Two stocks A and B both have a current price of $100 and are identical in every way except that the risk-neutral probability of default of A in three months is 10%,and that of B is zero.Assume a CRR-style jump-to-default model in which the volatility of both stocks is 30%.The risk-free rate is 2%.Consider the price of three-month at-the-money call options on these two stocks in a one-period jump-to-default tree model.Which of the following statements is valid?

Question

Quizplus · Accepted Answer

Let's go ahead and price the options on A and B.The calculations are as follows:
> S=100
> u=exp(0.3*sqrt(1/4))
> d=1/u
> R=exp(0.02/4)
> q=(R-d)/(u-d)
> q
[1] 0.4792161
> call_B = (S*u-100)*q/R
> call_B
[1] 7.716677
> q=(R/(1-0.10)-d)/(u-d)
> q
[1] 0.8500507
> call_A = (S*u-100)*q/R
> call_A
[1] 13.68812
We see that the call on A is worth more than the call on B even though it has more default risk.Why? In the risk-neutral world if stock A has default risk,then for its discounted expected value to be equal to $100 after default,the probability of the up outcome must be higher to compensate for the loss in the default state.Hence,the tree for A has a positive skew relative to the tree for B that leads to a higher call price for A.

Two Stocks a and B Both Have a Current Price