ارزش گذاری اختیار معاملات آمریکایی با استفاده از فرآیندهای garch

abstract: in the paper of black and scholes (1973) a closed form solution for the price of a european option is derived . as extension to the black and scholes model with constant volatility, option pricing model with time varying volatility have been suggested within the frame work of generalized autoregressive conditional heteroskedasticity (garch) . these processes can explain a number of empirical regularities found in asset return series, the most important of which are leptokurtosis and volatility clustering phenomenon. one of the most important problems in option pricing theory is the valuation and optimal exercise of derivative with american style exercise feature . in this paper we consider the problem of pricing options when the underlying asset exhibits time varying volatility. we model time varying in the context of garch processes, since this processes have been used extensively in the literature to model asset returns , and in many cases evidence has been found in favor of them . when volatility is allowed to vary through time in a way that depends on lagged innovations to the return process and lagged volatility, it is generally not possible to derive the future of underlying asset. the main technical problem when using a garch pricing model is that the distribution of future asset can not derive be derived in closed form. thus generally no analytical option pricing formula exists, and alternative method must be used to price derivative. simulation types are obvious candidates to be used whit the garch models. in the present paper, we show a powerful new simulation technique, using simple least square regression (the lsm method of long staff and schwartz), can be use to price american options. the key to this approach is the use of least square to estimate the conditional expected payoff to option holder from continuation. we will show how simulation methods can be used to price american options in a garch frame work by using the model of duan (1995) together with the lsm method of long staff and schwartz (2001). in the long staff and schwartz (2001) the conditional expectation of the payoff from continuing to hold the option is estimated from a cross sectional regression on a set of simulated paths. simulation allows state variable to fallow a garch or egarch or gjr process. the fitted values of the regression provides a direct estimate of the conditional expectation function . to understand the intuition behind this approach, re call that at any exercise time, the holder of an american option optimally compares the pay off from immediate exercise with the expected payoff from continuation, and then exercise if the immediate payoff is higher. thus the optimal exercise strategy is fundamentally determined by the conditional expectation of the payoff from continuing to keep the option alive. we show how flexible and simple the new method is.