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In the book "Mathematics of Financial Markets" by R. J. Elliott and P. E. Kopp, Second Edition there is a claim made in the proof of Lemma 2.2.5., namely that $V_0(\theta)=V_0(\phi)$. I don't see why this would hold without making a rather long argument about EMM, which the book has not gotten to yet. The assumptions on the trading strategies $\phi$ and $\theta$ are that they generating strategies of the contingent claim $H$ in a viable market model in discrete time $\mathbb T = \{0,1,...,T\}$. Using the definitions from the book this means that for $\xi\in\{\theta,\phi\}$ the following assumptions hold

  • $V_T(\xi)=H$
  • $\xi_{t+1}\cdot S_t = \xi_t\cdot S_t$ for all $t\in\{1,...,T-1\}$ (self financing)
  • $V_t(\xi)\geq 0$ for all $t\in\mathbb T$
  • $H$ is a non-negative random variable
  • There exists no strategy $\eta$ with $V_0(\eta)=0, P(V_T(\eta) > 0) > 0$ and $V_T(\eta)\geq 0$ (no arbitrage)

So my question is: Is there a simple argument here? The proof states it in the form of "since $V_0(\theta)=V_0(\phi)$", so I would suspect there is such a simple argument.

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L. t.
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  • I don't have the book. If both strategies $\theta$ and $\phi$ replicate the contingent claim I find it a rather natural assumption that they bot start at the same value which is typically the price of the option at which it was sold. – Kurt G. Mar 19 '22 at 15:33
  • @Kurt G. The Lemma is about the law of one price: Every generating function of H has the same value process. Your assumption is never mentioned and I think it would weaken the Lemma considerably because the initial fair price is then not proven to be unique. – L. t. Mar 19 '22 at 21:29
  • As far as I can tell : when they introduce $V_t(\theta)$ they call $V_0(\theta)$ the investors initial endowment. Strictly speaking the Lemma shows: When $V_0(\theta)=V_0(\phi)$ and $V_T(\theta)=H=V_T(\phi)$ then $V_t(\theta)=V_t(\phi)$ for all $t\in\mathbb T$. It does not seem like a very strong Lemma. The uniqueness of the initial fair price as we know it follows from another result using the fact that the discounted value price is a martingale and replicates $H$. This implies that every such martingale must start at the same value. – Kurt G. Mar 20 '22 at 10:37

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