Option Valuation

Can Black-Scholes option values be derived via the Capital Asset Pricing Model, without resort to the use of a risk-free portfolio being created from the option and a Delta determined quantity of the underlying instrument?

Answer

From Frequently Asked Questions in Quantitative Finance (2009) by Paul Wilmott, p. 416:

This derivation, originally due to Cox & Rubinstein (1985) starts from the Capital Asset Pricing Model in continuous time. In particular it uses the result that there is a linear relationship between the expected return on a financial instrument and the covariance of the asset with the market. The latter term can be thought of as compensation for taking risk.

But the asset and its option are perfectly correlated, so the compensation in excess of the risk-free rate for taking unit amount of risk must be the same for each.

For the stock, the expected return (dividing by $dt$) is $\mu$. Its risk is $\sigma$.

From Ito we have

$$dV = \frac{\partial V}{\partial t}dt + \frac{1}{2}\sigma^2S^2\frac{\partial ^2V}{\partial S^2}dt + \frac{\partial V}{\partial S}dS$$ Therefore the expected return on the option is $$\frac{1}{V}\left( \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial ^2V}{\partial S^2} + \mu S \frac{\partial V}{\partial S}\right)$$ and the risk is $$\frac{1}{V} \sigma S \frac{\partial V}{\partial S}$$ Since both the underlying and the option must have the same compensation, in excess of the risk-free rate, for unit risk $$\frac{\mu-r}{\sigma}= \frac{\frac{1}{V}\left( \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial ^2V}{\partial S^2} + \mu S \frac{\partial V}{\partial S}\right)}{\frac{1}{V} \sigma S \frac{\partial V}{\partial S}}$$ Now rearrange this. The $\mu$ drops out and we are left with the Black–Scholes equation.