Question

The current spot price of a stock is $34, the expected rate of return of the stock is 8%, and the volatility of the stock is 20%. the risk-free rate is 3%. compute the price of a derivative whose payoff in 6 months is • $8 if the stock price in 6 months, s6/12, is below $35, • $5 if 35 ≤ s6/12 ≤ 55, and • nothing otherwise.

Answer 1

To calculate the price of a derivative with given conditions, we use the risk-neutral pricing formula. This involves calculating the risk-neutral probability of the stock price being below $35 in 6 months and then using it to determine the expected payoff of the derivative. Finally, the expected payoff is discounted back to the present using the risk-free rate to find the price of the derivative.

Step-by-step explanation:

To calculate the price of a derivative with the given conditions, we need to use the concept of present discounted value (PDV). The PDV is the amount you should be willing to pay in the present for a stream of expected future payments. We will use the risk-neutral pricing formula to find the price of the derivative.

1. Calculate the risk-neutral probability of the stock price being below $35 in 6 months. This can be calculated using the Black-Scholes formula: N(d2), where d2 = (ln(S/K) + (r - q + σ^2/2) * t) / (σ * sqrt(t)), S is the spot price, K is the strike price, r is the risk-free rate, q is the continuous dividend yield (assumed to be 0), σ is the volatility, and t is the time to expiration.

2. Calculate the expected payoff of the derivative using the risk-neutral probability: expected payoff = (probability of stock price below $35) * $8 + (probability of stock price between $35 and $55) * $5.

3. Discount the expected payoff back to the present using the risk-free rate: price of the derivative = expected payoff / (1 + r)^t, where r is the risk-free rate and t is the time to expiration.