Deriving Black–Scholes from CAPM and Risk Compensation
Summary
The document outlines a derivation of the Black–Scholes equation from the Capital Asset Pricing Model rather than from a risk-free portfolio formed by delta hedging. It starts from CAPM’s relation between expected return and covariance-based risk compensation. Under the stated assumptions, the underlying asset and its option have the same risk compensation per unit of risk because their returns are perfectly correlated.
Applying Ito’s lemma to the option value gives its expected return and volatility exposure in terms of the option’s time and price sensitivities. Equating its excess return per unit of risk with the stock’s corresponding quantity eliminates the stock’s expected return and yields the Black–Scholes equation. The document cites a textbook explanation and points to another derivation, but gives no worked example or detailed assumptions. The argument relies on the model’s continuous-time framework and perfect correlation between the option and underlying.
Key ideas
- CAPM links an asset’s expected return to its exposure to market risk.
- Ito’s lemma expresses the option’s expected return and risk using its sensitivities to time and the underlying price.
- The derivation equates excess return per unit of risk for the option and underlying.
- Under the stated assumptions, this equality removes the underlying’s expected return and yields the Black–Scholes equation.
- The argument assumes a continuous-time model and perfect correlation between option and underlying returns.
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Full text
# Option Valuation
# 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 by vonjd (score 6, accepted)
https://quant.stackexchange.com/a/38303
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.
## Answer by Daneel Olivaw (score 3)
https://quant.stackexchange.com/a/38300
Yes, see page 16 from the below paper:
> "Four Derivations of the Black-Scholes Formula" (F. Rouah)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.