A firm has a single issue of a zero coupon debt that promises to pay $90 in 4 years,and the A₀ = $100,r = 5%,σ = 15%,and δ = 0.If the asset has no chance of total default,what is the value of the debt?

Question

Quizplus · Accepted Answer

Using the Black-Scholes formula for pricing a zero coupon bond:
V₀ = N(-d₂) × F
where N() is the standard normal cumulative distribution function, F is the face value of the debt (in this case $90), and d₂ is the term in the Black-Scholes formula:
d₂ = (ln(A₀/F) + (r - δ + 0.5σ²)t) / (σ√t)
Substituting the given values in the formula, we get:
d₂ = (ln(100/90) + (0.05 - 0 + 0.5×0.15²×4)) / (0.15√4) = 0.8192
Using a standard normal distribution table, N(-0.8192) = 0.2059
Therefore, V₀ = 0.2059 × 90 = $75.19, which is option B.

A Firm Has a Single Issue of a Zero Coupon