Applying the Black-Scholes option pricing formula to the value of an equity share, should r be defined as the expected increase in X0 rather than the continuously compounded risk free rate?
No, I don't believe so. r is the risk-free rate as normal in the BS model. The company growth rate (ie increase in X) is irrelevant as the model is risk-neutral.
Thanks Simon. Paul Sweeting’s book in section 14.5.3 (pg 358) applies the Black-Scholes model to estimate the probability of default at time T. E is substituted for B (value of the bond) and r (risk-free rate) for rx (the expected increase in the current value of the asset). Does this mean that this formula and the one in solution 23.12 are referring to different things? I was linking the two as the sentence preceding Question 23.12 states that the Black-Scholes formula will be applied to the value of the equity shares.
Hi. The growth rate and the volatility are connected (higher expected growth implies higher variability). Intuitively you can either consider the growth rate as the risk-free rate with a 'normal' level of volatility, or have a 'real' expected growth rate with a higher sigma reflecting the additional risk in pursuit of higher growth than the risk-free.
