American Claim Pricing and Its Relation to European Options

Summary

The document poses three questions about pricing an American contingent claim (ACC): how to decompose its discounted fair value, why the matching European claim should not be worth more, and when the two prices coincide. The supplied partial argument sketches a Doob decomposition, representing discounted value as a martingale plus a predictable, nondecreasing process. It does not define all notation clearly or establish the requested inequalities and equality, so it is an incomplete proof prompt rather than a worked explanation.

The pricing questions concern the general relationship between early exercise and a European payoff with the same terminal value. The text provides no numerical example, market data, or complete derivation for the comparison results. Readers would need the missing assumptions and a rigorous argument—particularly the claim that the European value dominates every interim ACC payoff—to assess or finish the proof. Its value is mainly as a statement of a mathematical problem and a partial decomposition, not as a self-contained pricing method.

Key ideas

• The ACC is compared with a European option having the same terminal payoff.
• The discounted ACC value is presented as a martingale plus a predictable nondecreasing component.
• The partial solution does not fully define its notation or prove the two requested price comparisons.
• Additional assumptions and a complete argument are needed to establish when the prices are equal.

Tags

# American Contingent Claim vs European Option pricing


# American Contingent Claim vs European Option pricing












Suppose $Y$ is an American Contingent Claim (ACC) defined as $Y = \{Y_t, t \in 0,1,...,T\}$ and asssume $U_t$ is its fair price. Also suppose $C_t$ is the arbitrage-free price at time $t$ of a European option with maturity $T$ and payoff $Y_T$.

i) Write the Doob decomposition of the discounted fair price $U_t := U_t(1 + r)^{-t}$ for $t \in \{0,1,...,T\}$ of the ACC.

ii) Show that $C_t \leq U_t$ for all $t$.

iii) Show that if $C_t \geq Y_t$ for all $t$, then $C_t = U_t$.

I have received the following partial solution. What about the other two points?

i) $V_t(\Phi) = \alpha_tS_t + \beta_tB_t + \frac{\delta_n}{B_{n-1}}$. Then we have $\hat{V}_t(\Phi) = \hat{U}_t + \frac{\delta_n}{B_{n-1}}$.

Here we set $M_t := \hat{V}_t(\Phi)$ and $A_t := \frac{\delta_n}{B_{n-1}}$.

By the definition of $\delta_n$, we know it is a predictive process and it is non-decreasing, and $M_t$ is a martingale as $\Phi$ here is a self-financing strategy.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.