Why Risk-Neutral Pricing Discounts Expected Payoffs at the Risk-Free Rate
Summary
The document explains why derivative pricing in a risk-neutral framework discounts expected future payoffs at the risk-free rate. Its central idea is that changing from real-world probabilities to risk-neutral probabilities accounts for risk premia in the probability measure. Under that measure, the expected return on traded investments is the risk-free rate, so expected payoffs can be discounted at that same rate.
The answer contrasts this with real-world valuation, where risky cash flows require risk-sensitive treatment. It also addresses a conceptual question about whether the risk-free rate is simply the time value of money: the risk-neutral measure and risk-free discounting work together in pricing, rather than requiring a separate risk premium in the discount rate. The exchange is an intuitive explanation, not a formal derivation, and assumes the standard risk-neutral pricing setup.
Key ideas
- Risk-neutral pricing changes probabilities to incorporate risk adjustment.
- In the risk-neutral measure, traded assets have expected returns equal to the risk-free rate.
- Expected derivative payoffs under that measure are discounted at the risk-free rate.
- Risk premia are reflected in the probability measure rather than added to the discount rate.
Tags
Full text
# Why use the risk-free rate for discounting in a risk neutral world?
# Why use the risk-free rate for discounting in a risk neutral world?
I am reading Options, Futures, and other derivatives by John C. Hull. In the chapter on Binomial trees, he remarks:
> A risk-neutral world has two features that simplify the pricing of derivatives: The expected return on a stock (or any other investment) is the risk-free rate. The discount rate used for the expected payoff on an option (or any other instrument) is the risk-free rate.
I understood the first point. The second point seems easier and almost obvious, though when I tried to come up with a written justification for it, I was at a lack of words. I am hoping someone could justify the second point for me. Thank you!
Edit: On further thought, it seems that in a world with only two possible investments - a risky stock and a riskless bond - it is the riskless bond that will represent the time value of money, hence we should use that as the discount rate.
But this raises another question: Based on the above argument, the fact that we use the risk-free rate for discounting is not a consequence of the risk neutral world, but this is not what Hull suggests.
## Answer by Kevin (score 4, accepted)
https://quant.stackexchange.com/a/46759
The first statement is kind of clear. If all investors are risk-neutral, they simply do not care about risk and do not pay more or less regardless how risky an asset is. As a consequence, the return of all assets is the risk-free rate.
Regarding the second statement. What does risk-neutral pricing really do? We change the probabilities from the real world, $\mathbb{P}$, to risk-adjusted ones, $\mathbb{Q}$ which capture all this risk. In a way, we take the risk out of the equation. However, as we only need ``real'' discount factors for stochastic, risky cashflows, we can simply discount with the risk-free rate since the risk-adjustment are done in $\mathbb{Q}$. So whilst there is a risk premium in the real worl and hence the need to use an appriopriate discount factor, there is no risk premium in the risk-neutral world - since no one cares about risk in this world. Thus, we may simply discount with the risk-free rate.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.