Good morning,
I am trying to understand the state-price density in a Black-Scholes Economy.
Model Setup
Consider a filtered probability space $( \Omega, \mathcal{F}, \mathcal{F}_t, P ). $ Furthermore let's define a standard Brownian Motion $W_t$ under the real world measure $P$. In the market we have 2 assets: $dB_t = rB_t dt$ and $dS_t = \mu S_tdt+\sigma S_tdW_t$. $B_0 =1$ and $S_0 =s$. I guess it is important that $\mu \geq r$, but that's not the point here.
Question
In this given economy it is widely known that the state price density process is uniquely determined by $H_t = exp\left\{-rt-\frac{1}{2}\left\{\frac{\mu-r}{\sigma}\right\}^2 t - \frac{\mu-r}{\sigma} W_t \right\}$. How do I derive this formula?
My own approach/ Ideas
By the state price density we are talking on the one hand of riskless pricing. Hence denote by Q a riskless measure. On the other hand the word density tells me that now it has to hold: $Q(X) = \int_{X} H_t dP, \forall X \in \mathcal{F}$, or in other words. Q has a density with respect to the measure P. Furthermore I guess the term $\frac{\mu - r}{\sigma}$ results from the idea, replacing $W_t$ by a "riskless" $W_t^*$. Then we have to replace the $\mu$ by $r$ and rearrange $dS_t$. I don't know wheter it is helpful or only an application(at least for my purpose its important), but the important implication is, that it holds now for the expectation: $E^Q[X] = E^P[H_tX]$, which is the popular formula for arbitrage-free/riskless pricing.
Thank you very much for your answers/hints in advance.
I got it: Applying Girsanov's Theorem with the (constant) mapping of the sharp ratio. Then you got the density of the Measure-switching. The first term "-rt" comes from the discounting.