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Consider the one period market model $\left(\overline{\pi},\overline{S}\right)$ consisting of a risk-free asset $\left(\pi^{0},S^{0}\right)=(1,1+r)$ and a risky $\left(\pi^{1},S^{1}\right)$

Let $ r > -1$ and $\pi^{1}>0$. $S^{1}$ has strictly positive density function $f: (0,\infty)\to \left(0,\infty\right)$, i.e. $P(S^{1}\leq x)=\int_{0}^{x}f(y)\mathrm{d}y$ for $x > 0$.

Find the risk-neutral measure $\mathbb Q$.

My idea:

I think we need to characterize $\mathbb Q$ in terms of its Radon-Nikodym derivative with respect to $\mathbb P$, i.e. $\frac{\mathrm{d}\mathbb Q}{\mathrm{d}\mathbb P}$. By definition of being risk-neutral, $\mathbb Q$ has to satisfy:

$$(1+r)\pi^{1}=E_{\mathbb Q}[S^{1}]$$

By the Radon-Nikodym derivative approach, we get

$$(1+r)\pi^{1}=E_{\mathbb Q}[S^{1}]=E_{\mathbb P}\left[S^{1}\frac{d\mathbb Q}{d\mathbb P}\right]=\int_{\Omega}S^{1}(\omega)\frac{\mathrm{d}\mathbb Q}{\mathrm{d}\mathbb P}(\omega)\mathbb P(\mathrm{d}\omega)=\int _{0}^{\infty}\left(S^{1}\circ f\right)(y)\left(\frac{d\mathbb Q}{d\mathbb P}\circ f\right)(y)f(y)\mathrm{d}y$$

I have no idea how to proceed to get an expression of $\frac{\mathrm{d}\mathbb Q}{\mathrm{d}\mathbb P}$. Any ideas? Am I on the right track?

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  • $\begingroup$ I don’t see how this can be achieved. There’s not enough information to define a unique risk neutral measure I think $\endgroup$
    – dm63
    Commented Dec 18, 2020 at 15:35
  • $\begingroup$ I added the source of the exercise. $\endgroup$
    – MinaThuma
    Commented Dec 18, 2020 at 15:45

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