The Review And Numerical Analysis On The Pricing Of European Basket Options

Due to the diversity of trading markets, the difference between customer demands,the perfection of financial derivatives markets and the development of financial theories,financial institutions have designed a variety of exotic options. Basket options are oneclass of exotics options, multi-asset options, which are usually used for hedging in thetrades with many foreign currencies. The payoff of basket options, depend on theweighted average of the price of various underlying assets. According to the portfoliotheory, the volatility of the portfolio of various underlying assets is relatively small,which makes the price of the basket options lower than the sum of the price ofunderlying asset, with a high efficiency in cost. Many scholars have explored andstudied the pricing of basket options based on different models, different marketassumptions and methods. Based on the theoretical and practical significance of thepricing of basket options, in this thesis, we focuses on review of analyzing pricingmodels, methods and formulas of the basket options for better application in actualtransactions.On the one hand, we conduct a review analysis on the pricing of European basketoptions under the Black-Scholes model. For geometric basket options, we solve theBlack-Scholes formula directly by introducing a combination of independent variable,we also get the pricing formula of geometric basket options based on the problem of thepricing of geometric basket options can be transformed into one-dimensional. Forarithmetic basket options, five different analytical approximate pricing formulas arelisted by summarizing the relevant research literature in China and abroad, and provesome of all pricing formula.On the other hand, we conduct a review analysis on the pricing of European basketoptions under different market assumptions, by relaxing the assumptions ofBlack-Scholes model. Based on the relaxation of the underlying asset price process,compile the pricing formula of European basket options under jump-diffusion andfractional Brownian motion. Based on the relaxation of the constant volatilityassumption, the approximation pricing formula of the European basket options with twounderlying assets is given under the Heston stochastic volatility model. Based on the relaxation of the no default risky assumption, we list the pricing model and formula ofthe default risky geometric basket options. Based on the relaxation of the frictionlessmarket, we derive the model of basket options pricing with transaction cost, by usingthe riskless hedging principle.We also carry on some numerical simulations on the pricing of the Europeanbasket call options under Black-Scholes assumption. Due to the law of large numbers,we estimate the European arithmetic basket call options by using Monte Carlo method,at the same time, use control variates method to improve the simulation efficiency, anduse low discrepancy Halton and Faure sequences to narrow the sampling scope, this iscalled Quasi-Monte Carlo method.