# For one-period model, construct a risk-neutral measure $\mathbb P^{*}$ such that the density is constant on $\{S^{1} (<,>,=)c\}$

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 $$\{S^{1} c\}, \{S^{1} =c\}$$.

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:

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

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