FX smile extrapolation

Question

Typically, 5 points data is available for smile construction : 25D RR, 25D SM, 10D RR, 10D SM and ATM. Questions: 1. How is a smooth smile curve generated with the help of these? 2. How is extrapolation done beyond 10D points?

Answer 1

This is a big industry, but here are some alternatives(as usual, the best choice depends on purpose and desired accuracy):

1. Fit a quadratic in delta space: $\sigma_{\Delta}=a + b \left( \Delta - \Delta_{ATM} \right) + c \left( \Delta - \Delta_{ATM} \right)^2$. When you have fitted this equation, you can input delta, and the function will return volatility. This is known as Malz quadratic approach, Malz actually solved this algebraically: you have three unknowns and you can use the ATM, RR (25 Delta), and SS(25 Delta) quotes. In the end, you will get the following expression (you can see the detailed steps here):

   $\sigma_{\Delta}=\sigma_{ATM} -2 RR_{25 Delta} \left( \Delta - \Delta_{ATM} \right) + 16 SS_{25 Delta} \left( \Delta - \Delta_{ATM} \right)^2$

2. You can also try other forms of interpolation - e.g., polynomial, cubic spline etc, but one needs to check that this does not introduce arbitrage.

3. You will find that a number of smaller firms use the Vanna Volga approach. You take the Back Scholes price, and add to it the cost of Vega-Vanna-Volga hedge - under the presumption that Black Scholes price only reflects cost of delta hedging. This gives the price of an arbitrary strike, which you can then invert to get the volatility. As this is going to be computationally intensive, one can use Taylor series expansion (first order or second order), which then simplifies, and you get an expression for volatility in terms of the market quotes (volatility, RR and SS) and other inputs (Black Scholes price inputs). The derivation is simple conceptually but the algebraic formulae are long. Again you can find the detailed derivation here.

4. For pricing purposes etc, these quotes are used to calibrate local/stochastic volatility models, with sophistication depending on the instruments/products being priced.

Re-extrapolation, anything beyond 10D is going to be quite dangerous as there is not enough liquidity in the extreme.

Hope this helps!