By the symmetry of the standard normal distribution N(−d) = (1−N(d)) so the formula for the put option is usually written as p(0) = e−rT KN(−d2) − S(0)N(−d1). Rewrite the Black-Scholes formula as c(0) = e−rT (S(0)erT N(d1) − KN(d2)). The formula can be interpreted as follows.
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The payoff ratio, also known as the profit factor is a metric that compares the average profit of winning trades to the average loss of losing trades. It helps traders assess the performance of their trading strategies and the potential profitability of their trades.
The payoff at time T from a European call option is (S(T)−K)+ and from a European put option is (K −S(T))+. In the case of American options, the payoff takes place at the moment of exercise t, where t ≤ T and we set t = T if the option is not exercised.
Combining the above two states of the world, we get the following expression for the long-put-option payoff: VP (T) = max(K − S(T),0) = (K − S(T))+. So, the payoff function for a put option is vP (s)=(K − s)+.
The payoff function is a function u i : S 1 × S 2 × ⋯ S m → R .