Valuation Of Barrier Option With Fuzzy Numbers

Barrier option is a kind of path-dependent option and its payoff depends onwhether the asset price has breached some barrier level during the life of the option.Its premium is lower than European option and investors’ subjective desire is involvedin the barrier contract flexibly, so it takes the fancy of the investors. When pricingoptions, the uncertainty contains not only the randomness but also the fuzziness.However, only the randomness is considered in general pricing models. They hardlydo research on the fuzziness. The models in this paper take into account both therandomness and the fuzziness.In this paper, we mainly study down-and-in and down-and-out call optionspricing with fuzzy numbers. First of all, we fuzzify the stock price at maturity and getnew payoff function. Then we use the risk neutral pricing principle and Girsanovtheory to research the pricing problems of barrier options with fuzzy numbers whilethe stock price follows the geometric Brownian motion and get the pricing formulas.Finally, we do research on the barrier option pricing problems under jump diffusionprocess with fuzzy numbers. Specific work is as follows:(1) Suppose that the stock price follows the geometric Brownian motion anddonate the stock price at maturity by trapezoidal fuzzy number. So the payoff isfuzzified. Then apply the risk neutral pricing principle together with fuzzy set theoryto solve the pricing formulas of down-and-in, down-and-out call options and find outthe relationship of their values with European call options’. Finally, use numericalexperiments to analyze the relationship between the option value and the stock price,demonstrate the rationality of this method and compare with B-S pricing formulas.(2) Suppose that the stock price follows the jump diffusion process, then applyfuzzy set theory to fuzzify the stock price at maturity date and solve the pricingformulas of down-and-in, down-and-out call options under jump diffusion processwith the risk neutral pricing principle. For the formulas are in form of series, wedemonstrate that the series in those pricing formulas is convergent. Finally, usenumerical experiments to analyze the relationship between the option value and thestock price, and compare with the general pricing formulas.