longer hold, risk-neutral valuation breaks down and returns to risky assets should have a risk premium. This is, therefore, one reason why Black-Scholes prices can differ from market prices of options. Engle and Mustafa (1992) examine the stochastic volatility that could be implied from observed market prices of options. Subsequently, Noh et al. (1994) and Engle and Rosenberg (1994, 1995) consider how the GARCH stochastic volatility can be incorporated into model prices of options. This is particularly important for short-maturity out-of-the-money equity options where practitioners may require more than a simple Black-Scholes valuation model. Amin and Ng (1993) and Hafner and Hardle (2000) have all found that the GARCH prices of such options are closer to the observed market price than the Black-Scholes prices.

Standard pricing methods for an option on a single asset assume that the underlying price S(t) follows the geometric Brownian motion diffusion process

dS(t)/S(t) = r dt + cs dZ(t),

where r is the risk-free rate of return and Z is a Wiener process. Since volatility a is assumed to be known, there is only one source of uncertainty, Z. If simulation methods are used at all it is only necessary to use Monte Carlo on the independent increments dZ to generate price paths S(t) over the life of the option. The simulation is based on the discrete form of GBM, adjusted for risk neutrality,22

S, = exp(r - 0.5a2 + az,),

where the z, are i.i.d. N(0, 1). Starting from the current price SQ, Monte Carlo simulation of an independent series on zt for t- 1, 2, . . ., T will generate a terminal price ST. Thousands of these terminal prices should be generated starting from the current price S0, and the discounted expectation of the option pay-off function at the risk-free interest rate gives the price of the option.

For example, a call option on an average price (known as an Asian call) has price

C(S0) = t~rT ( { - , 0}),

where T is the option maturity, is the strike and Am is the arithmetic average of m prices at times t\, t2, . . ., tm (and normally tm-T):

m=/2 StJm

The price of the Asian call may be estimated from the simulated distribution of average prices Amh for i =1, . . ., N and N very large, by

To derive this from the continuous form, use Itos lemma on InS and then make time discrete.

s, =crtz,

Random number generator Figure 4.11 Simulation with a GARCH process.

C(S0) = - ( ; { / - , 0}/N).

Engle and Rosenberg (1995) describe how to extend simulation methods for option pricing when the underlying returns have time-varying volatility generated by a GARCH process. A single random variable z, drives the two diffusion processes in discrete time:

S, = £, , ( -0.5 -2 + ,), g? = co-r-ae2 , + (3G2 b

where , - g,z, and the zt are independent standard normal variates. Monte Carlo simulation on z can be used to price the option exactly as described above and following the scheme shown in Figure 4.11.23

Corresponding GARCH option deltas and gammas may be calculated using Corresponding GARCH finite-difference approximations, such as the central differences: option deltas and

gammas may be

5 = [C(S0 + t) - C(S0 - )]/2 , calculated using finite-

differences

= [C(S„ + ti) - 2C(S0) + C(S0 - n)]/n2.

When calculating deltas and gammas by simulation, errors can be very large unless large numbers of simulations are used. Simulation errors will be reduced by using correlated random numbers because the variance of delta and gamma estimates will be reduced when C(S0 + r\) and C(S0 - r\) are positively correlated.

23Note that there is still only one source of uncertainty in this discrete time formulation. However, in continuous time the diffusion processes corresponding to GARCH models do have an additional error process. See Nelson (1990).

When volatility is stochastic a perfect hedge would not normally exist, so the risk-neutral pricing assumption will not hold

As noted by Engle and Rosenberg (1995), when volatility is stochastic a perfect hedge would not normally exist, so the risk-neutral pricing assumption will not hold. However, Duan (1995) explains that there is a local risk-neutral valuation relationship with the N-GARCH volatility model defined in §4.2.4. However, Duans option pricing model still depends on a risk premium parameter X. If this is non-zero there will be a risk premium in the asset return, indicating that a perfect hedge will not exist, and if the estimates of the risk premium parameter are not very robust the model will have to be used very carefully in practical applications. One of these is described in the next subsection.24

4.4.3 Smile Fitting

Instead of using time series data, a cross-section of market prices on options at different strikes and maturities is used to estimate the GARCH model parameters. Parameters are fitted by iterating on the root mean square error between the GARCH and the market implied volatility smile surfaces. How is a GARCH smile surface obtained?

GARCH option prices are put into the Black-Scholes formula, and the GARCH implied volatility is then backed out of the formula just as one would do with ordinary market implied volatilities

Initial values for the GARCH model parameters are fixed, and then GARCH option prices obtained, as explained above, for options of different strikes and maturity that also have reliable market prices. These GARCH option prices are put into the Black-Scholes formula, and the GARCH implied volatility is then backed out of the formula just as one would do with ordinary market implied volatilities, only this time the GARCH price is used instead of the price observed in the market. Comparison of the GARCH smile surface with the observed market smile surface leads to a refinement of the GARCH model parameters by iteration on the root mean square error between the two smiles, and so the GARCH smile is fitted, as in Figure 4.12.25

Duan (1996) reports that estimating the parameters of an N-GARCH model using current market data gives very similar results to those obtained using time series data. However, one must be very careful to check the stability of parameter estimates; parameter estimates that are based on a snapshot of option prices on one day, rather than a long historical series on the underlying price, may vary considerably from day to day. For example, Table 4.10 shows that comparing Duans estimates for the FTSE on 31 March 1995 with the estimates we obtained on 2 March 1998 gives quite different values for 9 + X and (3.

Looking at Figure 4.12, on 2 March 1998 the volatility term structure was sloping upwards, because the market had been relatively quiet for some time, and convergence to the 1-year volatility level was also relatively rapid.

24In later work Duan et at (1998) and Duan (1999) have derived an analytic approximation to the univariate N-GARCH option pricing model, and Duan and Wei (1999) have extended the framework to a bivariate N-GARCH model for pricing foreign exchange options and currency-protected options.

25Many thanks to Chris Leigh for providing this figure.