# Is there an error in this problem on pricing an asset using the true probability of an up move?

I'm self-studying for an actuarial exam and I encountered the following problem: The true probability of an up move, $p$, must satisfy: $$p = \frac{e^{{(\alpha - \delta})h} - d}{u - d},$$

where $\alpha$ is the continuously compounded annual expected return of the stock and $\delta$ is the continuous dividend rate.

But with $\alpha = 0.10$, $\delta = 0.03$, $u = 1.04$, and $d = 0.91$, we have $$p = \frac{e^{{(0.10 - 0.03})1} - d}{1.04 - 0.91} = 1.25,$$ which doesn't seem possible since I thought $0 \leq p \leq 1$.

How should I interpret this value of $p$ in terms of a probability, since $p > 1$?

Your formula for $p$ is $$p = \frac{e^{{(\alpha - \delta})h} - d}{u - d},$$ where $\alpha$ is not expected return on stock but continuous risk free rate, i.e. 1%.
If you use $\alpha$ as 1%, you will get $p=0.009125828$ which is within $[0,1]$
EDIT: With the information given in the question, it must satisfy following equality: $$S_0e^{\alpha - \delta}=uS_0p+dS_0(1-p)$$ where $S_0$ is initial price. Solving above equation provide $p$=1.25. This indicate that information provided in the question is incorrect. Either, your expected price after 1 year($S_0e^{\alpha - \delta}$) must be within $[S_0u,S_0d]$, so require change in $\alpha$ or $u$ must be sufficiently large such that $uS_0 > S_0e^{\alpha - \delta}$.
• Using the risk-free rate of return for $\alpha$ gives the risk-neutral probability measure $p^*$. The formula that I'm asking about in the original post is for the true probability measure $p$. How do I interpret that the true probability is greater than 1? – user2521987 Feb 11 '16 at 14:30
• @user2521987 option price does not depend on true probability measure. There are sufficient post on Quant. SE that explain why this is so. The formula provided by you for $p$ is probability of moving up under risk neutral measure. – Neeraj Feb 11 '16 at 14:38
• I'm copying the formula directly from my textbook. Letting $\alpha$ be the expected rate of return for the stock, the formula I have for $p$ is the true probability of an up-move. (letting $\alpha$ be the risk-free rate gives the risk-neutral probability of an up-move, $p^*$) My question is, what is the verbal interpretation that the true probability is greater than 1? – user2521987 Feb 11 '16 at 15:15