### FX Options summary

FX options are not any different from Stock options in so far as the Black Scholes model is concerned. One just needs to replace the current Stock price with the Spot FX rate, say GBPUSD=1.43, and the dividend yield with the yield/discount rate of the foreign currency (GBP in our GBPUSD example). The discount rate in the Black Scholes formula would be the domestic currency rate (USD interest rate in our GBPUSD example), and the volatility would be the implied volatility of the exchange rate of the currency pair. The Black Scholes Call option price would then represent the price of an FX option, paying 1 Unit of GBP for K units of USD (K being the strike rate), in USD. In other words, GBPUSD FX rate, which represents the price of 1 unit of GBP in terms of USD, should not be treated any different from a stock, which represents the price of 1 unit of, say Amazon stock, in terms of USD.

What makes FX options different though is that there is no natural numeraire. We are happy to talk about the price of a Stock in some currency, which again is true of FX rate, but here both sides of the exchange are currencies. And that leads to different quoting conventions: price, a) in which currency?, b) of an option paying one unit of which currency?. Similarly, delta of a stock option is naturally interpreted as the number of units of stocks (or fraction) that one needs to buy to hedge a short call option position, but in FX world, buying units of foreign currency is the same as selling units of domestic currency, and that leads to a variety of delta conventions.

To avoid repetition, we only highlight FX specific features in the FX section. The behaviour of FX option price as a function of an input, for example, is the same as that of a Stock option, and we do not repeat the analysis here, focusing instead on the incremental terminology/conventions used in the FX markets.

We also cover FX specific volatility terminology such as Risk Reversal and Strangles. The coverage of these conventions will take us a bit beyond the Black Scholes assumptions (of constant volatility), which might create a little bit of inconsistency against our coverage of stock options, as we have not explicitly covered the equivalent conventions, smile conventions, for stocks.

Before we delve into the details, we list the relevant articles and books for references and further reading:

- Castagna, A. (2010). FX options and smile risk. Chichester: John Wiley and Sons.
- Castagna, A. and Mercurio, F. (2006). Consistent Pricing of FX Options. SSRN Electronic Journal.
- Clark, I. (2013). Foreign exchange option pricing. Hoboken, N.J.: Wiley.
- Malz, A. (1997). Estimating the Probability Distribution of the Future Exchange Rate from Option Prices. The Journal of Derivatives, 5(2), pp.18-36.
- Reiswich, D. and Wystup, U. (2010). A Guide to FX Options Quoting Conventions. The Journal of Derivatives, 18(2), pp.58-68.

Ⅴ) gives a good summary of price and delta conventions, Ⅱ) contains details of the Vanna Volga method, Ⅳ) contains the Malz's quadratic smile, and Ⅰ and Ⅲ are the most popular FX options books.