline Whereas when dealing options you have to deal with time component T and the implied volatility (IV how far do traders expect the stock to move)
ATM How much would you willing to pay for an ATM option
OTM How much would you be willing to pay for an OTM option
Market participantsrsquo belief determines how far would the stock move
Implied Volatility (IV)-Task
When IV is high people are willing to pay more money for an option contract
When IV is low people are willing to not pay a lot of money for an option contract
Suppose a stock with an IV of 20 is trading at $100 The expected range is $80-$120
68 of the time the stock will trade between $80 and $120
IV How do we know if IV is high or low
Consider two stocks Google and FedEx with IVs equal to 15
How do we determine if 15 is a high or a low volatility for GoogleFedEx
Suppose from the historical data for Google the high was 40 and the low was 10 Whereas for FedEx the high was 25 and the low was 5 Which stock would you trade
Lookup the historical highs and lows for each stock and determine the relative ranking-percentile
Auto-correlated Heteroscedasticity
Heteroscedasticity or unequal variabilityvariance
Auto-correlated heteroscedasticity
hellip is called the ARCH effect
Estimating Volatility
Autoregressive conditional heteroscedasticity (ARCH)
Exponentially weight moving average (EWMA)
Generalized autoregressive conditional heteroscedasticity
(GARCH)
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Inputs of BS
Inputs
Current price 1198780
Strike price 119870
Risk-free rate 119903
Time to expiration 119879
Volatility 120590
Output
Price of the option
The Volatility
Volatility is a measure of the dispersion of returns ie a simple measure of asset return volatility is its variance over time
More formally volatility is standard deviation of the rate of return in 1 year
The standard deviation of the return in a short time period time Dt is approximately tD
Example Google stock prices last one year
In the BS model what value should be chosen for 120590 when pricing an option
Variance does not capture volatility clustering (periods of turbulence periods of calm impacts of economics events etc)
ndash It is a measure of unconditional variance
ndash It is a single number of a given sample
ndash It does not incorporate the past information
Volatility
Implied Volatility
The implied volatility of an option is the volatility for which the Black-Scholes-Merton price equals the market price
There is a one-to-one correspondence between prices and implied volatilities
Traders and brokers often quote implied volatilities rather than dollar prices
11
Implied Volatility
Consider a call option
119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873(1198892)119891 120590 = 119862 minus (1198780119873 1198891 minus 119870119890minus119903119879119873(1198892))
Find the zeros of 119891(120590)
Example
Example
Implied Volatility (IV)-Task
Why is IV important
Stocks vs Options You can theoretically hold a stock forever-no timeline Whereas when dealing options you have to deal with time component T and the implied volatility (IV how far do traders expect the stock to move)
ATM How much would you willing to pay for an ATM option
OTM How much would you be willing to pay for an OTM option
Market participantsrsquo belief determines how far would the stock move
Implied Volatility (IV)-Task
When IV is high people are willing to pay more money for an option contract
When IV is low people are willing to not pay a lot of money for an option contract
Suppose a stock with an IV of 20 is trading at $100 The expected range is $80-$120
68 of the time the stock will trade between $80 and $120
IV How do we know if IV is high or low
Consider two stocks Google and FedEx with IVs equal to 15
How do we determine if 15 is a high or a low volatility for GoogleFedEx
Suppose from the historical data for Google the high was 40 and the low was 10 Whereas for FedEx the high was 25 and the low was 5 Which stock would you trade
Lookup the historical highs and lows for each stock and determine the relative ranking-percentile
Auto-correlated Heteroscedasticity
Heteroscedasticity or unequal variabilityvariance
Auto-correlated heteroscedasticity
hellip is called the ARCH effect
Estimating Volatility
Autoregressive conditional heteroscedasticity (ARCH)
Exponentially weight moving average (EWMA)
Generalized autoregressive conditional heteroscedasticity
(GARCH)
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
The Volatility
Volatility is a measure of the dispersion of returns ie a simple measure of asset return volatility is its variance over time
More formally volatility is standard deviation of the rate of return in 1 year
The standard deviation of the return in a short time period time Dt is approximately tD
Example Google stock prices last one year
In the BS model what value should be chosen for 120590 when pricing an option
Variance does not capture volatility clustering (periods of turbulence periods of calm impacts of economics events etc)
