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We present a new put 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 put option. Should the option holder believe the true drift of the stock price to be unfavourable (based upon the observed price movements) he can substitute the true drift with the contract drift and minimize his losses. The practical implications of this protection feature are most remarkable 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 favourable he will generally receive higher returns at a lesser price. We derive a closed form expression for ...

Understanding the behaviour of the American put option is one of the classic problems in mathematical finance. Considerable efforts have been made to understand the asymptotic expansion of the optimal early exercise boundary for small time near expiry. Here we focus on the large-time expansion of the boundary. Based on a recent development of the convexity property, we are able to establish two integral identities pertaining to the boundary, from which the upper bound of its large-time expansion is derived. The bound includes parameter dependence in the exponential decay to its limiting value. In addition, these time explicit identities provide very efficient numerical approximations to the true solution to the problem.

We apply a singular perturbation analysis to some option pricing models. To illustrate the technique we first consider the European put option under the standard Black–Scholes model, with or without barriers. Then we consider the same option under the constant elasticity of variance (CEV) assumption, which is also called the square root process. In the CEV model the variability effects in the evolution of the asset, on which the option is based, are proportional to the square root of the asset value. We also consider the CEV model with barriers, and this leads to a rich asymptotic structure. The analysis assumes that the variability is small and employs the ray method of geometrical optics and matched asymptotic expansions.

The optimal exercise boundary near the expiration time is determined for an American put option. It is obtained by using Green's theorem to convert the boundary value problem for the price of the option into an integral equation for the optimal exercise boundary. This integral equation is solved asymptotically for small values of the time to expiration. The leading term in the asymptotic solution is the result of Barles et al. An asymptotic solution for the option price is obtained also.