Question

Suppose you are in a simple, one-period economy which has two assets. A risk-free asset that returns 5% and a risky asset that has a current price of $12. At the end of the period, the risky asset can be worth either $15 or $10.80. You are considering a derivative that has a payoff of $2.40 when the price of the risky asset increases and a payoff of 0 when the price of the risky asset goes down. Find the value of this derivative?

Answer 1

The value of the derivative in question, given a risk-free rate of 5%, is approximately $1.14 when calculated using the expected payoff method, which takes into account the risk-neutral probabilities of the derivative's possible payoffs and discounts them at the risk-free rate.

Step-by-step explanation:

The student is asked to find the value of a derivative in a simple, one-period economy with two assets, a risk-free asset and a risky asset. To find the value of the derivative, which pays off differently based on the movement of the risky asset's price, we can consider the possible outcomes and use the risk-free rate to discount the expected payoff.

The risky asset increases to $15: the derivative pays $2.40.

The risky asset drops to $10.80: the derivative pays $0.

Since the risk-free rate is 5%, the expected payoff of the derivative is the sum of the probabilities of each state times their respective payoffs, discounted back at the risk-free rate.

We can calculate the expected payoff as follows:

• Probability of increase: 0.5 (assuming a risk-neutral world)
• Probability of decrease: 0.5
• Expected Payoff = (0.5 * $2.40) + (0.5 * $0) = $1.20
• Value of derivative = $1.20 / (1 + 0.05) = $1.142857

The value of the derivative is approximately $1.14.