Why a Risk-Neutral Stock Process Is a Martingale
Summary
The document explains why a one-period stock price with up and down multipliers of 1.15 and 0.85 is a martingale under a risk-neutral measure when the risk-free rate is zero. Under that measure, the two outcomes each have probability 0.5, so the conditional expected next price equals the current price. The same reasoning applies at every step because the multipliers are independent and identically distributed. It also addresses integrability: the unconditional expectation remains finite and equals the initial price, as follows from the tower property. The explanation is a brief discrete-time argument, not a general proof for arbitrary price processes; it relies on the specified multipliers, independence, and zero interest rate.
Key ideas
- Under the stated risk-neutral measure, the up and down outcomes each have probability 0.5.
- The conditional expected price at the next step equals the current price because the weighted average multiplier is one.
- The same conditional expectation argument establishes the martingale property at each time step.
- The expected price remains finite and equals its initial value under the assumptions described.
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# Martingale under risk neutral probability
# Martingale under risk neutral probability
I have a question to prove martingale under risk-neutral measure:
Question
Consider a discrete time process $S$, which at time $n \in \mathbb{N}$ has value $S_n = S_{0}\prod_{j=1}^{n}Z_j$, $S_0>0$ and $Z_j$ a sequence of independent and identically distributed random variables to either 1.15 or 0.85. Assume a zero risk free rate of rate.
Show that under the risk neutral martingale measure $\mathbb{\hat{P}}$ the process $S_n$ is a martingale.
My thoughts
- I do understand that under the risk neutral measure $\mathbb{\hat{P}}$, which is that $e^{-rt}C_t = \mathbb{\hat{E}}[e^{-rt}C_T|\mathbb{F}_t]$.
- I understand that 1 - period $\mathbb{\hat{P}}$ is that: $\mathbb{\hat{P}}_u = \frac{1-0.85}{1.15-0.85}=0.5$, and so is $\mathbb{\hat{P}}_d$ = 0.5.
- I think I seek to prove the martingale property of $S_n$,but I am not sure how is that related to the $\mathbb{\hat{P}}$ if I just want to prove that: $\mathbb{E}|S_n|< \infty$ $\mathbb{E}[S_{n+1}|\mathbb{F}_t]=S_n$
## Answer by Arshdeep (score 1)
https://quant.stackexchange.com/a/76446
You already have it. Risk neutral measure is one where tosses are still independent but each individual toss has probability of 0.5 up or down.
Say you're at step n, with $S_{n}$ known.
$E(S_{n+1}|F_{n})=S_{n}*(0.5*1.15+0.5*0.85)=S_{n}$
and unconditional expectation is also finite using tower law as it equals $S0$.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.