ndash It is a measure of unconditional variance
ndash It is a single number of a given sample
ndash It does not incorporate the past information
Volatility
Implied Volatility
The implied volatility of an option is the volatility for which the Black-Scholes-Merton price equals the market price
There is a one-to-one correspondence between prices and implied volatilities
Traders and brokers often quote implied volatilities rather than dollar prices
11
Implied Volatility
Consider a call option
119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873(1198892)119891 120590 = 119862 minus (1198780119873 1198891 minus 119870119890minus119903119879119873(1198892))
Find the zeros of 119891(120590)
Example
Example
Implied Volatility (IV)-Task
Why is IV important
Stocks vs Options You can theoretically hold a stock forever-no timeline Whereas when dealing options you have to deal with time component T and the implied volatility (IV how far do traders expect the stock to move)
ATM How much would you willing to pay for an ATM option
OTM How much would you be willing to pay for an OTM option
Market participantsrsquo belief determines how far would the stock move
Implied Volatility (IV)-Task
When IV is high people are willing to pay more money for an option contract
When IV is low people are willing to not pay a lot of money for an option contract
Suppose a stock with an IV of 20 is trading at $100 The expected range is $80-$120
68 of the time the stock will trade between $80 and $120
IV How do we know if IV is high or low
Consider two stocks Google and FedEx with IVs equal to 15
How do we determine if 15 is a high or a low volatility for GoogleFedEx
Suppose from the historical data for Google the high was 40 and the low was 10 Whereas for FedEx the high was 25 and the low was 5 Which stock would you trade
Lookup the historical highs and lows for each stock and determine the relative ranking-percentile
Auto-correlated Heteroscedasticity
Heteroscedasticity or unequal variabilityvariance
Auto-correlated heteroscedasticity
hellip is called the ARCH effect
Estimating Volatility
Autoregressive conditional heteroscedasticity (ARCH)
Exponentially weight moving average (EWMA)
Generalized autoregressive conditional heteroscedasticity
(GARCH)
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Example Google stock prices last one year
In the BS model what value should be chosen for 120590 when pricing an option
Variance does not capture volatility clustering (periods of turbulence periods of calm impacts of economics events etc)
ndash It is a measure of unconditional variance
ndash It is a single number of a given sample
ndash It does not incorporate the past information
Volatility
Implied Volatility
The implied volatility of an option is the volatility for which the Black-Scholes-Merton price equals the market price
There is a one-to-one correspondence between prices and implied volatilities
Traders and brokers often quote implied volatilities rather than dollar prices
11
Implied Volatility
Consider a call option
119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873(1198892)119891 120590 = 119862 minus (1198780119873 1198891 minus 119870119890minus119903119879119873(1198892))
Find the zeros of 119891(120590)
Example
Example
Implied Volatility (IV)-Task
Why is IV important
Stocks vs Options You can theoretically hold a stock forever-no timeline Whereas when dealing options you have to deal with time component T and the implied volatility (IV how far do traders expect the stock to move)
ATM How much would you willing to pay for an ATM option
OTM How much would you be willing to pay for an OTM option
Market participantsrsquo belief determines how far would the stock move
Implied Volatility (IV)-Task
When IV is high people are willing to pay more money for an option contract
When IV is low people are willing to not pay a lot of money for an option contract
Suppose a stock with an IV of 20 is trading at $100 The expected range is $80-$120
68 of the time the stock will trade between $80 and $120
IV How do we know if IV is high or low
Consider two stocks Google and FedEx with IVs equal to 15
How do we determine if 15 is a high or a low volatility for GoogleFedEx
Suppose from the historical data for Google the high was 40 and the low was 10 Whereas for FedEx the high was 25 and the low was 5 Which stock would you trade
Lookup the historical highs and lows for each stock and determine the relative ranking-percentile
Auto-correlated Heteroscedasticity
Heteroscedasticity or unequal variabilityvariance
Auto-correlated heteroscedasticity
hellip is called the ARCH effect
Estimating Volatility
Autoregressive conditional heteroscedasticity (ARCH)
Exponentially weight moving average (EWMA)
Generalized autoregressive conditional heteroscedasticity
(GARCH)
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
In the BS model what value should be chosen for 120590 when pricing an option
