Primbs, MS&E 345, Spring 2002 1 The Analysis of Volatility.

Primbs, MS&E 345, Spring 2002 1

The Analysis of Volatility


Historical Volatility

Volatility Estimation (MLE, EWMA, GARCH...)

Implied Volatility

Smiles, smirks, and explanations

Maximum Likelihood Estimation


In the Black-Scholes formula, volatility is the only variable that is not directly observable in the market.

Therefore, we must estimate volatility in some way.

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

SdzSdtdS


(I am following [Hull, 2000] now)A Standard Volatility Estimate:

dzdtSd lnSdzSdtdS

Change to log coordinates

and discretize:),(~ln 2

)1(

ttNS

Su

ti

tii

Then, an unbiased estimate of the variance using the m most recent observations is

2

1

2 )(1

1ˆ

m

ii uu

m

m

iium

u1

1where


Note:

If m is large, it doesn’t matter which one you use...

Unbiased estimate means 22 ]ˆ[ E

2

1

2 )(1

1ˆ

m

ii uu

m

Max likelihood estimator

2

1

2 )(1

ˆ

m

ii uu

m

Minimum mean squared error estimator

2

1

2 )(1

1ˆ

m

ii uu

m


It is very small over small time periods, and this assumption has very little effect on the estimates.

Why is this okay?

Note:

u is an estimate of the mean return over the sampling period.

For simplicity, people often set and use:0u

m

iium 1

22

1

1̂

In the future, I will set as well.0u


Weighting Schemes

gives equal weight to each ui.

m

iium 1

22

1

1̂The estimate

Alternatively, we can use a scheme that weights recent data more:

m

iinin u

1

22 11

m

iiwhere

Furthermore, I will allow for the volatility to change over time. So n2 will denotes the volatility at day n.


An Extension

This is known as an ARCH(m) model

ARCH stands for Auto-Regressive Conditional Heteroscedasticity.

m

iinin uV

1

22ˆ 11

m

iiwhere

Assume there is a long run average volatility, V.

Weighting Schemes


Homoscedastic and Heteroscedastic

x xx

x x

x

x

x

x

xx

x

x

y

If the variance of the error e is constant, it is called homoscedastic.

However, if the error varies with x, it is said to be heteroscedastic.

regression:y=ax+b+ee is the error.


1

212 )1(i

ini

n u

2

2121 )1()1(

iin

in uu

21

21)1( nnu

weights die away exponentially

Weighting Schemes

Exponentially Weighted Moving Average (EWMA):


The (1,1) indicates that it depends on 21

21 and nnu

You can also have GARCH(p,q) models which depend on the p most recent observations of u2 and the q most recent estimates of 2.

Weighting Schemes

GARCH(1,1) Model

Generalized Auto-Regressive Conditional Heteroscedasticity

21

21

2 nnn uV

1 where




Implied Volatility




How do you estimate the parameters in these models?

One common technique is Maximum Likelihood Methods:

Idea: Given data, you choose the parameters in the model the maximize the probability that you would have observed that data.

)|(max parametersdatafparameters

where f is the conditional density of observing the data given values of the parameters.

That is, we solve:



Maximum Likelihood Methods:

Example:

Estimate the variance of a normal distribution from samples:

Let )2exp(2

1)|( 2 vu

vvuf ii

Given u1,...,um.

m

iim vu

vvuuf

1

21 )2exp(

2

1)|,...,(

)|( parametersdataf


It is usually easier to maximize the log of f(u|v).

m

i

im v

uvKKvuuf

1

2

211 )ln()|,...,(ln

where K1, and K2 are some constants.

To maximize, differentiate wrt v and set equal to zero:

m

iium

v1

21

m

iim vu

vvuuf

1

21 )2exp(

2

1)|,...,(

Example:




We can use a similar approach for a GARCH model:

12

1 nnn vuv

m

iiim vu

vuuf

1

21 )2exp(

2

1),,|,...,(

where

The problem is to maximize this over and

We don’t have any nice, neat solution in this case.

You have to solve it numerically...






Implied Volatility




Implied Volatility:

Let cm be the market price of a European call option.

Denote the Black-Scholes formula by:

)()(),,,,( 21 dNKedSNrTKSc rTBS

The value of that satisfies: mBS crTKSc ),,,,(

is known as the implied volatility

This can be thought of as the estimate of volatility that the “market” is using to price the option.


The Implied Volatility Smile and Smirk

Market prices of options tend to exhibit an “implied volatility smile” or an “implied volatility smirk”.

K/S0

ImpliedVolatility

smile

smirk


Where does the volatility smile/smirk come from?

Heavy Tail return distributions

Crash phobia (Rubenstein says it emerged after the 87 crash.)

Leverage: (as the price falls, leverage increases)

Probably many other explanations...


Why might return distributions have heavy tails?

Stochastic Volatility

Jump diffusion models

Risk management strategies and feedback effects

Heavy Tails


How do heavy tails cause a smile?

More probability under heavy tails

This option isworth more

This option isnot necessarilyworth more

Call optionstrike K

Out of the money call:

Call optionstrike K

At the money call:Probability balanceshere and here


Important Parameters of a distribution:

Gaussian~N(0,1)

0

1

0

3

)(XEMean

22 )( XEVariance

3

3)(

XESkewness

4

4)(

XE

Kurtosis


Mean Variance Skewness KurtosisRed (Gaussian) 0 1 0 3Blue 0 1 -0.5 3

Skewness tilts the distribution on one side.

-4 -3 -2 -1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


-6 -4 -2 0 2 4 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Large kurtosis creates heavy tails (leptokurtic)

Mean Variance Skewness KurtosisRed (Gaussian) 0 1 0 3Blue 0 1 0 5


Empirical Return Distribution

(Courtesy of Y. Yamada)

Mean Variance Skewness Kurtosis 0.0007 0.0089 -0.3923 3.8207

(Data from the Chicago Mercantile Exchange)


10 days to maturity


Volatility Smiles and Smirks

Mean Square Optimal Hedge Pricing


20 days to maturity





40 days to maturity





80 days to maturity




Date post:	18-Dec-2015
Category:	Documents
View:	215 times
Download:	0 times

Primbs, MS&E 345, Spring 2002 1 The Analysis of Volatility.

Documents