Reasoning About a Contingent Claim’s Lower Bound in an Incomplete Market

Summary

The document asks how to justify a lower bound for the price of a European contingent claim that is not directly marketable in an arbitrage-free incomplete market. It proposes taking the supremum of initial portfolio values over admissible strategies whose terminal value does not exceed the claim payoff. The author then sketches an arbitrage argument involving selling the claim, buying a cheaper portfolio, and investing the difference at the risk-free rate.

The question exposes a key issue in translating a supremum into an executable hedge: the proposed argument assumes a portfolio attaining the bound, while a supremum need not be attained. It also asks why a buyer would transact at the claimed bound if a lower-cost superhedging portfolio is available. No answer or resolution is included, so the document serves as a problem statement rather than a complete derivation. Details depend on the market’s admissibility and pricing assumptions.

Key ideas

• The proposed lower bound is defined as a supremum over admissible portfolios with terminal value bounded by the claim payoff.
• The argument sketches a trade using a short claim position, a cheaper portfolio, and investment of the price difference.
• A supremum need not be attained by a portfolio, which matters when turning the bound into a hedge.
• The document raises but does not resolve the pricing argument or its assumptions.

Tags

# Find the lower bound of a contingent claim in incomplete market


# Find the lower bound of a contingent claim in incomplete market












I'm trying to justify the lower bound for the price of a contingent claim (a European one) which is not marketable in an arbitrage free market. I would like to have your advice on my way to do it:

First, I start with the set $\{V_0(\theta) : V_T(\theta)\leq H\; and\;\theta\; admissible\}$ where $V_0(\theta)$ is the initial value of a portfolio with strategy $\theta$. I would like to show that $C_{-}:=\sup{\{V_0(\theta) : V_T(\theta)\leq H\; and\;\theta\; admissible\}}$ is a lower bound for the pricing of the contingent claim. Indeed, consider a price $V_0(\theta)<C_{-}$, then the buyer can short sell $C_{-}$, buy $V_0(\theta)$ and invest $(C_{-}-V_0(\theta))$ in a risk free asset such that at the maturity date $T$ the buyer can face the short selling by using $H$ and he receives $(C_{-}-V_0(\theta))(1+r)^T$ which is strictly positive.

However I am wondering if my arbitrage strategy is correct since why one will buy $C_{-}$ if there exists $V_0(\theta)<C_{-}$ ?

I think I have misunderstood something Thank you a lot

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