Variance does not capture volatility clustering (periods of turbulence periods of calm impacts of economics events etc)
ndash It is a measure of unconditional variance
ndash It is a single number of a given sample
ndash It does not incorporate the past information
Volatility
Implied Volatility
The implied volatility of an option is the volatility for which the Black-Scholes-Merton price equals the market price
There is a one-to-one correspondence between prices and implied volatilities
Traders and brokers often quote implied volatilities rather than dollar prices
11
Implied Volatility
Consider a call option
119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873(1198892)119891 120590 = 119862 minus (1198780119873 1198891 minus 119870119890minus119903119879119873(1198892))
Find the zeros of 119891(120590)
Example
Example
Implied Volatility (IV)-Task
Why is IV important
Stocks vs Options You can theoretically hold a stock forever-no timeline Whereas when dealing options you have to deal with time component T and the implied volatility (IV how far do traders expect the stock to move)
ATM How much would you willing to pay for an ATM option
OTM How much would you be willing to pay for an OTM option
Market participantsrsquo belief determines how far would the stock move
Implied Volatility (IV)-Task
When IV is high people are willing to pay more money for an option contract
When IV is low people are willing to not pay a lot of money for an option contract
Suppose a stock with an IV of 20 is trading at $100 The expected range is $80-$120
68 of the time the stock will trade between $80 and $120
IV How do we know if IV is high or low
Consider two stocks Google and FedEx with IVs equal to 15
How do we determine if 15 is a high or a low volatility for GoogleFedEx
Suppose from the historical data for Google the high was 40 and the low was 10 Whereas for FedEx the high was 25 and the low was 5 Which stock would you trade
Lookup the historical highs and lows for each stock and determine the relative ranking-percentile
Auto-correlated Heteroscedasticity
Heteroscedasticity or unequal variabilityvariance
Auto-correlated heteroscedasticity
hellip is called the ARCH effect
Estimating Volatility
Autoregressive conditional heteroscedasticity (ARCH)
Exponentially weight moving average (EWMA)
Generalized autoregressive conditional heteroscedasticity
(GARCH)
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Volatility
Implied Volatility
The implied volatility of an option is the volatility for which the Black-Scholes-Merton price equals the market price
There is a one-to-one correspondence between prices and implied volatilities
Traders and brokers often quote implied volatilities rather than dollar prices
11
Implied Volatility
Consider a call option
119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873(1198892)119891 120590 = 119862 minus (1198780119873 1198891 minus 119870119890minus119903119879119873(1198892))
Find the zeros of 119891(120590)
Example
Example
Implied Volatility (IV)-Task
Why is IV important
Stocks vs Options You can theoretically hold a stock forever-no timeline Whereas when dealing options you have to deal with time component T and the implied volatility (IV how far do traders expect the stock to move)
ATM How much would you willing to pay for an ATM option
OTM How much would you be willing to pay for an OTM option
Market participantsrsquo belief determines how far would the stock move
Implied Volatility (IV)-Task
When IV is high people are willing to pay more money for an option contract
When IV is low people are willing to not pay a lot of money for an option contract
Suppose a stock with an IV of 20 is trading at $100 The expected range is $80-$120
68 of the time the stock will trade between $80 and $120
IV How do we know if IV is high or low
Consider two stocks Google and FedEx with IVs equal to 15
How do we determine if 15 is a high or a low volatility for GoogleFedEx
Suppose from the historical data for Google the high was 40 and the low was 10 Whereas for FedEx the high was 25 and the low was 5 Which stock would you trade
Lookup the historical highs and lows for each stock and determine the relative ranking-percentile
Auto-correlated Heteroscedasticity
Heteroscedasticity or unequal variabilityvariance
Auto-correlated heteroscedasticity
hellip is called the ARCH effect
Estimating Volatility
Autoregressive conditional heteroscedasticity (ARCH)
Exponentially weight moving average (EWMA)
Generalized autoregressive conditional heteroscedasticity
(GARCH)
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Implied Volatility
The implied volatility of an option is the volatility for which the Black-Scholes-Merton price equals the market price
There is a one-to-one correspondence between prices and implied volatilities
Traders and brokers often quote implied volatilities rather than dollar prices
11
Implied Volatility
Consider a call option
119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873(1198892)119891 120590 = 119862 minus (1198780119873 1198891 minus 119870119890minus119903119879119873(1198892))
