Option Pricing without Black-Scholes

What does the Black-Scholes formula allow us to do?

What is the Black-Scholes formula?

What is an option payoff?

We would only exercise an option if it had a positive value at expiry, a positive payoff, irrespective of the premium that we originally paid for it.

For a call option, the payoff would be the maximum of the spot at expiry minus the strike, or zero. For a put option, it would be the maximum of the strike minus the spot at expiry, or zero. For example, if we take a strike of 500 and the expiry price of 520, the call would be exercised as it would have a payoff of 20. Whereas the put would be abandoned as the payoff would be zero, the maximum of -20 or zero.

How can we calculate the total probability weighted payoff/price of an option in the following example?

• Assume the option has a strike of 100, with one year to expiry. Let’s say that we could look into the future and know for certain that the price at expiry will be 110, the option payoff would be the maximum of the spot minus the strike (110-100 or zero) = a payoff of 10.
• The probability weighted playoff is the payoff multiplied by the probability. The payoff will be received in a year, so the call price today will be the present value of the expected payoff. Hence, we’ve used continuous rates and discounting.
• Let’s also assume that there are two possible price outcomes; 80 and 110, and arbitrarily assign a 50% probability to each outcome. Half the time the payoff is zero, and half the time the payoff is 10. Multiply each payoff by 50% and add, giving a probability weighted payoff of 5, and then discount for the call price today, which gives us the option price
• Call/Put Price = 5 x e^{-(0.1 x 1)} = 4.52