2 GARCH Option Pricing Model of Duan(1995) Asset returns follow the generalized autoregressive conditional heteroskedastic (GARCH) process. GARCH (1,1) model is the most commonly used GARCH process. S t is the stock price at time t r f is the constant one-period risk-free rate of return λ is constant unit risk premium
3 GARCH (1,1) Model To ensure covariance stationary of the GARCH (1,1) process The stationarity conditions are important to ensure that the moments of the normal distribution are finite. See Greene (2003) for more explanation.
4 GARCH (1,1) Model Estimation Maximum likelihood: Select likelihood function Estimated parameters from GARCH option Prices— Daily.xls written by Khanthavit (2007).
5 From GARCH process to LRNVR Duan (1995) used the locally risk-neutral valuation relationship (LRNVR) to derive GARCH option pricing. This is to ensure that the one-period ahead conditional variance is invariant with respect to a change to the risk-neutralized pricing measure.
6 The Terminal Asset Price The terminal asset price is as follows:
7 Option Pricing Model: Call Option European call option with exercise price X maturing at time T has to time-t value equal to For GARCH(1,1) model, X t and h t+1 serve as the sufficient statistics for Θ t. The delta of the call option equals to is an indicator function, i.e. equals 1if S T ≥ X and equals 0 otherwise.
8 Option Pricing Model: Put Option European put option with exercise price X maturing at time T has to time-t value can be derived from put- call parity relationship. The delta of the put option equals to
9 Advantage and Disadvantage of GARCH Option Pricing Model Volatility is observable from discrete asset price data and only a few parameters need to be estimated even in a long time series of options records. Unfortunately, the analytic solution for the GARCH option price is not available because the conditional distribution over more than one period cannot be analytically derived.
10 Option Pricing Model Estimation Monte Carlo simulation can be used. ( GARCH option Prices—Daily.xls) Simulate 10,000 times to get 10,000 values of possible terminal asset prices. Get 10,000 possible option values. Use expected value as the option price.
11 Option Pricing Model Estimation :Alternative Heston, S., and S. Nandi, 2000, “A Closed Form GARCH Option Pricing Model, ” The Review of Financial Studies, 13, 585-625.
12 Points for Consideration Duan (1995) pointed out that this model can only be used for the valuation of individual equity options. The market portfolio is expected to be highly correlated with aggregate assumption and the returns will not follow a GARCH process. How often do we need to estimate the parameters for GARCH model? When should we estimate the parameters from the option price directly? What model would be the best? Other GARCH-type model such as EGARCH and NGARCH could be explored in the future.