Suppose $u=2$ and $d=1/2$ in a binomial tree model.

Suppose that ${S_n^3}$ ($S_n$ represents stock price at time $n$) is a martingale under risk neutral probabilities. Find the risk free interest rate $r$.

I know how to find the risk neutral probabilities (note that $\sim$ means under risk neutral probabilities)

$$\tilde{p} = \frac {(1+r)-1/2}{2-1/2} = \frac {1+2r}{3}$$

$$\tilde{q} = 1-\tilde{p} = \frac{2-2r}{3}$$

After this I'm not really sure how to approach this. I understand the definition of a martingale, that is

$$\tilde{E}_n[S_{n+1}] = S_n$$

But I'm not sure how to find r.

  • $\begingroup$ Question: Isn't it $S_n^3$ that is a martingale? You didn't say $S_n$ was. $\endgroup$ – Raskolnikov Nov 7 '17 at 8:14

If we assume risk-neutral valuation holds, then


or in other words, the price of the stock at time $n$ is the discounted pay-off of the stock at time $n+1$. That's what you use to determine $\tilde{p}$ and $\tilde{q}$ plus the fact they add up to 1.

But you added to this that $S_n^3$ is a martingale for the risk-neutral measure thus


This gives an extra condition on $\tilde{p}$ and $\tilde{q}$, which can only be true for a special value of $r$.

Does this help?

  • $\begingroup$ Yes this does :) thank you so much $\endgroup$ – Niko L Nov 13 '17 at 7:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.