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Derivative InstrumentsFI6051
Finbarr MurphyDept. Accounting & FinanceUniversity of LimerickAutumn 2009
Week 8 Volatility & Exotics
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We will discuss lognormality, what exactly doesthis mean and what are the implications.
We will discuss the causes of volatility and the
significance of volatility in option pricing
We will examine implied volatility and how themarket forecasts volatility
We will look at implied volatility shapes, surfacesand implied volatility smiles
Lecture Summary
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This lecture also looks at non-vanilla (exotic)options. Each of the more recognised options areexamined in turn. We look at their uses anddiscuss how they might be valued.
Lecture Summary
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Underlying the Black-Scholes model is theassumption that the price of stock follows alognormal random walk, aka, GeometricBrownian Motion (GBM) with drift
This approach has been found to work very wellin industrial practice
We will look at why
Random Walk
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Changes in one time interval are independentover preceding time intervals
The size and direction of the change is (in some
way) random
Assumes market efficiency, I.e. the current stockprice reflects all available information and future
changes reflect future information
With Drift, refers to the fact that stock pricestend to increase over time
Random Walk
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Lognormality refers to the fact that the naturallog of the changes in the stock price areassumed to be normally distributed
Why do we use the natural log and not justsimply the changes in stock price?
Non-negative
Recombining
Works well in practice
Lognormality
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The relative rate at which the price of a securitymoves up and down. Volatility is found bycalculating the annualized standard deviation ofdaily change in price. If the price of a stock
moves up and down rapidly over short timeperiods, it has high volatility. If the price almostnever changes, it has low volatility
What causes volatility? Trading
News
Volatility
Source: InvestorWords.com
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A stock is priced at 50, its volatility is 15% In one week, a 1-STD DEV move is
= 1.04
A stock is priced at 50, its volatility is 45%
In one week, a 1-STD DEV move is
= 3.12
Volatility
52
115.050
52145.050
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How to measure volatility? We will examine data for the ISEQ taking daily
closing values for last three months 11th August 2005 to 11th October 2005
That is, 44 observations The Volatility of the stock , is given by
Where s = the standard deviation and is thetime interval (in years)
Volatility
s
=
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All we need to calculate is s
Where ui is ln(Si/Si-1 )
And is the average of the uis
Volatility
( )2
11
1 = =n
i iuu
ns
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A sample from the ISEQ daily returns
Volatility
i Date Close Si/Si-1 ln(Si/Si-1)
0 10-Aug-05 6,784.93
1 11-Aug-05 6,772.67 0.998193 -0.000785
2 12-Aug-05 6,739.24 0.995064 -0.002149
3 15-Aug-05 6,697.64 0.993827 -0.0026894 16-Aug-05 6,717.50 1.002965 0.001286
5 17-Aug-05 6,691.91 0.996191 -0.001658
6 18-Aug-05 6,668.44 0.996493 -0.001526
7 19-Aug-05 6,677.70 1.001389 0.000603
8 22-Aug-05 6,685.23 1.001128 0.000489
9 23-Aug-05 6,688.59 1.000503 0.000218
10 24-Aug-05 6,659.78 0.995693 -0.001875
11 25-Aug-05 6,639.90 0.997015 -0.001298
12 26-Aug-05 6,644.13 1.000637 0.000277
13 29-Aug-05 6,666.68 1.003394 0.001471
14 30-Aug-05 6,647.73 0.997158 -0.001236
15 31-Aug-05 6,677.28 1.004445 0.001926
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From these (44) observations, I calculated thestandard deviation of the daily return as
S = 0.0052659
Or 0.52659%
Assuming 260 trading days in the year, theannualised volatility is given as
Volatility
2600052659.0
=
0849101.0=%49101.8=
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Recall:
Where
Notice where volatility is embedded in the
equation
Volatility Use in Black Scholes
( ) ( )2100
dNKedNScrT=
( ) ( )
( ) ( )Td
T
TrKS
d
T
TrKSd
=
+
=
++=
1
2
0
2
2
0
1
2//ln
;2//ln
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If the call option price c0 is known from marketdata, along with the other variables
We can reverse out the volatility impliedby themarket price.
Although it is not possible to calculate from theBlack Scholes equation, we can get the value byiterative techniques
Volatility Use in Black Scholes
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At the time of writing, Apple Computers(symbol: AAPL) have just released quarterlyresults which were disappointing.Simultaneously, they announced the release to
market of a 60Gbyte iPod video recorder whichwas well received.
Implied Volatility
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Notice the subsequent erratic stock price andhigh volumes
It should be interesting to calculate the implied
volatility of AAPL and compare against historicalaverages
Maybe we can spot a trading opportunity!
