Theoretical Deduction And Computer Simulation For Several Typical Exotic Options

Option, which plays an indispensable role in current financial market, is a kind oftypical derivatives with enormous theoretical research value. Its significant influence onfinancial market mainly relies on the following four respects: creating a chance toearning money for speculators, controlling the price of underlying asset within areasonable range based on non-arbitrage principle, providing a way to get rid of risk forhedgers, and serving for make some specific portfolio.Option pricing is one of the core topics of Financial Engineering. With the adventand development of exotic options, it is necessary to make a deep discussion andthinking to refine the theory of exotic options’ pricing. This paper illustrates the methodof pricing several classical and widely-used exotic options based on mathematicaltheory and computer simulation.As a sort of derivatives, the price model of exotic option relies on the variation ofunderlying asset. In the circumstance of continuous time, this variation could bedescribed as a stochastic differential equation thus the price of exotic option is thesolution of a partial differential equation. Therefore, one of the key points of this paperis to price exotic options based on PDE theory, especially on the theory of heat equation.We assume that the stock price fits a geometric Brownian Motion and the marketcondition is congruent with B-S model. Under these premises, we derive the pricingformula of forward start option, cash-or-nothing option, asset-or-nothing option, barrieroption, look-back option, and geometric average Asian option. On the way of chasingthese conclusions, this paper emphasizes the structure of the solution, the techniques ofdelta-hedging and the property of heat equation. All of these would be helpful when wedealing with other related pricing problems.On the other hand, for the exotic options with complicated structure, it is difficultto find its theoretical approach through PDE approach. In this case, Monte-Carlo andQuasi Monte-Carlo method is the most efficient way to solve such problems. This paperarrives at the price of arithmetic average basket option with Monte=Carlo method anduse control variables to reduce the variation of MC method. Moreover, this paperpricing the arithmetic average Asian option through QMC method with Sobol’s sequence and Halton’s sequence and further discuss the dimensional stability amongdifferent methods.In addition, this paper also makes a brief introduction to the history of option, theclassification of exotic options, Black-Scholes model, jump diffusion model, randominterest rate model, and the core knowledge of MC and QMC methods.