Consistent Pricing of FX Options

Abstract

In the current markets, options with different strikes or maturities are usually priced with different implied volatilities. This stylized fact, which is commonly referred to asfsmile effect, can be accommodated by resorting to specific models, either for pricing exotic derivatives or for inferring implied volatilities for non quoted strikes or maturities. The former task is typically achieved by introducing alternative dynamics for the underlying asset price, whereas the latter is often tackled by means of statical adjustments or interpolations. In this article, we deal with this latter issue and analyze a possible solution in a foreign exchange (FX) option market. In such a market, in fact, there are only three active quotes for each market maturity (the 0Delta straddle, the risk reversal and the vega-weighted butterfly), thus presenting us with the problem of a consistent determination of the other implied volatilities. FX brokers and market makers typically address this issue by using an empirical procedure to construct the whole smile for a given maturity. Volatility quotes are then provided in terms of the option's Delta, for ranges from the 5Delta put to the 5Delta call. In the following, we will review this market procedure for a given currency. In particular, we will derive closed-form formulas so as to render its construction more explicit. We will then test the robustness (in a static sense) of the resulting smile, in that changing consistently the three initial pairs of strike and volatility produces eventually the same implied volatility curve. We will also show that the same procedure applied to Europeanstyle claims is consistent with static-replication results and consider, as an example, the practical case of a quanto European option. We will finally prove that the market procedure can also be justified in dynamical terms, by defining a hedging strategy that is locally replicating and self-financing.