Laocoön
10-20-2001, 11:08 AM
Consider a company with stock trading at $100 per share, with no debt, and with implied volatility of 20% annually. Black-Scholes prices for one-year European calls are as follows, based on exercise price, and assuming 0% interest rate:
50 50.00
70 30.25
90 13.59
110 4.29
130 1.01
Now suppose that the same company has debt such that $50 per share of stock must be repaid in one year, but otherwise the same assets and the same number of shares of stock (now worth $50.00 a share). If the value of debt plus equity has a lognormal future distribution (as implied by the prior, debt-free example), then one-year European call prices must be as follows (with Black-Scholes implied volatility, ignoring the capital structure, given last):
20 30.25 52.0%
40 13.59 42.6%
60 4.29 38.4%
80 1.01 35.9%
50 50.00
70 30.25
90 13.59
110 4.29
130 1.01
Now suppose that the same company has debt such that $50 per share of stock must be repaid in one year, but otherwise the same assets and the same number of shares of stock (now worth $50.00 a share). If the value of debt plus equity has a lognormal future distribution (as implied by the prior, debt-free example), then one-year European call prices must be as follows (with Black-Scholes implied volatility, ignoring the capital structure, given last):
20 30.25 52.0%
40 13.59 42.6%
60 4.29 38.4%
80 1.01 35.9%