Consider a one-period arbitrage-free model, it has one risky asset $(\pi^{1},S^{1})$ such that $\pi^{1}>0$, with interest rate on the risk-free asset $(\pi^{0},S^{0})$ at $r > -1$.Furthermore $E_{\mathbb P}[S^{1}]< \infty$. Construct a risk-neutral measure $\mathbb P^{*}$ such that on $c:=(1+r)\pi^{1}$ the density $\frac{d\mathbb P^{*}}{d\mathbb P}$ is constant on $\{S^{1} <c\}, \{S^{1} >c\}, \{S^{1} =c\}$.

My idea:

By the Fundamental Theorem of Asset Pricing, we know that there exists some to $\mathbb P$ equivalent risk-neutral measure $\mathbb Q$ such that it follows:

$c = E_{\mathbb Q}[S^{1}]=E_{\mathbb P}[S^{1}\frac{d\mathbb Q}{d\mathbb P}]=E_{\mathbb P}[(S^{1}\frac{d\mathbb Q}{d\mathbb P})1_{\{S^{1} <c\}}+(S^{1}\frac{d\mathbb Q}{d\mathbb P})1_{\{S^{1} >c\}}+(S^{1}\frac{d\mathbb Q}{d\mathbb P})1_{\{S^{1} =c\}}]$

I assume that we will need to find a measure $\mathbb P^{*}$ that is equivalent is $\mathbb Q$, so that we may use $\frac{d\mathbb P^{*}}{d\mathbb P}=\frac{d\mathbb P^{*}}{d\mathbb Q}\frac{d\mathbb Q}{d\mathbb P}$

I am still stuck on how to go about this. Any ideas?


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