THE BRITISH ASSET-OR-NOTHING PUT OPTION

Abstract

Following the economic rationale of Peskir & Samee [The British put option, Applied Mathematical Finance 18 (6), 537–563 (2011); The British call option, Quantitative Finance 13 (1), 95–109 (2013)], we present a new class of asset-or-nothing put option where the holder enjoys the early exercise feature of American asset-or-nothing put option whereupon his payoff is the ‘best prediction’ of the European asset-or-nothing put option payoff under the hypothesis that the true drift equals a contract drift. Based on the observed price movements, the option holder finds that if the true drift of the stock price is unfavorable, then he can substitute it with the contract drift and minimize his losses. The key to the British asset-or-nothing put option is the protection feature as not only can the option holder exercise at or above the strike price to a substantial reimbursement of the original option price (covering the ability to sell in a liquid option market completely endogenously) but also when the stock price movements are favorable he will generally receive high returns. We derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterized as the unique solution to a nonlinear integral equation. We also analyze the financial meaning of the British asset-or-nothing put option using the results above and show that with the contract drift properly selected, the British asset-or-nothing put option becomes a very attractive alternative to the classic European/American asset-or-nothing put option.