# Martingale under Risk Neutral Probabilities

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.

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

If we assume risk-neutral valuation holds, then

$$\mathbb{\tilde{E}}_n[S_{n+1}]=S_n(1+r)$$

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

$$\mathbb{\tilde{E}}_n[S_{n+1}^3]=S_n^3$$

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?

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