Pricing Foreign Currency and Cross-Currency Options Under GARCH

Abstract

Experience has shown that the returns volatility for most financial instruments is not constant, as assumed in the Black-Scholes model and similar valuation formulas. Rather, high-volatility and low-volatility days cluster together, and in many cases there is significant asymmetry between volatility movements in up and down markets. These regularities are well-represented by a GARCH model for the underlying asset's returns, and the theory of option pricing under GARCH has been developed by Duan in earlier wok. Here, Duan and Wei extend the GARCH option pricing paradigm to contracts that are affected by multiple stochastic variables, such as a quanto option on a foreign stock that is exposed to risk from changes in both the stock price and the exchange rate. In addition to presenting new valuation formulas, the authors demonstrate that taking GARCH effects into account can help to resolve the pricing anomalies uncovered by earlier research using simpler models.