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1. Introduction

The purpose of the present paper is to introduce and examine the British payo® mechanism (see [10] and [11]) in the context of lookback options. There are various types of lookback options that may be considered to this end: (i) calls and puts; (ii) those with ˉxed or °oating strike; (iii) those based on the maximum or minimum; (iv) the weighting in the maximum or minimum may be equal or °exible (for more details on each of these types see Section 6 below).This leads to an extensive programme of research that we open in this paper by focusing on a pure maximum lookback option with no strike that plays a canonical role among all other possibilities and has the advantage of (relative) tractability. For the reasons outlined in theˉnal paragraph below we refer to this option as the British Russian option.Following the economic rationale of [10] and [11] we thus introduce a new class of look-back options which endogenously provide their holder with a protection mechanism against unfavourable stock price movements. This mechanism is intrinsically built into the option con- tract using the concept of optimal prediction (see e.g. [2] and the References therein) and we refer to such contracts as `British' for the reasons outlined in [10] and [11] where the British put and call options were introduced. Similarly to [10] and [11] most remarkable about the British Russian option is not only that it provides a unique protection against unfavourable stock price movements (endogenously covering the ability of a European/American Russian holder to sell his contract in a liquid option market) but also when the stock price movements are favourable it enables its holder to obtain high returns. This rea±rms the fact noted in [10] and [11] that the British feature of optimal prediction acts as a powerful tool for generating ˉnancial instruments (in the context of lookback options as well) which aim at both providing protection against unfavourable price movements as well as securing high returns when these movements are favourable. We recall that these combined features appear to be especially appealing as they address problems of liquidity and return completely endogenously (reducing the need for exogenous regulation).
The rest of the paper is organised as follows. In Section 2 we present a basic motivation for the British Russian option. In Section 3 we formally deˉne the British Russian option and present some of its basic properties. The most surprising discovery in this context is that the rational exercise boundary turns out to be a monotone function of time despite the fact that the payo® function itself is rather complicated. This is continued in Section 4 where we derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary (the early-exercise premium representation) and show that the rational exercise boundary itself can be characterised as the unique solution to a nonlinear integral equation (Theorem 1). Using these results in Section 5 we present a ˉnancial analysis of the British Russian option (making comparisons with the European/American Russian options). This analysis provides more detail/insight into the full scope of the conclusions brie°y outlined above. We conclude in Section 6 by presenting the other British lookback options addressed in (i)-(iv) above and providing an agenda for future research.

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