# SDF derivation by a stochastic process

I have a stochastic process to model the stochastic discount factor (SDF) with M: $$\begin{equation} dM_t = aM_tdt + bM_t d Z_t \end{equation}$$ where, $$Z_t$$ is a standard brownian motion. How do I show that $$a = -r$$, where $$r$$ is the continuously compounded risk-free rate? I think that I should use the following equation that the SDF requires, $$\begin{equation} E(M_t e^{rt}) = 1 \end{equation}$$ or, $$\begin{equation} E(M_t) = e^{-rt}. \end{equation}$$ I am close to the answer, but I don't know how to put it all together to prove $$a = -r$$.

Also, we can setup a differential equation for a risky asset $$S_t$$: $$\begin{equation} dS_t = \mu S_t dt + \sigma S_t dZ_t. \end{equation}$$ How do I prove that $$b = - \frac{\mu - r}{\sigma}$$?

• If $dM_t = aM_tdt + bM_t d Z_t$, then $E[M_t]=M_0e^{at}$, see here. Typically, you impose $M_0=1$ and then you can conclude $a=-r$.
– Alex
Feb 6, 2021 at 13:50
• Thanks! I edited my question now, do you also know how to prove the expression for $b$? Feb 6, 2021 at 14:55
• @nithingong $\frac{\mu-r}{\sigma}$ is the market price of risk (Sharpe ratio). It appears as a result of Girsanov's theorem, see here, here, here and here Feb 6, 2021 at 15:19