Whenever the stock price movement is only known up to time t and the price at time *t* is *S*_{t}, the Black – Scholes formula gives that:*S*_{T} > *K*, where *K* is the strike price. This means:

(A.1)

provided it is a non-dividend paying stock. In the above formula y is normally distributed with mean 0 and variance 1. Now it is easy to calculate the probability that an option expires in the money. Take a call option. A call option expires in the money when Since *y* has a standard normal distribution:

(A.2)

In the same way one can derive:

(A.3)

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