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MGT 821/ECON 873

Volatility Smiles &

Extension of Models

What is a Volatility Smile?

It is the relationship between implied volatility and strike price for options with a certain maturity

The volatility smile for European call options should be exactly the same as that for European put options

The same is at least approximately true for American options

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Why the Volatility Smile is the Same for Calls and Put

Put-call parity p +S0e-qT = c +Ke–r T holds for market prices (pmkt and cmkt) and for Black-Scholes prices (pbs and cbs)

It follows that pmkt−pbs=cmkt−cbs

When pbs=pmkt, it must be true that cbs=cmkt

It follows that the implied volatility calculated from a European call option should be the same as that calculated from a European put option when both have the same strike price and maturity

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The Volatility Smile for Foreign The Volatility Smile for Foreign Currency Options Currency Options

ImpliedVolatility

StrikePrice

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Implied Distribution for Foreign Currency Options

Both tails are heavier than the lognormal distribution

It is also “more peaked” than the lognormal distribution

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The Volatility Smile for The Volatility Smile for Equity OptionsEquity Options

ImpliedVolatility

Strike

Price

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Implied Distribution for Equity Options

The left tail is heavier and the right tail is less heavy than the lognormal distribution

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Other Volatility Smiles?

What is the volatility smile if True distribution has a less heavy left tail and

heavier right tail True distribution has both a less heavy left

tail and a less heavy right tail

Ways of Characterizing the Volatility Smiles Plot implied volatility against K/S0 (The volatility

smile is then more stable) Plot implied volatility against K/F0 (Traders usually

define an option as at-the-money when K equals the forward price, F0, not when it equals the spot price S0)

Plot implied volatility against delta of the option (This approach allows the volatility smile to be applied to some non-standard options. At-the money is defined as a call with a delta of 0.5 or a put with a delta of −0.5. These are referred to as 50-delta options)

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Possible Causes of Volatility Smile Asset price exhibits jumps rather than

continuous changes Volatility for asset price is stochastic

In the case of an exchange rate volatility is not heavily correlated with the exchange rate. The effect of a stochastic volatility is to create a symmetrical smile

In the case of equities volatility is negatively related to stock prices because of the impact of leverage. This is consistent with the skew that is observed in practice

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Volatility Term Structure

In addition to calculating a volatility smile, traders also calculate a volatility term structure

This shows the variation of implied volatility with the time to maturity of the option

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Volatility Term Structure

The volatility term structure tends to be downward sloping when volatility is high and upward sloping when it is low

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Example of a Volatility Example of a Volatility SurfaceSurface

K/S0 0.90 0.95 1.00 1.05 1.10

1 mnth 14.2 13.0 12.0 13.1 14.5

3 mnth 14.0 13.0 12.0 13.1 14.2

6 mnth 14.1 13.3 12.5 13.4 14.3

1 year 14.7 14.0 13.5 14.0 14.8

2 year 15.0 14.4 14.0 14.5 15.1

5 year 14.8 14.6 14.4 14.7 15.0

Greek Letters

If the Black-Scholes price, cBS is expressed as a function of the stock price, S, and the implied volatility, imp, the delta of a call is

Is the delta higher or lower than

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S

c

S

c

imp

imp

BSBS

S

c

BS

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Three Alternatives to Geometric Three Alternatives to Geometric Brownian MotionBrownian Motion

Constant elasticity of variance (CEV) Mixed Jump diffusion Variance Gamma

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CEV Model

When = 1 the model is Black-Scholes When > 1 volatility rises as stock price

rises When < 1 volatility falls as stock price

risesEuropean option can be value analytically in terms of the cumulative non-central chi square distribution

dzSSdtqrdS )(

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CEV Models Implied Volatilities

imp

K

< 1

> 1

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Mixed Jump Diffusion

Merton produced a pricing formula when the asset price follows a diffusion process overlaid with random jumps

dp is the random jump k is the expected size of the jump dt is the probability that a jump occurs in the next interval of length dt

dpdzdtkqrSdS )(/

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Jumps and the Smile

Jumps have a big effect on the implied volatility of short term options

They have a much smaller effect on the implied volatility of long term options

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The Variance-Gamma ModelThe Variance-Gamma Model Define g as change over time T in a variable

that follows a gamma process. This is a process where small jumps occur frequently and there are occasional large jumps

Conditional on g, ln ST is normal. Its variance proportional to g

There are 3 parameters v, the variance rate of the gamma process the average variance rate of ln S per unit time a parameter defining skewness

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Understanding the Variance-Understanding the Variance-Gamma ModelGamma Model g defines the rate at which information arrives

during time T (g is sometimes referred to as measuring economic time)

If g is large the change in ln S has a relatively large mean and variance

If g is small relatively little information arrives and the change in ln S has a relatively small mean and variance

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Time Varying Volatility

Suppose the volatility is 1 for the first year and 2 for the second and third

Total accumulated variance at the end of three years is 1

2 + 222

The 3-year average volatility is2 2

2 2 2 1 21 2

23 2 ;

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Stochastic Volatility Models

When V and S are uncorrelated a European option price is the Black-Scholes price integrated over the distribution of the average variance

VL

S

dzVdtVVadV

dzVdtqrS

dS

)(

)(

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Stochastic Volatility Models Stochastic Volatility Models continuedcontinued

When V and S are negatively correlated we obtain a downward sloping volatility skew similar to that observed in the market for equities

When V and S are positively correlated the skew is upward sloping. (This pattern is sometimes observed for commodities)

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The IVF ModelThe IVF Model

SdztSSdttqtrdS

SdzSdtqrdS

),()]()([

by replaced is

)(

model

motion Brownian geomeric usual The prices.

option observed matchesexactly that price

asset for the process a create todesigned

is modelfunction y volatilitimplied The

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The Volatility FunctionThe Volatility Function

The volatility function that leads to the model matching all European option prices is

)(

)]()([)(2

)],([

222

2

KcK

KctqtrKctqtc

tK

mkt

mktmktmkt

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Strengths and Weaknesses of the IVF Model The model matches the probability distribution

of asset prices assumed by the market at each future time

The models does not necessarily get the joint probability distribution of asset prices at two or more times correct