Risk-Neutral Pricing and Equal Expected Returns in a One-Period Model
Summary
The document examines a one-period example in which a risky security has two possible returns, while a riskless asset offers the same expected return under the stated real-world probabilities. It asks whether risk-neutral valuation should make the risky security’s expected return lower, and whether the example is suitable for explaining why risk-neutral probabilities are used.
Applying the standard one-period up/down model to the supplied risk-free rate and security payoffs gives risk-neutral probabilities equal to the example’s stated probabilities. The risky asset therefore has the same expected return as the bond under both measures in this setup. This does not establish that the investments have the same risk or payoff distribution; risk-neutral probabilities are a pricing device chosen to align discounted asset prices with no-arbitrage valuation, not a general measure of investor risk preference. The post raises the conceptual distinction but does not include a full response or broader discussion of preferences, state prices, or model assumptions.
Key ideas
- Equal expected returns under the real-world measure can occur when the example’s probabilities already match risk-neutral probabilities.
- Risk-neutral probabilities are used to price payoffs consistently with no-arbitrage and the risk-free discount rate.
- Matching expected returns do not make a risky payoff distribution equivalent to a riskless bond.
- The example highlights a teaching limitation but does not develop a general account of risk-neutral valuation.
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Full text
# Risk neutral pricing - Example from a book is correct?
# Risk neutral pricing - Example from a book is correct?
I found the following example in a book on Model Risk, while trying to explain how risk-neutral pricing takes properly into account the risk involved in different investments. The Example is this.
Suppose you have a riskless asset ( a bond, say) promising you a riskless return of 50%. So, take a risk free interest rate $r=0.5$. Then, you have a security, which, under the actual probability measure can go up with probability $p=0.5$ and make a return of $125\%$, or can go down (with probability $q=0.5$) yielding a return of $-25\%$.
Under the actual probability the expected return of the security is then equal to $\frac{1}{2}125\%-\frac{1}{2}25\%=50\%$. On the other hand, this is the same return promised by the bond, so the two investments produce the same expected return (using the actual probability measure, that is).
The book goes on to say that, however, the two investments are not the same because in the security's case it is present a risk that is not present in the bond case and, unless you are a special investor, you will prefer the riskless bond yielding the same expected return. That is the reason why, the book adds, you need to introduce a different measure than the actual one, the risk-neutral measure, that takes into account the presence of risk in the investments, and you shall take the expectation of the return of the risky asset with respect to this measure, and not the actual measure, in order to compare this investment to the riskless investment.
Now all this makes perfect sense to me, however, if I understand well, the book implies that the two investments are not the same because in the risk neutral measure the expected return of the risky asset shall be less than the $50\%$ arising from the actual probability.
However, after some calculation, I realised that this is not the case: it appears that the risk-neutral probabilities $\tilde{p}, \tilde{q}$ in this case are the same as the actual probabilities postulated by the example, that is: $\tilde{p}=\tilde{q}=\frac{1}{2}$, as it is easy to see by letting $r=0.5$, $d=0.75$ and $u=2.25$, and applying the well-known equations for the one-period model.
So, in fact one obtains that the risky asset has the same return as the bond (i.e., $50\%$) also under the risk-neutral probability. Therefore, I have two questions:
(1) Am I right to claim that, from the risk-neutral perspective, the two investments are actually equivalent (contrary to what the book seems to claim)? Or am I missing something?
(2) Is the above example ill-chosen to illustrate the motivation behind risk-neutrality?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.