I'm working my way through an elementary stochastic calculus textbook. I'm having trouble with one of the questions:
Bachelier type stock price dynamics. Let the SDE for stock price $S$ be given by $dS(t) = \mu dt + \sigma dB(t)$, where $\mu$ and $\sigma$ are constant. Derive the SDE for $S$ under the money market account as the numeraire.
I interpret as follows: they want $dS^*$, where $S^* = S e^{-rt} $, where $r$ is the market rate. Applying the usual rules for stochastic differentials I arrive at:
$dS^* = (\mu e^{-rt} - r S^*) \ dt + \sigma dB $
or
$dS^* = (\mu - r S) e^{-rt} dt + \sigma dB$
Then turn this into a martingale by a suitable Girsanov transformation, and specify the expression for the Radon–Nikodym derivative.
Here I'm unsure how to proceed. Help much appreciated. What I've tried so far was defining a new Brownian that depended on the stock price but that was inconsistent / did not make sense in the end.
What I've worked with previously either had the stock price on every term (so that it could be taken out) or not at all. Rest of the question, if it helps below:
Thereafter derive the SDE for the undiscounted stock price and solve that SDE. Finally, using this latest stock price, derive the expected value of max[S(T ) − K, 0] where K is a constant.