Implied Volatility
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Ive selected a list of near-term, at-the-moneycall and put options from CBOE
Implied Volatility
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Now, using MatLabs in-built Black-Scholesfunction:
[CALL,PUT] = BLSPRICE(SO,X,R,T,,Q) SO= Current Stock Price
X = Strike Price
R = Risk Free Rate
T = Time to Maturity
= Volatility
Q = Asset dividend rate
We can calculate the theoretical value of the calloptions
Implied Volatility
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[CALL,PUT] = BLSPRICE(49.25,45,0.0375,(37/365),0.25,0) Call Option = 4.6385 [CALL,PUT] = BLSPRICE(49.25,50,0.0375,(37/365),0.25,0)
Call Option = 1.3081
[CALL,PUT] = BLSPRICE(49.25,55,0.0375,(37/365),0.25,0) Call Option = 0.1729
Compare these results with the actual market:
5.00, 2.30 and 0.75 respectively
The volatility we used (25%) is lower than thatimplied by the market prices
Implied Volatility
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Now, using another of MatLabs in-builtfunctions, we can calculate the implied volatility:
ImpVol = BLSIMPV(SO,X,R,T,CallPx)
CallPx = The actual call option value
ImpVol = BLSIMPV(49.25,45,0.0375,(37/365),5.0)
ImpVol = 34.55% ImpVol = BLSIMPV(49.25,50,0.0375,(37/365),2.3)
ImpVol = 40.89% ImpVol = BLSIMPV(49.25,55,0.0375,(37/365),0.75)
ImpVol = 40.14%
Implied Volatility
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In other words, the options prices tells us thatthe market believes volatility of Apple Sharesto be about 40%
IF, we believed that volatility was in fact lower,how could we exploit our belief and makemoney?
Implied Volatility
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Notice that the implied volatility was about40%.
The underlying stock is the same and so theimplied volatilities should be the same, right?
Volatility Smile
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Extend the previous graph to a 3D model withimplied volatility, moneyness and maturity
Volatility Surface
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These are options on options A call (option) on a call (option)
A put on a put
A call on a put
A put on a call
Used to hedge for a certain period. E.g. Buy a 3-month compound call option on oil futures
during a particularly volatile period. Leverage on leverage
But if both options are exercised, then thepremium for the compound option will be larger
than that for a single option
Compound Options
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Barrier is an umbrella name for a two differentoption types Knock-Out Calls and Puts
Down and Out
Up and Out Knock-In Calls and Puts
Down and In
Up and In
A knock-Out, causes the option to terminate ifthe underlying breaches a barrier
A knock-In, causes the option to activate if theunderlying breaches a barrier
Barrier Options
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The following diagram shows a Knock-In
If the barrier was set below 90 at initiation,these would become Down and Ins
The strike of the option should be judiciously set
Barrier Options
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An airline might buy a Up-And-In Call option onaviation fuel costs, cheaper than a standard call
The premium for a barrier is lower than that of a
normal option
A down-and-out call plus a down-and-in call equals? A
standard call!
AKA kick-outs, kick-ins, ins, outs, exploding options, extinguishing
options and trigger options
Barrier Options
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Barrier Options are path-dependent. I.e. atexpiration, it matters howthe underlying pricearrived at the final value
Barrier options are typically priced using Monte-Carlo techniques, these will be discussed insome detail next semester in FinancialEngineering
Barrier Options
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Binary Options
Binary Options
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Also known as Hindsight Options There are two (call) types
Fixed Strike Payoff is dependent on the Smax -X
Floating Strike Payoff is dependent on the ST-Smin
The equivalent put options have Fixed Strike Payoff is dependent on the X-Smin
Floating Strike Payoff is dependent on the Smax -ST
Lookback Options
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Consider the payback on each option using thefollowing graph
Lookback Options
SMAX
ST
SMIN
X
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Shout options are very similar to lookbackoptions but the option holder shouts (decides)during the life of the option when she believesthat the market has peaked or bottomed.
These are cheaper than lookback options
But no guarantee that you can pick the market
bottom or top!
Shout Options
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Asian Options have a payoff that depends on theaverage asset price during at least some part ofthe option life
Consider a pension fund manager with apredominantly equity fund. The fund is held for 5years. The manager should consider an Asianput option on the basket, this will protect the
fund from sudden drops in the fund value in thelast days of the fund
Asian Options
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There are two types of Asian Options Average Rate Options
The payoff is dependent on the averageunderlying asset price for a given period
Average Strike Options The strike (and therefore payoff) is dependent
on the average underlying asset price for agiven period
Both forms can be either calls or puts
Asian options are path-dependent
Asian Options
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Note how the payoff on a regular call option (ST)is less than that of the Asian payoff (ST
AVG )
described by the diagram below
Asian Options
ST
STAVG
Averaging Period
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Also known as Rainbow Options The payoff is dependent on the value of a
portfolio of assets (e.g., stocks)
The volatility of the basket is dependent on the
price correlation of the constituent stocks Lower correlation implies higher variability
(volatility)
Typical baskets belong to fund managers orindex of shares (FTSE, ISEQ, etc)
Basket Options
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An exotic option is a non-standard option
Weve looked at the more common exoticoptions
An option can be contrived to suit a particularclients needs
The pricing of these options is a financialengineering skill. The construction/engineeringof the options is a much prized skill used instructured products desks in investment banks
Other Exotic Options
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Hull, J.C, Options, Futures & Other Derivatives,2009, 7th Ed. Chapter 13, 18, 24
Further reading
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Hull, J.C, Options, Futures & Other Derivatives,2005, 6th Ed. Chapter 13
Further reading