# 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?

• What is that book? – Robert Jul 29 '16 at 11:55
• Morini, Understanding and Managing Model Risk. Example is on p. 24. So what is the answer in your opinion? – RandomGuy Jul 29 '16 at 13:25
• I don't think you could use a model for derivatives as a general model. The author is saying that a "normal" investor will adjust the discount rate by risk, so the security will have a discount rate greater than 50% (rf + risk premium) and then, considering positive risk premiums, less value in t=0. – Robert Jul 29 '16 at 14:07
• I disagree. The author clearly sates this example in the context of motivating the concept of risk-neutral measure as opposed to actual market measure. So it is the author himself that introduces the notion of risk-neutral measure in the context of a market with only a security and a bond (and no derivative at all). – RandomGuy Jul 29 '16 at 14:19
• "In order to avoid this we consider a special expectation, the risk-adjusted or risk-neutral one". "There may be different ways of taking risk into account" (final of p.24). So the author is not saying how the "risk-neutral probabilities" are computed because those are specific for each assets class? – Robert Jul 29 '16 at 14:32