GARCH Option Pricing

Volatility is a measure of the dispersion of an asset price about it’s mean level over a fixed time interval and it is a well known fact that the price of an option depends on the expected volatility of the underlying asset during the life of the option. Therefore, a great deal of effort is put into forecasting the future volatility.

Black-Scholes’ (BS) option pricing model assumes that proportional changes in the underlying future prices are identically independent distributed and normal. Also, the volatility is found by the help of past returns and is kept as constant for the future prices. However, in fact, price movements exhibit excess kurtosis and time-varying volatility. Also, estimating the parameters for the option pricing formula by using historical asset prices i.e. past information is meaningless for pricing the future prices. Because, the information found by historical values could be very different from the expectations embedded in option prices about the future evolution of the asset price. For these reasons, BS’s model results in systematic option pricing errors. (Hauser and Neff)

Many studies have shown that the variance of returns can be modeled with GARCH process and volatility forecasts found by GARCH performed better results according to other forecasting methods. (Bollerslev, Chou, Kroner, 92)

The objective of the present part of the thesis is to find the conditional expected variance from a GARCH model for the variance input in the BS formula. Since the GARCH model is defined in discrete time, using BS formula with GARCH conditional variance will not work in American options. Because, BS formula is defined under the assumption that securities prices move continuously in time and it could be hedged any time. Although there are some methods such as Monte Carlo simulations, so that an early-exercise premium could be added to the European price derived from a GARCH model, they are impractical due to the large amounts of computing power is required to compute option prices.

The investigation performed here can be regarded as a study how well a continuous time pricing formula works in a discrete environment. In other words, the aim is to find out the effectiveness of the combination of GARCH volatility forecasts with the BS formula. To sum up, in the discrete-time GARCH option pricing model, conditional variances are predicted by the help of GARCH modeling and they are directly put into the BS formula on a daily basis.

The GARCH (1,1) Model and Volatility Forecasting

There are several reasons that you may want to model and forecast volatility. All economic events including risk should consider the volatility or even future volatility. In BS option pricing formula, volatility is found by using past asset returns and is kept as constant through the life of the option. However, if we can expect the following days volatility by using the information up to today and use this term in pricing the following days options, it would be better that using the today’s volatility.

Autoregressive Conditional Heteroscedasticity (ARCH) models are specifically designed to model and forecast conditional variances. The variance of the dependent variable is modeled as a function of past values of the dependent variable and exogenous variables.

ARCH models were first introduced by Engle (1982) and further developed and generalized as GARCH (generalized ARCH) by Bollerslev (1986). One of the most commonly used GARCH models is the GARCH (1,1) model. In the standard GARCH (1,1) specification, there are two parts:

The mean equation is written as a function of exogenous variables with an error term

(1) y_t = g x_t + e _t

and,

the conditional variance of asset return at time t, h_t obeys the following process

(2) h_t = w + a e ²_t-1 + b h_t-1

where e _t is the error term in the return process that is assumed to be

r_t = j + e _t

and the error term is assumed to have the form

e _t = z_t h_t^1/2,

where z_t is i.i.d with an expected “0” mean and unit variance i.e. normally distributed.

This specification means that, this period’s variance (ht) is formed as a weighted of a long term average (the constant, w ), the forecasted variance from last period (the GARCH term, h_t-1), and information about volatility observed in the previous period measured as the lag of the squared residual from the mean equation (the ARCH term, e ²_t-1). This way of interpreting the variance term is helpful when the volatility is time varying.

By this way, conditional variances can be found for each day by using the previous day’s information. If there are no volatility shocks, it is reasonable to update

the volatility estimates daily by using the most current information. Then, this forecasted variance term could be used in the BS formula.

References:

Bollerslev, Tim, Ray Chou, and Kenneth Kroner, “ARCH Modeling in Finance,” Journal of Econometrics, 52(1992), 5-59

Hauser, R. J., and D. Neff. “Pricing Options on Agricultural Futures: Departures from Traditional Theory.” d. Fut. Mkts. 5(1985): 539 - 577

Engle, Robert F. (1982), “Autoregressive Conditional Heteroscedasticity With Estimates of the Variance of United Kingdom Inflation”, Econometrica, 50, 987-1007

Bollerslev, Tim (1986), “Generalized Autoregressive Conditional Heteroscedasticity”, Journal of Econometrics, 31, 307-327