Primbs, MS&E 345, Spring The Analysis of Volatility

Primbs, MS&E 345, Spring Historical Volatility Volatility Estimation (MLE, EWMA, GARCH...) Implied Volatility Smiles, smirks, and explanations Maximum Likelihood Estimation

Primbs, MS&E 345, Spring 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 (I am following [Hull, 2000] now)A Standard Volatility Estimate: Change to log coordinates and discretize: Then, an unbiased estimate of the variance using the m most recent observations is where

Primbs, MS&E 345, Spring Note: If m is large, it doesn’t matter which one you use... Unbiased estimate means Max likelihood estimator Minimum mean squared error estimator

Primbs, MS&E 345, Spring It is very small over small time periods, and this assumption has very little effect on the estimates. Why is this okay? Note: is an estimate of the mean return over the sampling period. For simplicity, people often set and use: In the future, I will set as well.

Primbs, MS&E 345, Spring Weighting Schemes gives equal weight to each u i. The estimate Alternatively, we can use a scheme that weights recent data more: where Furthermore, I will allow for the volatility to change over time. So  n 2 will denotes the volatility at day n.

Primbs, MS&E 345, Spring An Extension This is known as an ARCH(m) model ARCH stands for Auto-Regressive Conditional Heteroscedasticity. where Assume there is a long run average volatility, V. Weighting Schemes

Primbs, MS&E 345, Spring Homoscedastic and Heteroscedastic x x x x x x x x x x x 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+e e is the error.

Primbs, MS&E 345, Spring weights die away exponentially Weighting Schemes Exponentially Weighted Moving Average (EWMA):

Primbs, MS&E 345, Spring The (1,1) indicates that it depends on You can also have GARCH(p,q) models which depend on the p most recent observations of u 2 and the q most recent estimates of  2. Weighting Schemes GARCH(1,1) Model Generalized Auto-Regressive Conditional Heteroscedasticity where

Primbs, MS&E 345, Spring Historical Volatility Volatility Estimation (MLE, EWMA, GARCH...) Implied Volatility Smiles, smirks, and explanations Maximum Likelihood Estimation

Primbs, MS&E 345, Spring 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. where f is the conditional density of observing the data given values of the parameters. That is, we solve:

Primbs, MS&E 345, Spring Maximum Likelihood Methods: Example: Estimate the variance of a normal distribution from samples: Let Given u 1,...,u m.

Primbs, MS&E 345, Spring It is usually easier to maximize the log of f(u|v). where K 1, and K 2 are some constants. To maximize, differentiate wrt v and set equal to zero: Example: Maximum Likelihood Methods:

Primbs, MS&E 345, Spring We can use a similar approach for a GARCH model: 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... Maximum Likelihood Methods:

Primbs, MS&E 345, Spring Historical Volatility Volatility Estimation (MLE, EWMA, GARCH...) Implied Volatility Smiles, smirks, and explanations Maximum Likelihood Estimation

Primbs, MS&E 345, Spring Implied Volatility: Let c m be the market price of a European call option. Denote the Black-Scholes formula by: The value of  that satisfies: 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.

Primbs, MS&E 345, Spring The Implied Volatility Smile and Smirk Market prices of options tend to exhibit an “implied volatility smile” or an “implied volatility smirk”. K/S 0 Implied Volatility smile smirk

Primbs, MS&E 345, Spring 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 Why might return distributions have heavy tails? Stochastic Volatility Jump diffusion models Risk management strategies and feedback effects Heavy Tails

Primbs, MS&E 345, Spring How do heavy tails cause a smile? More probability under heavy tails This option is worth more This option is not necessarily worth more Call option strike K Out of the money call: Call option strike K At the money call: Probability balances here and here

Primbs, MS&E 345, Spring Important Parameters of a distribution: Gaussian~N(0,1) Mean Variance Skewness Kurtosis

Primbs, MS&E 345, Spring Mean Variance Skewness Kurtosis Red (Gaussian) Blue Skewness tilts the distribution on one side.

Primbs, MS&E 345, Spring Large kurtosis creates heavy tails (leptokurtic) Mean Variance Skewness Kurtosis Red (Gaussian) Blue

Primbs, MS&E 345, Spring Empirical Return Distribution (Courtesy of Y. Yamada) Mean Variance Skewness Kurtosis (Data from the Chicago Mercantile Exchange)

Primbs, MS&E 345, Spring days to maturity (Courtesy of Y. Yamada) Volatility Smiles and Smirks Mean Square Optimal Hedge Pricing

Primbs, MS&E 345, Spring days to maturity (Courtesy of Y. Yamada) Volatility Smiles and Smirks Mean Square Optimal Hedge Pricing

Primbs, MS&E 345, Spring days to maturity (Courtesy of Y. Yamada) Volatility Smiles and Smirks Mean Square Optimal Hedge Pricing

Primbs, MS&E 345, Spring days to maturity (Courtesy of Y. Yamada) Volatility Smiles and Smirks Mean Square Optimal Hedge Pricing