# Foreign currency options: deltas, market conventions and volatility smiles

**Practitioner Lecture Series **

Foreign currency as an asset class is noted by the presence of not one but two possible numeraires: the so-called domestic and foreign currencies. Since market participants can regard the risk free money market account in either of these currencies as their natural numeraire, the risk-neutral option pricing technique can proceed in one of two ways: either we price options on the foreign currency from the domestic investor’s point of view, or we price options on the domestic currency from the foreign investor’s point of view.

In this talk we show using the standard Black-Scholes model that the two approaches are mathematically equivalent in price terms, which is reassuring. It means that investors in both domestic and foreign economies will naturally agree on the model price. However, unless volatilities are negligible they will absolutely not agree on the risk numbers, in particular the delta (which can be either spot delta or forward delta). This is of particular importance in foreign currency options, which naturally have volatility surfaces described using strangles and risk reversals expressed in terms of deltas -- most commonly 25- and 10-delta. It turns out that the discrepancy is directly related to which currency the premium for the option is paid in, a cashflow which will naturally be regarded as risky for one investor but riskless for the other.

We conclude by giving an overview of the way in which FX volatility surfaces are constructed, taking into account the ATM backbone, single-vol broker strangles and risk reversals – where we see the surprising feature that a convex smile can actually have a negative broker strangle if the skew is large enough. Such considerations are not only an interesting exploration of the symmetries implicit in foreign currency option pricing but have direct practical relevance for the practitioner, which need to be understood and handled correctly when attempting to handle volatility surfaces correctly in pricing and risk managing an options book.

Location: | |

Speaker(s): | Iain Clark (Unicredit) |

Date: | Monday, January 16, 2012 - 18:00 |