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The Analysis of Volatility. Historical Volatility. Volatility Estimation (MLE, EWMA, GARCH...). Maximum Likelihood Estimation. Implied Volatility. Smiles, smirks, and explanations. In the Black-Scholes formula, volatility is the only variable that is not directly observable in the market.

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the analysis of volatility

The Analysis of Volatility

Primbs, MS&E 345, Spring 2002

slide2

Historical Volatility

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

Maximum Likelihood Estimation

Implied Volatility

Smiles, smirks, and explanations

Primbs, MS&E 345, Spring 2002

slide3

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.

Primbs, MS&E 345, Spring 2002

slide4

Change to log coordinates

and discretize:

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

where

A Standard Volatility Estimate:

(I am following [Hull, 2000] now)

Primbs, MS&E 345, Spring 2002

slide5

Unbiased estimate means

Max likelihood estimator

Minimum mean squared error estimator

Note:

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

Primbs, MS&E 345, Spring 2002

slide6

For simplicity, people often set and use:

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

In the future, I will set as well.

Note:

Why is this okay?

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

Primbs, MS&E 345, Spring 2002

slide7

The estimate

gives equal weight to each ui.

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

where

Weighting Schemes

Furthermore, I will allow for the volatility to change over time. So sn2 will denotes the volatility at day n.

Primbs, MS&E 345, Spring 2002

slide8

Assume there is a long run average volatility, V.

where

Weighting Schemes

An Extension

This is known as an ARCH(m) model

ARCH stands for

Auto-Regressive Conditional Heteroscedasticity.

Primbs, MS&E 345, Spring 2002

slide9

y

regression:

y=ax+b+e

x

x

x

x

e is the error.

x

x

x

x

x

x

x

x

x

Homoscedastic and Heteroscedastic

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.

Primbs, MS&E 345, Spring 2002

slide10

Exponentially Weighted Moving Average (EWMA):

weights die away exponentially

Weighting Schemes

Primbs, MS&E 345, Spring 2002

slide11

GARCH(1,1) Model

Generalized Auto-Regressive Conditional Heteroscedasticity

where

The (1,1) indicates that it depends on

Weighting Schemes

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 s2.

Primbs, MS&E 345, Spring 2002

slide12

Historical Volatility

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

Maximum Likelihood Estimation

Implied Volatility

Smiles, smirks, and explanations

Primbs, MS&E 345, Spring 2002

slide13

That is, we solve:

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

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.

Primbs, MS&E 345, Spring 2002

slide14

Let

Maximum Likelihood Methods:

Example:

Estimate the variance of a normal distribution from samples:

Given u1,...,um.

Primbs, MS&E 345, Spring 2002

slide15

where K1, and K2 are some constants.

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

Maximum Likelihood Methods:

Example:

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

Primbs, MS&E 345, Spring 2002

slide16

where

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

You have to solve it numerically...

Maximum Likelihood Methods:

We can use a similar approach for a GARCH model:

The problem is to maximize this over w, a, and b.

Primbs, MS&E 345, Spring 2002

slide17

Historical Volatility

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

Maximum Likelihood Estimation

Implied Volatility

Smiles, smirks, and explanations

Primbs, MS&E 345, Spring 2002

slide18

Denote the Black-Scholes formula by:

The value of s that satisfies:

is known as the implied volatility

Implied Volatility:

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

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

Primbs, MS&E 345, Spring 2002

slide19

Implied

Volatility

smile

smirk

K/S0

The Implied Volatility Smile and Smirk

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

Primbs, MS&E 345, Spring 2002

slide20

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...

Primbs, MS&E 345, Spring 2002

slide21

Why might return distributions have heavy tails?

Heavy Tails

Stochastic Volatility

Jump diffusion models

Risk management strategies and feedback effects

Primbs, MS&E 345, Spring 2002

slide22

Out of the money call:

Call option

strike K

More probability

under heavy tails

At the money call:

Probability balances

here and here

Call option

strike K

How do heavy tails cause a smile?

This option is

worth more

This option is

not necessarily

worth more

Primbs, MS&E 345, Spring 2002

slide23

Mean

Variance

Skewness

Kurtosis

Important Parameters of a distribution:

Gaussian~N(0,1)

0

1

0

3

Primbs, MS&E 345, Spring 2002

slide24

Mean Variance Skewness Kurtosis

Red (Gaussian) 0 1 0 3

Blue 0 1 -0.5 3

Skewness tilts the distribution on one side.

Primbs, MS&E 345, Spring 2002

slide25

Mean Variance Skewness Kurtosis

Red (Gaussian) 0 1 0 3

Blue 0 1 0 5

Large kurtosis creates heavy tails (leptokurtic)

Primbs, MS&E 345, Spring 2002

slide26

Empirical Return Distribution

Mean Variance Skewness Kurtosis

0.0007 0.0089 -0.3923 3.8207

(Data from the Chicago Mercantile Exchange)

(Courtesy of Y. Yamada)

Primbs, MS&E 345, Spring 2002

slide27

Volatility Smiles and Smirks

10 days to maturity

Mean Square Optimal Hedge Pricing

(Courtesy of Y. Yamada)

Primbs, MS&E 345, Spring 2002

slide28

Volatility Smiles and Smirks

20 days to maturity

Mean Square Optimal Hedge Pricing

(Courtesy of Y. Yamada)

Primbs, MS&E 345, Spring 2002

slide29

Volatility Smiles and Smirks

40 days to maturity

Mean Square Optimal Hedge Pricing

(Courtesy of Y. Yamada)

Primbs, MS&E 345, Spring 2002

slide30

Volatility Smiles and Smirks

80 days to maturity

Mean Square Optimal Hedge Pricing

(Courtesy of Y. Yamada)

Primbs, MS&E 345, Spring 2002

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