American Options under Proportional Transaction Costs: Pricing, Hedging and Stopping Algorithms for Long and Short Positions

TL;DR

This paper develops comprehensive algorithms for pricing, hedging, and stopping American options under proportional transaction costs in discrete markets, applicable to both buyers and sellers with arbitrary payoffs.

Contribution

It extends existing models by allowing arbitrary payoffs and transaction costs, providing efficient computational algorithms for both long and short positions in discrete markets.

Findings

Pricing algorithms grow polynomially with time steps

Buyer's and seller's stopping times can differ significantly

The approach applies to arbitrary payoffs and transaction costs

Abstract

American options are studied in a general discrete market in the presence of proportional transaction costs, modelled as bid-ask spreads. Pricing algorithms and constructions of hedging strategies, stopping times and martingale representations are presented for short (seller's) and long (buyer's) positions in an American option with an arbitrary payoff. This general approach extends the special cases considered in the literature concerned primarily with computing the prices of American puts under transaction costs by relaxing any restrictions on the form of the payoff, the magnitude of the transaction costs or the discrete market model itself. The largely unexplored case of pricing, hedging and stopping for the American option buyer under transaction costs is also covered. The pricing algorithms are computationally efficient, growing only polynomially with the number of time steps in a recombinant tree model. The stopping times realising the ask (seller's) and bid (buyer's) option prices can differ from one another. The former is generally a so-called mixed (randomised) stopping time, whereas the latter is always a pure (ordinary) stopping time.