Option pricing with simulation of fuzzy stochastic variables

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Technická Univerzita v Liberci
Technical university of Liberec, Czech Republic
During last decades the stochastic simulation approach, both via MC and QMC has been vastly applied and subsequently analyzed in almost all branches of science. Very nice applications can be found in areas that rely on modeling via stochastic processes, such as finance. However, since financial quantities as opposed to natural processes depend on human activity, their modeling is often very challenging. Many scholars therefor suggest to specify some parts of financial models by means of fuzzy set theory. Many financial problems, such as pricing and hedging of specific financial derivatives, are too complex to be solved analytically even in a crisp case, it can be efficient to apply (Quasi) Monte Carlo simulation. In this contribution a recent knowledge of fuzzy numbers and their approximation is utilized in order to suggest fuzzy-MC simulation to option price modeling in terms of fuzzy-random variables. In particular, we suggest three distinct fuzzy-random processes as an alternative to a standard crisp model. Application possibilities are shown on illustrative examples assuming a plain vanilla European put option under Brownian motion with fuzzy parameter (volatility), Brownian motion with fuzzy subordinator and Brownian motion with fuzzyfied subordinator. In each case the model result into a whole set of prices – thus, since we assume one of the input data as LU fuzzy number, we get the price in terms of the LU fuzzy number as well. The payoff function of analyzed put option can be obviously replaced by more complex payoff structure.
random variable, fuzzy variable, option, simulation