Pricing a European Basket Option in the Presence of Proportional Transaction Costs

Abstract

A crucial assumption in the Black–Scholes theory of options pricing is the no transaction costs assumption. However, following such a strategy in the presence of transaction costs would lead to immediate ruin. This paper presents a stochastic control approach to the pricing and hedging of a European basket option, dependent on primitive assets whose prices are modelled as lognormal diffusions, in the presence of costs proportional to the size of the transaction. Under certain assumptions on the individual preferences, it is able to reduce the dimensionality of the resulting control problem. This facilitates considerably the study of the value function and the characterisation of the optimal trading policy. For solution of the problem a perturbation analysis scheme is utilized to derive a non‐trivial, asymptotically optimal result. The findings reveal that this result can be expressed by means of a small correction to the corresponding solution of the frictionless Black–Scholes type problem, resembling a multi‐dimensional ‘bandwidth’ around the vanilla case, which, moreover, is readily tractable.

Keywords:

• Option pricing
• transaction costs
• utility function
• asymptotic expansion
• Hamilton–Jacobi–Bellman equation
• closed form solution

Notes

1. The HJB equation is derived here only heuristically. A complete development and rigorous justification can be found in Karatzas and Shreve (1998) or Korn and Korn (2001).

2. See also Whalley and Wilmott (1997) and Korn (1998) for this point.

3. See also Dixit (1991) and Dumas (1991) on this issue.