European Options under Proportional Transaction Costs: An Algorithmic Approach to Pricing and Hedging

Abstract

The paper is devoted to optimal superreplication of European options in the discrete setting under proportional transaction costs on the underlying asset. In particular, general pricing and hedging algorithms are developed. This extends previous work by many authors, which has been focused on the binomial tree model and options with specific payoffs such as calls or puts, often under certain bounds on the magnitude of transaction costs. All such restrictions are hereby removed. The results apply to options with arbitrary payoffs in the general discrete market model with arbitrary proportional transaction costs. Numerical examples are presented to illustrate the results and their relationships to the earlier work on pricing options under transaction costs.

American Options Under Proportional Transaction Costs: Seller's Price Algorithm, Hedging Strategy and Optimal Stopping

“…Because of this correspondence, Algorithm 3.1 can be regarded as a natural extension of the Snell envelope construction. Similar results, algorithms and representations have been established for European options in the general discrete setting by Roux, Tokarz and Zastawniak [RTZ06], and for American options under small transaction costs in the binomial tree model by Tokarz and Zastawniak [TZ05]. The case of American options in an arbitrary discrete market model subject to arbitrary transaction costs turns out to be significantly more challenging due to the appearance of mixed stopping times.…”

“…The case of American options in an arbitrary discrete market model subject to arbitrary transaction costs turns out to be significantly more challenging due to the appearance of mixed stopping times. Nevertheless, the underlying idea is similar to that in the European options paper [RTZ06].…”

“…We shall use similar notational conventions and model assumptions as in Roux, Tokarz and Zastawniak [RTZ06].…”

“…which feature in Theorem 2.1, will be denoted by P. Probability measures P of this kind will be called martingale probabilities. This family of pairs (P, S) can be used to represent the prices of European options under transaction costs, see Jouini and Kallal [JK95] or Roux, Tokarz and Zastawniak [RTZ06]. To represent the prices of American options we need certain larger families than P.…”

“…In effect, to find the seller's price, one needs to compute the function Z 0 , which amounts to solving a large convex optimisation problem over the set of all superhedging self-financing strategies. In Algorithm 3.1 this is achieved by means of a dynamic programming type backward iterative procedure for computing Z t , given Z t+1 and the option payoff at time t. Similarly as for European options [RTZ06], this pricing algorithm offers considerable efficiency. In the case of path-independent stock prices and American options with path-independent payoff, the polyhedral concave functions Z t are also path-independent.…”