The Estimation Of Upper And Lower Bounds For Option Pricing

Option is one of the basic financial derivatives.Any financial product,after its being invented, has problems of pricing that practitioners in financial markets are trying to find a fair price for the financial product.Therefore,pricing option properly is especially important. It is of practical significance to study problems of option pricing.B-S model is the essence of theory of option pricing.In order to eliminate the limitation of the assumptions,Merton introduced upper and lower boundary values for option pricing, basing which many researchers do some further studies in the field.This paper first reviews several kinds of bound values for option pricing,then considers the factor of dividend in option pricing,and discusses its effects on the option pricing bounds.The content of this paper consists of three chapters as follow:The first chapter mainly introuduced option and bounds of option pricing.The second chapter expounds the basic theory of option pricing and B-S option pricing model.B-S option pricing model is the key and foundation for the theoretic system of option pricing.In this chapter,we analyze the conditions for B-S formulae ,and deduce the B-S equation throughΔ-hedge and It(?) formula, solution of which is the expression of option pricing.In the third chapter,several kinds of upper and lower bounds for option pricing are briefly introduced. The first one is the famous option pricing bounds by Merton.In terms of stochastic dominance ,Merton derived the conclusion that the value of option could never be negative , by which he determined a lower bound for the option price.Furthermore, he also deduced an upper bound for European call options.In other words,the Merton's option bound isThe second one is the option pricing bounds by Perrakis and Ryan. Perrakis and Ryan, assuming that time of transaction is discrete, the independence of distribution for stock price is valid,and marginal utility of consumption does not increase with value of Y changing,derived upper and lower values of option bounds in a discrete form by Rubinstein methodThe third one is the Levy and Ritchken bounds of option pricing.Peter H.Ritchken set up a target function of a linear programming through the method of Rubinstein's equation for uncertain income flow, and thus derived option bounds of incomplete market and the ones of incomplete market where risk aversion exists through linear programming problem.1. The linear programming upper bound of in a single period option price is given by:2. The linear programming lower bound of in a single period option price is given by: where j^*satisfies3. Passing to the limit of continuous states, the upper bound remains unchanged,but the lower bound converges to:The option price bound of incomplete market where risk aversion exists through linear programming problem is1. The upper bound is by working out a maximum problem of linear programming2. The lower bound is by working out a maximum problem of linear programming where j^* satisfies3. Limiting to the continuous state,the lower bound converges to where s_j^*satisfies E((?)_T|(?)_T < s_j^*)B₀ =S₀/B₀. The fourth one is the option pricing bounds by Ricardo Rodriguez. Ricardo J.Rodriguez deduced a formula of option pricing boundsL(S) = max[0,S-XR_f+V_k(S)I(S)] through the discount factor of a put option corresponding to a call option. Analyzing the attributions of and selecting various Ricardo J.Rodriguez obtained some traditional upper bound and a more precise bound-R bound.The conclusion is1. Merton's lower bound2. Peter H.Ritchken's lower bound3. Levy and Ritchken lower bound4. Rodriguez boundWe discuss the effects of dividend on option pricing in Chapter 4.Adding the dividend paid discretely to the formula of option pricing by Ricardo J.Rodriguez,we obtain the formula for option pricing with dividendWe derive various pricing bounds with dividend through selecting various values of V'_k(S):1.Merton's lower bound with dividend 2.Peter H.Ritchken lower bound with dividend3.Levy and Ritchken lower bound with dividend4.Rodriguez bound with dividendIn terms of the formula above,the lower bound of the European call option decreases when we consider the factor of dividend.In fact,the effects of dividend on the value of option are executed by affecting the prices of underlying assets.Generally speaking,the price of underlying assets drops due to the exclusion of claim after dividend provision. It is negative for a call option that the price of underlying assets drops,which makes the value of the contingent claim fall.