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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 $b = - \frac{\mu - r}{\sigma}$?

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  • $\begingroup$ 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 $a=-r$. $\endgroup$
    – Alex
    Feb 6, 2021 at 13:50
  • $\begingroup$ Thanks! I edited my question now, do you also know how to prove the expression for $b$? $\endgroup$
    – nit hingong
    Feb 6, 2021 at 14:55
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    $\begingroup$ @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 $\endgroup$
    – Kevin
    Feb 6, 2021 at 15:19

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