Basic notation

The following symbols are used throughout:

         S₀ = stock price today (time 0 = today)

         X = exercise price
         T = time to expiration as defined below
         r = risk-free rate as defined below
         S_T = stock price at expiration
         C(S₀, T, X) = price of call option in which the stock price is S_0, the time to expiration is T, and the exercise price is X.
         P(S₀, T, X) = price of put option in which the stock price is S₀, the time to expiration is T, and the exercise price is X.

In some cases, we may need to distinguish an American call from a European call. If so the call price will be denoted with either an 'a' subscript or 'e' subscript for American and European call respectively. For most of the examples we shall assume that the stock pays no dividends. If, during the life of the option, the stock pays dividends Dl, D2, ...etc., then we can make a simple adjustment and obtain similar results. To do so, we simply subtract the present value of the dividends

     

where there are N dividends and tj is the time to each ex-dividend day from the stock price. We assume the dividends are known ahead of time. The time to expiration is expressed as a decimal fraction of a year. The risk-free rate, r, is the rate earned on a riskless investment. The rate of return on a Treasury bill of comparable maturity would be a good proxy for the risk-free rate.