Abstract
Considering the uncertainty of a financial market includes two aspects: risk and vagueness; in this paper, fuzzy sets theory is applied to model the imprecise input parameters (interest rate and volatility). We present the fuzzy price of compound option by fuzzing the interest and volatility in Geske’s compound option pricing formula. For eachα, theα-level set of fuzzy prices is obtained according to the fuzzy arithmetics and the definition of fuzzy-valued function. We apply a defuzzification method based on crisp possibilistic mean values of the fuzzy interest rate and fuzzy volatility to obtain the crisp possibilistic mean value of compound option price. Finally, we present a numerical analysis to illustrate the compound option pricing under fuzzy environment.
Highlights
Compound options are options with other options as underlying assets
According to Wu [21] and Nowak and Romaniuk [29], we replace r and σ in (8) by fuzzy numbers rand σ, respectively, and replace the arithmetics by fuzzy arithmetics to get the analytical formula for the compound option under fuzzy environment, which is given in Theorem 4
Considering the uncertainty of the financial market includes both risk and vagueness; the compound option price will turn into a fuzzy number
Summary
Compound options are options with other options as underlying assets. Since Geske [1] derived the closed form pricing formula using the method of partial differential equations for the first time, some scholars have extended the pricing model and proposed some new pricing methods. The work of [7, 8] introduced time dependent volatility and an interest rate to the pricing model of compound options. The existing literature on option pricing under the fuzzy stochastic model mainly studied the European option, based on the Black-Scholes model. The work of [24] introduced a crisp weighted possibilistic mean value Black-Scholes option pricing formula. The main contribution of this paper is that we present the α-level set of fuzzy prices for each α and give a sensitivity analysis of the crisp possibilistic mean value of compound option price with respect to the core value of fuzzy interest rate and fuzzy volatility.
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