At expiry, the asset-or-nothing call option pays 0 if S <= X and S if S > X. Similarly, a put option pays 0 if S >=X and S if S < X.
TheoV
Call
Put
where
b – the cost-of-carry.
b = r to price options on stocks.
b = r – q to price options on stocks and stock indexes paying a continuous dividend yield q
b = 0 to price options on futures.
b = r – rf to price currency options (rf – risk-free rate of the foreign currency).
Delta
Gamma
Vega
Theta
Rho
Implied volatility
To find implied volatility the following equations should be solved for the value of sigma:
Call
Put
System numerically solves these equations.