Find the zeros of 119891(120590)
Example
Example
Implied Volatility (IV)-Task
Why is IV important
Stocks vs Options You can theoretically hold a stock forever-no timeline Whereas when dealing options you have to deal with time component T and the implied volatility (IV how far do traders expect the stock to move)
ATM How much would you willing to pay for an ATM option
OTM How much would you be willing to pay for an OTM option
Market participantsrsquo belief determines how far would the stock move
Implied Volatility (IV)-Task
When IV is high people are willing to pay more money for an option contract
When IV is low people are willing to not pay a lot of money for an option contract
Suppose a stock with an IV of 20 is trading at $100 The expected range is $80-$120
68 of the time the stock will trade between $80 and $120
IV How do we know if IV is high or low
Consider two stocks Google and FedEx with IVs equal to 15
How do we determine if 15 is a high or a low volatility for GoogleFedEx
Suppose from the historical data for Google the high was 40 and the low was 10 Whereas for FedEx the high was 25 and the low was 5 Which stock would you trade
Lookup the historical highs and lows for each stock and determine the relative ranking-percentile
Auto-correlated Heteroscedasticity
Heteroscedasticity or unequal variabilityvariance
Auto-correlated heteroscedasticity
hellip is called the ARCH effect
Estimating Volatility
Autoregressive conditional heteroscedasticity (ARCH)
Exponentially weight moving average (EWMA)
Generalized autoregressive conditional heteroscedasticity
(GARCH)
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Implied Volatility
Consider a call option
119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873(1198892)119891 120590 = 119862 minus (1198780119873 1198891 minus 119870119890minus119903119879119873(1198892))
Find the zeros of 119891(120590)
Example
Example
Implied Volatility (IV)-Task
Why is IV important
Stocks vs Options You can theoretically hold a stock forever-no timeline Whereas when dealing options you have to deal with time component T and the implied volatility (IV how far do traders expect the stock to move)
ATM How much would you willing to pay for an ATM option
OTM How much would you be willing to pay for an OTM option
Market participantsrsquo belief determines how far would the stock move
Implied Volatility (IV)-Task
When IV is high people are willing to pay more money for an option contract
When IV is low people are willing to not pay a lot of money for an option contract
Suppose a stock with an IV of 20 is trading at $100 The expected range is $80-$120
68 of the time the stock will trade between $80 and $120
IV How do we know if IV is high or low
Consider two stocks Google and FedEx with IVs equal to 15
How do we determine if 15 is a high or a low volatility for GoogleFedEx
Suppose from the historical data for Google the high was 40 and the low was 10 Whereas for FedEx the high was 25 and the low was 5 Which stock would you trade
Lookup the historical highs and lows for each stock and determine the relative ranking-percentile
Auto-correlated Heteroscedasticity
Heteroscedasticity or unequal variabilityvariance
Auto-correlated heteroscedasticity
hellip is called the ARCH effect
Estimating Volatility
Autoregressive conditional heteroscedasticity (ARCH)
Exponentially weight moving average (EWMA)
Generalized autoregressive conditional heteroscedasticity
(GARCH)
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Example
Example
Implied Volatility (IV)-Task
Why is IV important
Stocks vs Options You can theoretically hold a stock forever-no timeline Whereas when dealing options you have to deal with time component T and the implied volatility (IV how far do traders expect the stock to move)
ATM How much would you willing to pay for an ATM option
OTM How much would you be willing to pay for an OTM option
Market participantsrsquo belief determines how far would the stock move
Implied Volatility (IV)-Task
When IV is high people are willing to pay more money for an option contract
When IV is low people are willing to not pay a lot of money for an option contract
Suppose a stock with an IV of 20 is trading at $100 The expected range is $80-$120
68 of the time the stock will trade between $80 and $120
IV How do we know if IV is high or low
Consider two stocks Google and FedEx with IVs equal to 15
How do we determine if 15 is a high or a low volatility for GoogleFedEx
Suppose from the historical data for Google the high was 40 and the low was 10 Whereas for FedEx the high was 25 and the low was 5 Which stock would you trade
Lookup the historical highs and lows for each stock and determine the relative ranking-percentile
Auto-correlated Heteroscedasticity
Heteroscedasticity or unequal variabilityvariance
Auto-correlated heteroscedasticity
hellip is called the ARCH effect
Estimating Volatility
Autoregressive conditional heteroscedasticity (ARCH)
