OPTIONS PRICING AND HEDGING WITH GARCH

1. OPTIONS PRICING AND HEDGING WITH GARCH .THE PRICING KERNEL .HULL AND WHITE .THE PLUG-IN ESTIMATOR AND GARCH GAMMA .ENGLE-MUSTAFA – IMPLIED GARCH .DUAN AND EXTENSIONS .ENGLE AND MEZRICH .ROSENBERG AND ENGLE

2. THE PRICING KERNEL • Theoretical Pricing of all risky assets which excludes arbitrage opportunities • P is price of a put with strike K, interest rate r, maturity T, underlying S • M is the pricing kernel which is a random variable • E* is the expectation with respect to the “risk neutral density

3. Risk Premium • Risk premium is the difference between the expected value of an asset in the empirical distribution and in the risk neutral measure which is just its price

4. Black Scholes Option Prices • When underlying prices are geometric brownian motion with constant volatility and the pricing kernel prices the underlying asset then:

5. HULL AND WHITE(1987) • When volatility is stochastic options are no longer redundant assets • If the risk neutral density of S given realized volatility is log normal, then:

6. PLUG-IN PRICING • For at-the-money options, the Black Scholes formula is nearly linear in volatility, • THEREFORE • And it makes sense to price options with expected volatility • In this setting implied volatilities are the option market forecast of realized “risk neutral” volatility

7. Assumptions Required • Expected Volatility not Variance • Risk Neutral Expectation - not empirical • Lognormal Conditional Distribution • Volatility and returns uncorrelated • At the money options

9. GARCH GAMMA • Engle and Rosenberg(1995) examine the gamma and vega hedges in a “plug in” setting • When expected volatility is approximated by GARCH, it depends upon daily squared returns. • Hence second derivative of option price with respect to the underlying has two terms:

10. GARCH GAMMA • For long maturity options, the vega is very large but the vega multiplier calculated from GARCH declines due to the mean reversion in volatility • Empirically - traditional vega hedging is a bad idea - use Gamma or better yet GARCH GAMMA

11. Engle and Mustafa(1992)IMPLIED GARCH • What GARCH model can best explain a collection of option prices? • Simulate the risk neutral distribution with hypothetical parameters and estimate the option prices as discounted payoff • Adjust the parameters to get the best approximation to observed option prices • Find that this is similar to the historical GARCH except right after ‘87 crash

12. DUAN(1995)(1996) • How to risk neutralize a GARCH simulation? • Proposes “local risk neutralization” which sets the mean to the riskless rate and the variance equal to the GARCH • Uses the NGARCH model

13. SIMULATING GARCH* • Z is a set of gaussian random numbers • The asymmetry is increased • If then c is increased by delta • Duan most successfully implies this parameter from options data

14. Engle and Mezrich - An Alternative Simulation • Standardized residuals • Bootstrap from these rather than random numbers • Oversample the extreme negatives

15. Risk Neutralization • Adjust the mean to price the underlying exactly • Adjust the variance of log returns to equal the GARCH empirical variance

16. GARCH TREES

18. Rosenberg and Engle(1999) Empirical Pricing Kernels

25. SIMULATION PRICING WITH GARCH • We will simulate GARCH models • We will risk neutralize the outcome several ways • We will compare with observed prices • We will use a new Longstaff and Schwarz method to estimate delta and gamma which is really GARCH GAMMA

26. LONGSTAFF AND SCHWARTZ LEAST SQUARES METHOD • To estimate the value of an option before expiration we want to find • Use regression where state variables are the simulated prices at this point • This gives GARCH GREEKS or other volatility process greeks

27. RISK MANAGEMENT:FIND VaR FOR AN OPTION • 1) Find Risk Neutral Expected discounted value of option payoff conditional on the state variable tomorrow • 2) Find sets which have probability of containing the state variable tomorrow • 3) Find the worst expected option value in each set • 4) Find the maximum of such values over all sets

28. MORE PRECISELY • Find VaR to satisfy: • Notice that the first part is simply the option price as a function of the state variable tomorrow • This will typically be just an interval of low or high prices.

29. AN EXAMPLE:SHORT A PUT OPTION • Simulate a risk neutral GARCH process • Calculate the discounted payoff on each path • Regress this payoff on the underlying on day 1 • Evaluate the regression at the 1% point of the underlying on day one (since negative returns are the worst case)

30. Value at Risk for Portfolios of Options • If all options are on the same underlying, then the strategy above will still work, even if the maturities are different • If the options are on different underlyings then a multivariate approach is necessary

31. VaR for Portfolio of Options • Letting Payoff be the payoff of all the options and letting s be a vector of state variables, the definition remains the same • Now however we must simulate a multivariate process to price options, regress portfolio payoff on vector of state variables and compute VaR

32. REQUIREMENTS • MULTIVARIATE GARCH OR OTHER SIMULATION TYPE MODEL • RISK NEUTRALIZATION FOR MULTIVARIATE PROCESS