Asian Options Pricing On GARCH Stochastic Interest Rate Model

Asian option is a kind of typical path-dependent option which income depends on the mean of underlying assets’prices in a specific period.By comparing with the standard option, the option premium and risk of asian option are both less. The design of asian option is primarily used to limit the artificial manipulation of underlying assets’prices which is widely applied in interna-tional trade and foreign exchange market.Furthermore, the hypotheses of classical Black-Scholes option pricing model are rigour and idealization (such as the risk-free interest rate r and stock re-turns variance σ are both constants), which is disagreed with practice. In actual financial markets, both volatility and interest rate is non-combatants. The interest rate is part of the most important factors that various financial assets and financial derivatives are affected by it. The interest rate risk is originated from the randomness of the interest rate.In addition, GARCH Stochastic Interest Rate Models can fit Heteroscedastic Model with Long Memory effectively. In order to broaden the use of GARCH models, many scholars construct multiple variants of GARCH model from dif-ferent angle embarking. GARCH Model was applied widely, in the case of stochastic interest rate study. Therefore, in order to work around this problem people established the stochastic interest option pricing model under the hypothesis of interest rate obeys stochastic process variable.This paper studies on Asian options on GARCH Stochastic Interest Rate Models which have certain theoretical significance and realistic importance.In this paper, the pricing of Asian option was studied by using the GARCH model in the case when a stochastic interest rate was considered.The main work as follows:First, European type ge-ometric mean Asian call option with fixed strike price on GARCH Stochastic Interest Rate Models was studied.First of all, multi-dimensional GARCH model was introduced. Thus it is assumed that underlying assets’prices and interest rates satisfy GARCH (1,1) Model. Secondly, seeking the characteristic function of the two-dimensional GARCH (1,1) Model, and pricing European type geometric mean Asian call option with fixed strike price by utilizing stochastic differentiation like Fourier inverse transformation.Secondly, based on the first part of the study further research on European type geometric mean Asian call option with fixed strike price on GARCH Stochastic In-terest Rate Models was developed.So, by means of generalized Edgeworth expansion to obtain the density function of distribution function, it is mainly approximately obtained by using lognormal distribution.The results are as follows:(1) Comparison on Models. Comparing with classical BS model, the price of the Asian option is smaller which is calculated by using GARCH Stochastic Interest Rate Models, further to explain the volatility affects the pricing of Asian option.(2) Comparison on methods. Geometric average Asian options adopt Fourier inverse transfor-mation, arithmetic mean Asian options adopt Generalized Edgeworth Expansion which also use MonteCarlo to simulate, found that the differences of price are small, suggesting that Fourier in-verse transformation and Generalized Edgeworth Expansion are reasonable and effective for Asian option pricing.(3) Comparison on α0r, α1r the two parameters of interest rate model, found that both of them are an increasing function of option pricing, and the changing speed of variable α0r is faster than α1r. So the randomness of interest rates affect the change of option price, further to explain interest rates is a crucial factor to affect option pricing.