# Using return on equity instead of risk free rate when pricing an equity call option

I am currently a second year university student studying business, so excuse my lack of knowledge regarding the subject.

I am currently studying the binomial options pricing model, which involves working out the risk neutral probabilities involving states: up/down. A risk neutral-measure implies that there is no arbitrage in the market.

The formula for this would be: Sd * P + Su * (1-P) = S * (1 + Rf)

If we were pricing an equity call option, for example, why wouldn't we use the return on equity derived from the capital asset pricing model for that particular stock, instead of using the risk free rate?

EDIT:

I partly understand that the principle of no arbitrage breaks down, but what assumptions may be broken if we use CAPM instead of Rf?

• You can actually do it. See "Derivation using the CAPM" here: frouah.com/finance%20notes/Black%20Scholes%20PDE.pdf Roughly speaking $P$ and $(1+r_f)$ are interchangeable. If you want to discount by $(1+r_s)$ simply multiply and divide by it, i.e. $\frac{P}{(1+r_f)} = \underbrace{\left(\frac{P(1+r_s)}{1+r_f}\right)}_{P'}\frac{1}{1+r_s} = \frac{P'}{1+r_s}$ – fni Mar 13 '18 at 15:33
• Isn't the question about the expected stock return under the risk neutral measure ? – Antoine Conze Mar 13 '18 at 16:19

The risk neutral measure is such that all assets have same expected return under that measure (hence the name), therefore equal to the risk free rate $r_f$. Existence of a risk neutral measure is equivalent (under some technical conditions) to no arbitrage opportunity. The measure is unique iif the market is complete, meaning any derivative can be hedged. In that case a derivative PV, set by a no arbitrage argument to the initial value of its hedging portfolio, is equal to the risk neutral expectation of its discounted payoff, hence the need to work out the risk neutral probabilities in the binomial model, or any model, for pricing purposes.
Also note that for the binomial model the no arbitrage condition implies $S_d \leq S(1+r_f) \leq S_u$ (otherwise you can make money with certainty starting from a zero endowment by setting up either a long position or a short position funded at the risk free rate $r_f$), which guarantees that the computed $p = \frac{S_u - S(1+r_f)}{S_u - S_d}$ is such that $0 \leq p \leq 1$, therefore qualifies as a probability measure.