Arbitrage-Free Pricing with an American Derivative's Price Process

Summary

The document asks how to define an arbitrage-free price process for an American-style derivative when the buyer may make payments over its lifetime and choose a random exercise time. It proposes selecting a process so the supremum over exercise times of expected discounted payoff minus discounted payments is zero, and compares this with the familiar valuation expression when payment occurs only at inception.

The material presents the question and its intuition but supplies no answer, derivation, or supporting evidence. In particular, it does not resolve whether the proposed condition characterizes a unique no-arbitrage price process or what assumptions on admissible payment processes and exercise strategies are required. It is therefore a useful framing of a pricing problem, rather than an established pricing method.

Key ideas

• The question concerns an American-style derivative with payments potentially spread across its lifetime.
• The proposed pricing condition equates the greatest expected discounted net payoff across exercise times to zero.
• A single upfront payment reduces the setup to the usual discounted payoff valuation question.
• The document leaves uniqueness and the assumptions needed for no-arbitrage pricing unresolved.

Tags

# Arbitrage free price of a derivative when the price is collected over the lifetime of the derivative


# Arbitrage free price of a derivative when the price is collected over the lifetime of the derivative












Let $X_t$ be an american style financial derivative with random exercise time $T$ where $t$ and $T$ belongs to some finite set $A$. Buying this derivative requires the buyer to pay $p_t$ up to time $T$. Let $\Omega$ be the sample space of $X_t$, $p=(p_t)_{t \in A}$ the price process and $B={\left(C^A \right)}^\Omega$ the value space of $p$ for some set $C \subset \mathbb{R}$. Assume expectation are taken under the risk-neutral measure with $R_t$ as the risk-free discounting factor from times $0$ to time $t$. Is the no-arbitrage pricing process of the derivative given by $$ \arg_{p \in B} \left( \sup_{T \in A} E(X_T R_T - \int_0^T p_t R_tdt)=0 \right) \text{ (1)} $$

when $B$ requires that $p_t(\omega) \ne 0, t>0$ for some $\omega \in \Omega$?

My knowledge of finance tells me the no-arbitrage price would be $$ \sup_{T \in A} E(X_T R_T) \text{ (2)} $$ when B is degenerated to $p_0=k$ for some $k \in \mathbb{R}$ and $p_t=0,t \ne 0$. Intuitively, I would expect (1) to be the natural extension to (2). But is it theoretically true? I searched, but I couldn't find any source confirming my hypothesis.

Thanks for any help.

Note: I asked this question on mathoverflow and I was guided here.

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