European type Currency Binary Option Pricing with 3 methods and implied smile

• (calls) PHLX (Philadelphia Stock Exchange (PHLX)) bid ask quotes for March XDE calls and puts on 1/22/08.
• (puts) PHLX bid ask quotes for March XDE calls and puts on 1/22/08.
• (euro) - Daily $/€ 1/03/2000 – 1/22/2008
• (libor) - LIBOR rates EUR and USD. (22-Jan)

The contract is a European call option, written on currency (dollar/euro FX), which has a payoff similar to a Heaviside step function, H(x). Pricing date: 1/22/08
Underlying (1/22/208 contemporaneous with option quotes): S0 = $/€ = 145.88
Derivatives: European, Expiration date = 3/21/08, K = Strike price = 146

• Analytical solution
• Binomial
• Monte Carlo
• Deriving implied volatility from the option prices

1. Pricing formula:

Source: [Tomas Bjork] Arbitrage Theory in Continuous Time, Chpater 17.1