A stock has a current price of $100. Assume a CRR-style jump-to-default model in which the volatility is 30%. Let the risk-neutral probability of default in three months be 10%. The 3-month risk-free rate is 2% in continuously-componded and annualized terms. What is the price of a three-month at-the-money put option on this stock in a one-period jump-to-default tree model?
A) $7.22
B) 7.72
C) $11.82
D) $13.42

Question

Quizplus · Accepted Answer

To solve for the price of the put option, we need to construct a one-period jump-to-default tree. 
At time t=3 months, the stock price can either go up or down. Assuming the up probability is p and the down probability is 1-p, we can calculate p from the risk-neutral probability of default as follows: 
(1-p) * (1+0.3*sqrt(0.25)) + p * (1-0.3*sqrt(0.25)) = 1-0.1 = 0.9 
Solving for p, we get p = 0.4273. 
At the end of three months, the stock price can either be $139.29 (up-up) or $84.86 (down-down) with probabilities p^2 and (1-p)^2, respectively. The option payoff at expiration is the maximum of the strike price and the stock price minus the strike price. At-the-money means the strike price is $100. 
Thus, the option values at the final nodes of the tree are: 
$39.29 for the up-up node 
$0 for the down-up and up-down nodes 
$15.14 for the down-down node 
We can now work backwards from the final nodes to calculate the option values at the middle node (i.e., the current stock price of $100). Using risk-neutral valuation, the option value at the middle node is: 
[P(up) * $39.29 + P(down) * $15.14]/(1+r) = [0.4273^2*$39.29 + (1-0.4273)^2*$15.14]/(1+0.02*0.25) = $11.82 
Therefore, the price of the three-month at-the-money put option is $11.82. 
The correct answer is thus C.

A Stock Has a Current Price of $100