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Why Risk-Neutral Option Pricing Uses the Risk-Free Rate

Article Quant Q&A · Author: Gleb Pilipenko
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Summary

In a binomial model, the risk-neutral probability is chosen so that the stock’s expected return under that pricing measure equals the risk-free rate. The document explains why a stock’s CAPM expected return is not substituted: option valuation uses no-arbitrage pricing under a risk-neutral measure, rather than forecasting the stock’s real-world expected return.

For a one-period binomial model, no arbitrage requires the risk-free grown stock price to lie between the down and up prices. This condition makes the implied risk-neutral probability fall between zero and one. Under technical conditions, a risk-neutral measure exists when there is no arbitrage; it is unique when the market is complete, meaning derivatives can be hedged. The explanation assumes this framework and does not address incomplete-market pricing choices or practical frictions.

Key ideas

  • Risk-neutral expected returns for assets equal the risk-free rate.
  • CAPM expected return is a real-world return estimate, not the rate used for risk-neutral valuation.
  • No arbitrage requires the risk-free grown price to fall between the binomial up and down prices.
  • That no-arbitrage condition makes the implied probability a valid probability.
  • A complete market gives a unique risk-neutral measure and derivative price under the stated assumptions.

Tags

OptionsDerivatives pricingArbitrage
Full text
# Using return on equity instead of risk free rate when pricing an equity call option


# 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?

## Answer by Antoine Conze (score 1)

https://quant.stackexchange.com/a/38754

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.