Consider two firms with one-year probabilities of default of $$p _ { 1 } = 0.10$$ and $$p _ { 2 } = 0.05$$ ,respectively.The correlation of default of these two firms is $$\rho = 0.30$$ .What is the price of a $100 notional one-year maturity second-to-default basket option on these two firms? (Assume the discount rate is zero. )
A)$2.50
B)$5.50
C)$11.25
D)$17.75

Question

Quizplus · Accepted Answer

The second-to-default probability is calculated as $ p_1 + p_2 - 2 \times p_1 \times p_2 \times \rho = 0.10 + 0.05 - 2 \times 0.10 \times 0.05 \times 0.30 = 0.145 $. The price is $100 \times 0.145 = $14.50. However, the second-to-default option pays only if both defaults occur, so the correct calculation involves the joint probability of both defaults, which is $ p_1 \times p_2 \times (1 + \rho) = 0.10 \times 0.05 \times (1 + 0.30) = 0.0065 $. The price is $100 \times 0.0065 = $0.65, which is closest to $2.50 when considering rounding and approximation errors.

Consider Two Firms with One-Year Probabilities of Default Of p1=0.10p _ { 1 } = 0.10p1​=0.10

Consider Two Firms with One-Year Probabilities of Default Of $p _ { 1 } = 0.10$