1 / 30

The Analysis of Volatility

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.

karlyn
Download Presentation

The Analysis of Volatility

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The Analysis of Volatility Primbs, MS&E 345, Spring 2002

  2. Historical Volatility Volatility Estimation (MLE, EWMA, GARCH...) Maximum Likelihood Estimation Implied Volatility Smiles, smirks, and explanations Primbs, MS&E 345, Spring 2002

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. Exponentially Weighted Moving Average (EWMA): weights die away exponentially Weighting Schemes Primbs, MS&E 345, Spring 2002

  11. 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

  12. Historical Volatility Volatility Estimation (MLE, EWMA, GARCH...) Maximum Likelihood Estimation Implied Volatility Smiles, smirks, and explanations Primbs, MS&E 345, Spring 2002

  13. 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

  14. Let Maximum Likelihood Methods: Example: Estimate the variance of a normal distribution from samples: Given u1,...,um. Primbs, MS&E 345, Spring 2002

  15. 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

  16. 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

  17. Historical Volatility Volatility Estimation (MLE, EWMA, GARCH...) Maximum Likelihood Estimation Implied Volatility Smiles, smirks, and explanations Primbs, MS&E 345, Spring 2002

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. Mean Variance Skewness Kurtosis Important Parameters of a distribution: Gaussian~N(0,1) 0 1 0 3 Primbs, MS&E 345, Spring 2002

  24. 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

  25. 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

  26. 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

  27. Volatility Smiles and Smirks 10 days to maturity Mean Square Optimal Hedge Pricing (Courtesy of Y. Yamada) Primbs, MS&E 345, Spring 2002

  28. Volatility Smiles and Smirks 20 days to maturity Mean Square Optimal Hedge Pricing (Courtesy of Y. Yamada) Primbs, MS&E 345, Spring 2002

  29. Volatility Smiles and Smirks 40 days to maturity Mean Square Optimal Hedge Pricing (Courtesy of Y. Yamada) Primbs, MS&E 345, Spring 2002

  30. Volatility Smiles and Smirks 80 days to maturity Mean Square Optimal Hedge Pricing (Courtesy of Y. Yamada) Primbs, MS&E 345, Spring 2002

More Related