A study of Greek letters of currency option under uncertainty environments

Owing to the fluctuations in the financial markets from time to time, some input variables, in particular the interest rate, spot exchange rate and volatility, in the Garman–Kohlhagen model [M.B. Garman, S.W. Kohlhagen, Foreign currency option values, Journal of International Money and Finance 2 (1983) 231–237] cannot be expected in a precise sense. Therefore, the fuzzy set theory introduced by Zadeh [L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353] can provide a useful tool to deal with this kind of impreciseness. In this paper, three different versions of the G–K model, the put-call parity relationship and the calculation formulas of the Greek letters according to these three different G–K models are then proposed based on fuzzy set theory. Under the considerations of fuzzy foreign and domestic interest rates, fuzzy spot exchange rate and fuzzy volatility, the European currency option prices and the Greek letters turn into fuzzy numbers or closed intervals. This means that the investors can pick any European currency option price and Greek letters with an acceptable belief degree or belief degree weighting parameter for his later use. The empirical results indicate that the Greeks calculated under fuzzy environment can be considered as a useful tool for managing option risk for an option writer.