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The Analysis of VolatilityPowerPoint Presentation

The Analysis of Volatility

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### The Analysis of Volatility

Primbs, MS&E 345, Spring 2002

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

Maximum Likelihood Estimation

Implied Volatility

Smiles, smirks, and explanations

Primbs, MS&E 345, Spring 2002

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

Change to log coordinates variable that is not directly observable in the market.

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

Unbiased estimate means variable that is not directly observable in the market.

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

For simplicity, people often set and use: variable that is not directly observable in the market.

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

The estimate variable that is not directly observable in the market.

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

Assume there is a long run average volatility, V. variable that is not directly observable in the market.

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

y variable that is not directly observable in the market.

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

Exponentially Weighted Moving Average (EWMA): variable that is not directly observable in the market.

weights die away exponentially

Weighting Schemes

Primbs, MS&E 345, Spring 2002

GARCH(1,1) Model variable that is not directly observable in the market.

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

Historical Volatility variable that is not directly observable in the market.

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

Maximum Likelihood Estimation

Implied Volatility

Smiles, smirks, and explanations

Primbs, MS&E 345, Spring 2002

That is, we solve: variable that is not directly observable in the market.

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

Let variable that is not directly observable in the market.

Maximum Likelihood Methods:

Example:

Estimate the variance of a normal distribution from samples:

Given u1,...,um.

Primbs, MS&E 345, Spring 2002

where K variable that is not directly observable in the market. 1, 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

where variable that is not directly observable in the market.

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

Historical Volatility variable that is not directly observable in the market.

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

Maximum Likelihood Estimation

Implied Volatility

Smiles, smirks, and explanations

Primbs, MS&E 345, Spring 2002

Denote the Black-Scholes formula by: variable that is not directly observable in the market.

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

Implied variable that is not directly observable in the market.

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

Where does the volatility smile/smirk come from? variable that is not directly observable in the market.

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

Why might return distributions have heavy tails? variable that is not directly observable in the market.

Heavy Tails

Stochastic Volatility

Jump diffusion models

Risk management strategies and feedback effects

Primbs, MS&E 345, Spring 2002

Out of the money call: variable that is not directly observable in the market.

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

Mean variable that is not directly observable in the market.

Variance

Skewness

Kurtosis

Important Parameters of a distribution:

Gaussian~N(0,1)

0

1

0

3

Primbs, MS&E 345, Spring 2002

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

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

Empirical Return Distribution Skewness Kurtosis

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

Volatility Smiles and Smirks Skewness Kurtosis

10 days to maturity

Mean Square Optimal Hedge Pricing

(Courtesy of Y. Yamada)

Primbs, MS&E 345, Spring 2002

Volatility Smiles and Smirks Skewness Kurtosis

20 days to maturity

Mean Square Optimal Hedge Pricing

(Courtesy of Y. Yamada)

Primbs, MS&E 345, Spring 2002

Volatility Smiles and Smirks Skewness Kurtosis

40 days to maturity

Mean Square Optimal Hedge Pricing

(Courtesy of Y. Yamada)

Primbs, MS&E 345, Spring 2002

Volatility Smiles and Smirks Skewness Kurtosis

80 days to maturity

Mean Square Optimal Hedge Pricing

(Courtesy of Y. Yamada)

Primbs, MS&E 345, Spring 2002

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