We use an asymptotic expansion to study the behavior of shout options close to expiry. Series solutions are obtained for the location of the free boundary and the price of the option in that limit. 1. Introduction Since the seminal work of Black and Scholes [1] and Merton [2] on the pricing of options appeared forty years ago, there has been a dramatic growth in both the role and complexity of financial contracts. The world’s first organized options exchange, the Chicago Board of Options Exchange (CBOE), opened in 1973, the same year as [1, 2] appeared in print, and trading volumes for the standard options traded on exchanges such as the CBOE exploded in the late 1970’s and early 1980’s. Around the same time as the growth in standard options, financial institutions began to look for alternative forms of options, termed exotic options, both to meet their needs in terms of reallocating risk and also to increase their business. These exotics, which are usually traded over-the-counter (OTC), became very popular in the late 1980’s and early 1990’s, with their users including big corporations, financial institutions, fund managers, and private bankers. One such exotic, which is the topic of the current study, is a shout option [3, 4]. This option has the feature that it allows an investor to receive a portion of the pay-off prior to expiry while still retaining the right to profit from further upsides. In order to use this feature, the investor must shout, meaning exercise the option, at a time of his choosing, and this leads to an optimization problem wherein the investor must decide the best time at which to shout, which in turn leads to a free boundary problem, with the free boundary dividing the region where it is optimal to shout from that where it is not. In practice, shouting should of course only take place on the free boundary. This sort of free boundary problem is of course common in the pricing of options with American-style early option features, and this aspect of vanilla American options has been studied extensively in, for example, the recent studies of [5–13], although American-style exotics have received somewhat less attention. In the present study, we will use a technique developed by Tao [14–22] for the free boundary problems arising in melting and solidification; such problems are termed Stefan problems. Tao used a series expansion in time to find the location of the moving surface of separation between two phases of a material, and, in almost all of the cases he studied, he found that the location of the interface was proportional to ,
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