Delta Hedging a Payoff Based on the Hedge Portfolio’s Own Returns

Summary

The document poses a replication problem in the Black–Scholes setting for a payoff built from a sequence of annual hedge-portfolio returns. Each period’s payoff factor includes a guaranteed return and any portfolio return above that guarantee. This makes the portfolio both the source of the payoff and the instrument used to replicate it, unlike a standard option whose payoff depends on the underlying asset’s terminal price.

The author asks how to derive an exact delta-hedging strategy and notes that a closed-form price may not be available. No solution, derivation, numerical method, or evidence is provided in the text. The material therefore identifies a challenging endogenous-payoff problem rather than presenting a usable hedge. Any approach would need to address how the strategy’s own returns determine its future liability, an aspect not resolved here.

Key ideas

• The proposed payoff compounds period-by-period factors based on the hedge portfolio’s returns.
• The portfolio acts both as the payoff-generating process and as the replicating strategy.
• The question seeks an exact delta hedge in a Black–Scholes market.
• The document presents the problem but supplies no replication method or pricing result.

Tags

# Exact delta-hedging for endogenous payoffs


# Exact delta-hedging for endogenous payoffs












I would like to derive the exact delta-hedging strategy in the Black-Scholes market to replicate the following non-standard endogenous payoff. The particularity is that the payoff does not only depend on the final value of the stock but on the yearly returns of the hedge porfolio.

More specifically, it is given by $$ L_T= \prod_{t=1}^{T} (1+g+\max [R_t-g,0]) $$ where $g$ is a constant guaranteed return rate and $R_t=\frac{\Pi_{t+1}-\Pi_{t}}{\Pi_{t}}$ is the yearly returns of the hedging portfolio.

So, contrary to the Black-Scholes call option where the payoff is fixed and there is a dynamic portfolio of stocks and bonds, here the hedging portfolio serves both as the underlying security and the replicating portfolio.

In this case, I don't think there is a closed-form for the price but I would like to still find the perfect replicating strategy for this payoff. Any help or reference is welcome !

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