Practical Option Pricing and Related Topics

Abstract

The BS (Black-Scholes) option theory and its generalized version given by Harrison and Pliska (1981) provide mathematically beautiful and powerful results on option pricing. However, as has been observed in Chapters 2 and 3, a price process {S_t} in reality will not satisfy the assumptions the theory requires. In fact, the time series structure of return series{X_t} with x,=logS_t–logS_t-1 will not admit a (dominating) measure with respect to which the discounted process {e ^-rt S _t } becomes a martingale. Hence we will not be able to develop an arbitrage pricing theory by forming an equivalent (replicated) portfolio. In other words, in such a nonlinear model as Taylor model which is consistent with empirical features of returns, it is difficult to replicate an option and price it by the fundamental lemma in Chapter 12. As has been discussed, in such a case we are interested in the distribution of the present value X _T ^* = exp(— r τ) X _T of a contingent claim X _T and often regard the expected value E (X_T ^* ) as a proxy for pricing maybe with help of a risk neutrality argument, where E(•) is evaluated at t and τ = T-t. In this case, evaluations of the variance Var(X_T ^* ) and of such probabilities asP(X _T ^* =0), P(X _T ^* ≧E(X _T ^* )) etc. will be also important for practical purposes, as discussed in Chapter 12. In this chapter we consider such practical pricing problems and related topics in some specific models.