Exponentially weight moving average (EWMA)
Generalized autoregressive conditional heteroscedasticity
(GARCH)
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Example
Implied Volatility (IV)-Task
Why is IV important
Stocks vs Options You can theoretically hold a stock forever-no timeline Whereas when dealing options you have to deal with time component T and the implied volatility (IV how far do traders expect the stock to move)
ATM How much would you willing to pay for an ATM option
OTM How much would you be willing to pay for an OTM option
Market participantsrsquo belief determines how far would the stock move
Implied Volatility (IV)-Task
When IV is high people are willing to pay more money for an option contract
When IV is low people are willing to not pay a lot of money for an option contract
Suppose a stock with an IV of 20 is trading at $100 The expected range is $80-$120
68 of the time the stock will trade between $80 and $120
IV How do we know if IV is high or low
Consider two stocks Google and FedEx with IVs equal to 15
How do we determine if 15 is a high or a low volatility for GoogleFedEx
Suppose from the historical data for Google the high was 40 and the low was 10 Whereas for FedEx the high was 25 and the low was 5 Which stock would you trade
Lookup the historical highs and lows for each stock and determine the relative ranking-percentile
Auto-correlated Heteroscedasticity
Heteroscedasticity or unequal variabilityvariance
Auto-correlated heteroscedasticity
hellip is called the ARCH effect
Estimating Volatility
Autoregressive conditional heteroscedasticity (ARCH)
Exponentially weight moving average (EWMA)
Generalized autoregressive conditional heteroscedasticity
(GARCH)
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Implied Volatility (IV)-Task
Why is IV important
Stocks vs Options You can theoretically hold a stock forever-no timeline Whereas when dealing options you have to deal with time component T and the implied volatility (IV how far do traders expect the stock to move)
ATM How much would you willing to pay for an ATM option
OTM How much would you be willing to pay for an OTM option
Market participantsrsquo belief determines how far would the stock move
Implied Volatility (IV)-Task
When IV is high people are willing to pay more money for an option contract
When IV is low people are willing to not pay a lot of money for an option contract
Suppose a stock with an IV of 20 is trading at $100 The expected range is $80-$120
68 of the time the stock will trade between $80 and $120
IV How do we know if IV is high or low
Consider two stocks Google and FedEx with IVs equal to 15
How do we determine if 15 is a high or a low volatility for GoogleFedEx
Suppose from the historical data for Google the high was 40 and the low was 10 Whereas for FedEx the high was 25 and the low was 5 Which stock would you trade
Lookup the historical highs and lows for each stock and determine the relative ranking-percentile
Auto-correlated Heteroscedasticity
Heteroscedasticity or unequal variabilityvariance
Auto-correlated heteroscedasticity
hellip is called the ARCH effect
Estimating Volatility
Autoregressive conditional heteroscedasticity (ARCH)
Exponentially weight moving average (EWMA)
Generalized autoregressive conditional heteroscedasticity
(GARCH)
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Implied Volatility (IV)-Task
When IV is high people are willing to pay more money for an option contract
When IV is low people are willing to not pay a lot of money for an option contract
Suppose a stock with an IV of 20 is trading at $100 The expected range is $80-$120
68 of the time the stock will trade between $80 and $120
IV How do we know if IV is high or low
Consider two stocks Google and FedEx with IVs equal to 15
How do we determine if 15 is a high or a low volatility for GoogleFedEx
Suppose from the historical data for Google the high was 40 and the low was 10 Whereas for FedEx the high was 25 and the low was 5 Which stock would you trade
Lookup the historical highs and lows for each stock and determine the relative ranking-percentile
Auto-correlated Heteroscedasticity
Heteroscedasticity or unequal variabilityvariance
Auto-correlated heteroscedasticity
hellip is called the ARCH effect
Estimating Volatility
Autoregressive conditional heteroscedasticity (ARCH)
Exponentially weight moving average (EWMA)
Generalized autoregressive conditional heteroscedasticity
(GARCH)
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
IV How do we know if IV is high or low
Consider two stocks Google and FedEx with IVs equal to 15
How do we determine if 15 is a high or a low volatility for GoogleFedEx
Suppose from the historical data for Google the high was 40 and the low was 10 Whereas for FedEx the high was 25 and the low was 5 Which stock would you trade
Lookup the historical highs and lows for each stock and determine the relative ranking-percentile
Auto-correlated Heteroscedasticity
Heteroscedasticity or unequal variabilityvariance
