I am valuing a binary FX option (european) with a defined strike and term (2Y). I'm using a closed form solution based on Black-Scholes framework. How can I derive the appropriate volatility to use from the market data I have?
Market Data (all quoted in implied volatility):
ATM
25D Risk Reversal
25D Butterfly
10D Risk Reversal
Binary options can be replicated (in theory) by trading long and short call options with very close strikes. Take the Black-Scholes formula and differentiate it over the strike. You will need to know the slope of the implied volatility skew around the strike of the binary option. This you can do by fitting a parametric formula (I don't know exactly what is used in FX, also SABR?) to your market data. If your option's strike is not too far away from ATM, you should get a reasonable number.
Disclaimer: I don't specialise in FX.
It might be easier to use the Black-Scholes formula for binary options:
http://en.wikipedia.org/wiki/Binary_option#Black-Scholes_Valuation
then add the distributions for each leg:
Heuristics for calculating theoretical probabilities of being ITM at time T for listed options
and then use numerical methods to calculate what volatility makes the legs match the quotes prices.
