The British call option

Alongside the British put option (Peskir and Samee [ Appl. Math. Finance , 2011, 18 , 537--563]) we present a new call option where the holder enjoys the early exercise feature of American options whereupon his payoff (deliverable immediately) is the ‘best prediction’ of the European payoff under the hypothesis that the true drift of the stock price equals a contract drift. Inherent in this is a protection feature which is key to the British call option. Should the option holder believe the true drift of the stock price to be unfavourable (based upon observed price movements) he can substitute the true drift with the contract drift and minimise his losses. The practical implications of this protection feature are most remarkable as not only can the option holder exercise at or below 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 favourable 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 characterised as the unique solution to a nonlinear integral equation. In addition we derive the ‘British put--call symmetry’ relations which express the arbitrage-free price and the rational exercise boundary of the British call option in terms of the arbitrage-free price and the rational exercise boundary of the British put option where the roles of the contract drift and the interest rate have been swapped. These relations provide a useful insight into the British payoff mechanism that is of both theoretical and practical interest. Using these results we perform a financial analysis of the British call option that leads to the conclusions above and shows that with the contract drift properly selected the British call option becomes a very attractive alternative to the classic European/American call.