Further Study Of Option Pricing

The problem of option pricing on the base of no-arbitrage principleand risk-neutral valuation was further researched in this thesis. In whichinclude follow contents:a) Derive a general pricing equation of European option.b) Derive the pricing formula of European chooser option whoseinterest rate and volatility rate are constants.c) Extend European compound options and European chooser option tothe cases in which interest rate and volatility rate change as time, andanalyze the sensitivity of Black-Scholes formula using an example.Firstly, the general equation for an arbitrary underlying asset of aEuropean call option was derived based on the risk-neutral methodology.The derivation is valid for and makes no assumptions regarding thedistribution of the underlying asset's price when the option expires. Thenapplying acquired general equation to the special case of a lognormallydistributed underlying asset, the general pricing equation was gotten forthe class of all European options written on lognormal distributedunderlying assets, and applying the pricing formula to European calloptions written on stocks, currencies, and futures.Secondly, the pricing formula of European chooser option by themethod of risk-neutral valuation was set up. The result we get is samewith the result by resolve the Black-Scholes differential equation.At last, we know that interest rate and volatility rate change as time in financial market. The model in which the interest rate and volatilityrate changed as time was established. We extend the pricing formulas ofEuropean compound options and chooser option in which interest rateand volatility rate are constants to the cases in which parameters aretime-dependent, and let r(t)=r,σ(t)=σ, we have the ordinary compoundoptions and chooser option formulas.Finally, the sensitivity analyses of Black-Scholes formula wereshowed. It based on the Black-Scholes formula of stock option, analyzeits Delta, Gamma and Theta, and explain them by chart, at last deducedthe equation that they satisfied.