Question

Suppose that, in each period, the cost of a security either goes up by a factor of u = 2 or down by a factor d = 1/2. Assume the initial price of the security is $100 and that the interest rate r is 0.

Compute the risk neutral probabilities p (price moves up) and q = 1 − p (price moves down) for this model.

Answer 1

Risk neutral probabilities p and q for the given financial model are 2/3 and 1/3, respectively, reflecting the expected return equaling the risk-free rate of zero.

Step-by-step explanation:

The student's question pertains to the computation of risk neutral probabilities in a simplified financial model where a security's price can only go up by a factor of 2 (u=2) or down by a factor of 1/2 (d=1/2), starting at an initial price of $100 and with an interest rate of 0 (r=0).

Considering the interest rate is zero, the expected return of the security should be equal to its initial price to avoid arbitrage.

In this context, the risk-neutral probability p that the price will go up satisfies the equation p*u + (1-p)*d = 1.

Solving for p gives p=2/3 (the security is more likely to go up) and therefore q=1-p=1/3 (the probability that the price will go down).

These probabilities reflect a risk-neutral world where the expected return of the security is the same as the risk-free rate, which is zero in this case.