Proving a Discounted Stock Price Is a Risk-Neutral Martingale

Express the asset's dynamics under the risk-neutral measure

\mathbb Q

, divide its price by the money-market account, and verify that the result is a true martingale. In a no-dividend constant-coefficient model with risk-free rate

r

and volatility

\sigma

, the discounted price is

e^{-rt}S_t=S_0e^{\sigma W_t^{\mathbb Q}-\frac12\sigma^2t}.

It has conditional expected growth of one by independence of the future Brownian increment and its normal moment-generating function.

Discount the traded value and verify the conditional mean

Assume

dS_t=rS_t\,dt+\sigma S_t\,dW_t^{\mathbb Q}

with

S_0>0

. The discount factor is deterministic and of finite variation, so the product rule yields

d(e^{-rt}S_t)=e^{-rt}dS_t-re^{-rt}S_t\,dt =\sigma e^{-rt}S_t\,dW_t^{\mathbb Q}.

This shows that the discounted process $\widetilde S$ is a local martingale. To prove the stronger claim, solve the driftless geometric SDE or use its exponential representation. At any $s<t$ ,

\mathbb E_{\mathbb Q}[\widetilde S_t\mid\mathcal F_s] =\widetilde S_s\, \mathbb E_{\mathbb Q}\!\left[ e^{\sigma(W_t^{\mathbb Q}-W_s^{\mathbb Q})-\frac12\sigma^2(t-s)} \right] =\widetilde S_s.

The process is adapted, and its finite-time first moments equal $S_0$ . You could alternatively verify $\mathbb E_{\mathbb Q}\int_0^T \sigma^2\widetilde S_u^2\,du<\infty$ , which ensures its stochastic integral is a true martingale on the finite interval.

To verify the martingale directly, write the discounted stock at

t

as its value at

s

times

e^{-\sigma^2(t-s)/2+\sigma(W_t^{\mathbb Q}-W_s^{\mathbb Q})}

. The second factor is independent of the information at

s

and has mean one by the Gaussian exponential moment. This checks both why the discounted value is fair under

\mathbb Q

and why the undiscounted conditional mean still grows at

r

. When the asset pays dividends, the traded gains include those cash flows and the same statement requires accounting for them.

For an original exercise, assume $r=0.04$ and $\sigma=0.25$ , and suppose $S_{0.5}=108$ is observed. Then $\mathbb E_{\mathbb Q}[S_1\mid\mathcal F_{0.5}]=108e^{0.02}\approx110.18$ . Multiplying by $e^{-0.04}$ gives $108e^{-0.02}\approx105.86$ , precisely the discounted time- $0.5$ value. In units of time $0.5$ , the equivalent equality is $S_{0.5}=e^{-0.02}\mathbb E_{\mathbb Q}[S_1\mid\mathcal F_{0.5}]$ .

For a stock with continuous proportional dividend yield

q

, the ex-dividend price drifts at

r-q

under

\mathbb Q

. In that case the martingale is the discounted gain process,

e^{-rt}S_t+\int_0^t e^{-ru}qS_u\,du

. The dividend integral offsets the negative

-q

drift of the discounted ex-dividend price.

A deterministic time-varying rate works similarly. Replace $e^{rt}$ by $\exp(\int_0^t r_u\,du)$ and use $r_tS_t$ as the drift under $\mathbb Q$ . The product rule removes the local rate at each instant.

Proving a Discounted Stock Price Is a Risk-Neutral Martingale

Discount the traded value and verify the conditional mean

Turn a fair random walk into a biased one

References and next steps