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A Self-Financing Portfolio Hedges a Claim by Matching Its Payoff

Article Quant Q&A · Author: user32091
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Summary

The document gives a concise mathematical criterion for calling a trading portfolio a hedge of a terminal claim. The portfolio must be self-financing, meaning its holdings are adjusted without adding or withdrawing outside funds, and its value at maturity must equal the claim’s payoff. This frames hedging as a property of the portfolio’s value process and terminal outcome rather than as a general intention to reduce risk.

The answer states a defining condition but does not explain how to construct such a portfolio, whether one exists in a particular market, or how to handle imperfect hedges. It also gives no examples or evidence beyond the definition. In practice, the criterion depends on the chosen claim, market model, and assumptions about trading and financing; matching the terminal payoff alone does not describe the pathwise risks or the cost of the hedge.

Key ideas

  • A hedge portfolio for a terminal claim must be self-financing.
  • Its value at maturity must equal the claim’s payoff.
  • The definition specifies a terminal payoff condition but does not provide a construction method.
  • The criterion does not discuss imperfect hedging or risk along the path to maturity.

Tags

Derivatives pricingRisk management
Full text
# Mathematical definition of a hedge?


# Mathematical definition of a hedge?












For two given portfolios/trading strategies I want to know what criteria need to fulfilled in order to call the one portfolio a hedge to the other. In other words; what is the mathematical definition of a hedge given in terms of the value processes of the portfolios? before down voting this question for being to basic please google ‘hedge definition’ and the first many hits won’t give you an mathematical definition but rather oral explanations. Thanks

## Answer by Sanjay (score 3, accepted)

https://quant.stackexchange.com/a/38593

If you have a T-claim $X$, then $h$ is a hedge portfolio if and only if $h$ is self-financing and the value $V^h(t)$ at time $T$ is as following: $$V^h(T)=X$$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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