# Is it possible to calculate the equity required (or expected) return using Black-Scholes option pricing model?

I know the method of calculating the equity value as a European call option (using Black-scholes formula). My question is: Is it possible to calculate the expected (or required) return of equity when we assume the equity to be a call option on the firms assets?It should be mentioned that no call or put option has been offered in the market.

• With equity value you mean Merton's firm value model? What is the expected/required return on equity in this context? Do you mean the drift term in the SDE used in the model? If so, then you could clarify this, maybe even write some formulas? – Richard Jan 31 at 7:39

With $$E=C(V,D)$$, $$V$$ being firm value, and $$D$$ being face value of debt, and $$r_{ROE}$$ being continuously compounded Return on Equity, the following should hold
$$E*e^{r_{ROE}T}=C(V,D)*e^{r_{ROE}T}=V$$, and therefore:
$$r_{ROE}=\frac{1}{T}ln(\frac{V}{C(V,D)})$$
For the sake of making a numerical example, I use $$V=100$$, $$D=80$$, $$\sigma=20\%$$ (determined via index or peer firms with outstanding options), $$T=10$$, a risk-free rate of $$r=1\%$$, and get $$C(V,D)=30.45$$. $$r_{ROE}=\frac{1}{10}ln(\frac{100}{30.45})=11.9\%$$