Jeff Dewynne

Differential equations and asymptotic solutions for arithmetic Asian options: 'Black-Scholes formulae' for Asian rate calls

In this article, we present a simplified means of pricing Asian options using partial differential equations (PDEs). We first provide a concise derivation of the well-known similarity reduction and exact Laplace transform solution. We then analyse the problem afresh as a power series in the volatility-scaled contract duration, with a view to obtaining an asymptotic solution for the low-volatility limit, a limit which presents difficulties in the context of the general Laplace transform solution. The problem is approached anew from the point of view of asymptotic expansions and the results are compared with direct, high precision, inversion of the Laplace transform and with numerical results obtained by V. Linetsky and J. Vecer. Our asymptotic formulae are little more complicated than the standard Black-Scholes formulae and, working to third order in the volatility-scaled expiry, are accurate to at least four significant figures for standard test problems. In the case of zero risk-neutral drift, we have the solution to fifth order and, for practical purposes, the results are effectively exact. We also provide comparisons with the hybrid analytic and finite-difference method of Zhang.

THE END-OF-THE-YEAR BONUS: HOW TO OPTIMALLY REWARD A TRADER?

Traders are compensated by bonuses, in addition to their basic salary. However, little is known about how to optimally reward a trader. In this article we build a framework for the study of this problem and explore a variety of possible compensation structures.

THE VALUATION OF SELF-FUNDING INSTALMENT WARRANTS

We present two models for the fair value of a self-funding instalment warrant. In both models we assume the underlying stock process follows a geometric Brownian motion. In the first model, we assume that the underlying stock pays a continuous dividend yield and in the second we assume that it pays a series of discrete dividend yields. We show that both models admit similarity reductions and use these to obtain simple finite-difference and Monte Carlo solutions. We use the method of multiple scales to connect these two models and establish the first-order correction term to be applied to the first model in order to obtain the second, thereby establishing that the former model is justified when many dividends are paid during the life of the warrant. Further, we show that the functional form of this correction may be expressed in terms of the hedging parameters for the first model and is, from this point of view, independent of the particular payoff in the first model. In two appendices we present approximate solutions for the first model which are valid in the small volatility and the short time-to-expiry limits, respectively, by using singular perturbation techniques. The small volatility solutions are used to check our finite-difference solutions and the small time-to-expiry solutions are used as a means of systematically smoothing the payoffs so we may use pathwise sensitivities for our Monte Carlo methods.

The Black–Scholes Model

Introduction We begin this chapter with a discussion of the concept of arbitrage, a concept which, in certain circumstances, allows us to establish precise relationships between prices and thence to determine them. We then discuss option strategies in general and use arbitrage, together with the model for asset price movements that we discussed in the previous chapter, to derive the celebrated Black–Scholes differential equation for the price of the simplest options, the so-called European vanilla options. We also discuss the boundary conditions to be satisfied by different types of option, and we set the scene for the derivation of explicit solutions. This chapter is fundam,ental to the whole subject of option pricing and should be read with care . Arbitrage One of the fundamental concepts underlying the theory of financial derivative pricing and hedging is that of arbitrage . This can be loosely stated as “theres no such thing as a free lunch.” More formally, in financial terms, there are never any opportunities to make an instantaneous risk-free profit. (More correctly, such opportunities cannot exist for a significant length of time before prices move to eliminate them.) The financial application of this principle leads to some elegant modelling. Almost all finance theory, this book included, assumes the existence of risk-free investments that give a guaranteed return with no chance of default. A good approximation to such an investment is a government bond or a deposit in a sound bank.