An option gives its buyer the right to buy (call option) or sell (put option) something in the future to the option seller at a predetermined price (exercise price). For example, if we buy a European call option to acquire a stock for X dollars, such as $30, at the end of three months, our payoff on maturity day will be the one calculated using the following formula:

Here, is the stock price at the maturity date (T), and the exercise price is X (X is equal to 30 in this case). Assume that three months later the stock price will be $25. We would not exercise our call option to pay $30 in exchange for the stock, since we could buy the same stock with $25 in the open market. On the other hand, if the stock price is $40, we will exercise our right to reap a payoff of $10, that is, buy the stock at $30 and sell it at $40. The following program presents the payoff function for a call:

>>>def payoff_call(sT,x):
      return...