Auto-correlated heteroscedasticity
hellip is called the ARCH effect
Estimating Volatility
Autoregressive conditional heteroscedasticity (ARCH)
Exponentially weight moving average (EWMA)
Generalized autoregressive conditional heteroscedasticity
(GARCH)
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Auto-correlated Heteroscedasticity
Heteroscedasticity or unequal variabilityvariance
Auto-correlated heteroscedasticity
hellip is called the ARCH effect
Estimating Volatility
Autoregressive conditional heteroscedasticity (ARCH)
Exponentially weight moving average (EWMA)
Generalized autoregressive conditional heteroscedasticity
(GARCH)
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Estimating Volatility
Autoregressive conditional heteroscedasticity (ARCH)
Exponentially weight moving average (EWMA)
Generalized autoregressive conditional heteroscedasticity
(GARCH)
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Estimating Volatility
Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define 119906119894 = ln(119878119894
119878119894minus1)
20
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Estimating Volatility
Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894
Set 119906119894 =119878119894minus119878119894minus1
119878119894minus1
Assume that the mean value of ui is zero
Replace mminus1 by m
This gives
21
n n ii
m
mu2 2
1
1
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Example
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Weighting Scheme
Instead of assigning equal weights to the observations we can assign more weight to recent data
n i n ii
m
ii
m
u2 2
1
1
1
where
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate VL
m
i
i
m
i iniLn uV
1
1
22
1
where
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
EWMA Model
In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time
This leads to
25
2
1
2
1
2 )1( nnn u
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
EWMA Weights Decline Exponentially
1205901198992 = 120582 120590119899minus1
2 + 1 minus 120582 119906119899minus12
= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2
2 ] + 1 minus 120582 119906119899minus12
= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2
2 + 1205822 120590119899minus22
Continue substituting for 120590119899minus22 120590119899minus3
2 and so on we have
1205901198992 = 1 minus 120582 119906119899minus1
2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3
2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +
120582119898120590119899minus1198982
Weights decline at the rate of 120582
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
094 is a popular choice for
27
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
GARCH (11)
In GARCH (11) we assign some weight to the long-run average variance rate
1205901198992 = 120574119881119871 + 120572119906119899minus1
2 + 120573120590119899minus12
such that
120574 + 120572 + 120573 = 1
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
GARCH (11)
Let 120596 = 120574119881119871 then GARCH(11) becomes
1205901198992 = 120596 + 120572119906119899minus1
2 + 120573120590119899minus12
Where
119881119871 =120596
1minus120572minus120573
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
GARCH (pq)
30
2
1 1
22
jn
p
i
q
j
jinin u
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Maximum Likelihood Methods
How do we estimate the parameters in the models discussed above
Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Example
Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day
Itrsquos 01
Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9
We maximize 119891(119901) to obtain a maximum likelihood estimate
Itrsquos p = 01
32
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Example Estimating a constant variance
Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean
Assume that the observations are 1199061 1199062 ⋯ 119906119898
Denote the variance by 120584
The likelihood of 119906119894 being observed is defined as the probability density function for
119883 when 119883 = 119906119894 That is 119891 119906119894 =1
2120587120584119890minus
1199061198942
2120584 for 119894 = 12⋯ 119898
The likelihood of m observations occurring in the order in which they are observed
is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Example
Estimate of 120584 ∶
m
i
i
m
i
i
m
i
i
um
v
v
uv
v
u
v
1
2
1
2
1
2
1
)ln(
2exp
2
1
Result
maximizing to equivalent is this logarithms Taking
Maximize
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Application to GARCH
We choose parameters that maximize
35
m
i i
ii
i
im
i i
v
uv
v
u
v
1
2
2
1
)ln(
2exp
2
1
or
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
Example
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg
References
Options Futures and Other Derivatives (10th Edition) 10th Edition
by John C Hull
Handbook of Volatility Models and Their Application
by Luc Bauwens Christian M Hafner Sebastien Laurent
Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition
by Sheldon Natenberg