FX Options and Smile Risk

3

Dynamic Hedging and Volatility Trading

3.1 PRELIMINARY CONSIDERATIONS

We have shown in Chapter 2 how to build a portfolio replicating the terminal payoff of a contingent claim via a dynamic trading strategy, with the additional requirement for it to be self-financing. The effectiveness of the strategy hinges on the starting assumptions regarding the dynamics of the FX spot rate. If the assumptions are met in the real world, in the sense that they correctly describe the basic features and behaviour of the market, then it can be argued that the theoretical results hold and the profits and losses originated by the strategy exactly match those arising from a position in the contingent claim. If the assumptions do not hold (or only hold partially), then residual profits and losses result from the strategy so that the contingent claim’s replication will be inaccurate.

In this chapter we present a general framework³¹ to examine the possible sources of profits and losses when a dynamic trading strategy is implemented and the starting assumptions do not (fully) hold. We will extend the range of possible financial instruments entering into the trading strategy, by including not only the FX spot rate and the domestic deposit but other (actively) traded contingent claims (i.e., in practice, other plain vanilla options). The reason why we are interested in analysing such strategies is that market makers hedge their books not by adhering strictly to how, in theory, the replication strategy ...