Question

Traditional call and put options are traded on a portfolio basis to match investor expectations about the underlying asset. But identifying the appropriate portfolio can be complicated and having the portfolio payoff match your expectations can be imperfect, In this example, you will illustrate to a venture capitalist from whom you seek funding for a fintech options trading platform, how you can create and price exotic options designed to more accurately match investor expectations about the underlying asset. Assume the annual continuously compounded rate is 10%, the annual standard deviation of stock returns is 0.30 (i.e., 30%), and the Time-to-Maturity is 1 year. The current stock price is $100. What is the value of an option that pays $1 if the stock price at maturity falls within the range $50 to $150, and 0 otherwise. Use the Binomial Option Pricing Model for N (the number of periods) = 7.

Answer 1

To value the exotic option, use the Binomial Option Pricing Model. Calculate the up and down factors, then the risk-neutral probabilities. Finally, calculate the option value.

Step-by-step explanation:

To value this exotic option, we can use the Binomial Option Pricing Model. In the model, we need to calculate the probability of the stock price falling within the range of $50 to $150 at maturity, and then discount the payoff to its present value.

First, we calculate the up and down factors:

Up factor = e^(σ√(T/N)) = e^(0.30√(1/7)) = 1.0664

Down factor = 1/up factor = 1/1.0664 = 0.9388

Next, we calculate the risk-neutral probabilities:

Probability of up movement = (e^(rt) - down factor)/(up factor - down factor) = (e^(0.10*1) - 0.9388)/(1.0664 - 0.9388) = 0.5595

Probability of down movement = 1 - probability of up movement = 1 - 0.5595 = 0.4405

Now, we calculate the option value:

Option value = e^(-rt) * [(probability of stock price in range) * payoff in range + (probability of stock price not in range) * payoff not in range]

Payoff in range = $1

Payoff not in range = $0

Option value = e^(-0.10*1) * [(0.5595 * $1) + (0.4405 * $0)] = 0.9332