Buyer's price in terms of risk-neutral measures

Question

Let us consider a finite arbitrage-free market model $(B,S)$, where $B$ is a bank account and $S$ is a share. Let $X$ be a claim. We define a buyer's price of $X$ as follows:$$\Pi^b_0(X)=\sup \lbrace V_0(\varphi) : V_T(\varphi) \le X \rbrace .$$ Show that $$\Pi^b_0(X)=\inf_{\mathbb{P}\in \mathcal{P(M)}}\mathbb{E}_{\mathbb{P}}\left(\frac{X}{B_T} \right),$$ where $\mathcal{P(M)}$ is a set of all risk-neutral measures. It is easy to show that $\Pi^b_0(X)\le\inf_{\mathbb{P}\in \mathcal{P(M)}}\mathbb{E}_{\mathbb{P}}\left(\frac{X}{B_T} \right)$, since for any $\mathbb{P}\in \mathcal{P(M)}$ we have $$\Pi^b_0(X)=\sup \lbrace V_0(\varphi) : V_T(\varphi) \le X \rbrace=\sup \lbrace \mathbb{E}_{\mathbb{P}}\left(\frac{V_T(\varphi)}{B_T} \right) : \frac{V_T(\varphi)}{B_T} \le \frac{X}{B_T} \rbrace \le$$ $$\le \mathbb{E}_{\mathbb{P}}\left(\frac{X}{B_T} \right)$$ and the left-hand side does not depend on $\mathbb{P}$ and therefore we get $\Pi^b_0(X) \le \inf_{\mathbb{P}\in \mathcal{P(M)}}\mathbb{E}_{\mathbb{P}}\left(\frac{X}{B_T} \right)$. I have absolutely no idea how to show equality. Could anyone help?

Answer 1

Sketch for the one-period case: Denote $\Pi(X) = \inf_{\mathbb{P}\in \mathcal{P(M)}}\mathbb{E}_{\mathbb{P}}\left(\frac{X}{B_1} \right)$. By FTAP, this is the set of non-arbitrage prices for $X$. Therefore, any $\pi<\Pi(X)$ must be an arbitrage price. Then, denoting $Y = S_1/B_1-S_0$ the increment of discounted share, we have for some $(\xi,\xi_{1})$ that $\xi \cdot Y + \xi_1 (X/B_1-\pi)\ge 0$ a.s. and it is not identically zero. $\xi_1$ cannot be zero, as otherwise $\xi$ is an arbitrage portfolio in the original market. Moreover, taking the expectation with respect to some $\mathbb{P}\in \mathcal{P(M)}$, we get that $\xi_1>0$. Therefore, $X/B_1\ge \pi + \zeta\cdot Y$ with $\zeta = \xi/\xi_1$. The last is equivalent to $\pi\in \Pi_0^b$.

In a multi-period model, one may recall that the existence of arbitrage implies that some of corresponding one-period models admits arbitrage and use the above argument. For more information, you may consult Follmer and Schied's book.