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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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    $\begingroup$ If I understand you correctly the terminal payoff, written in continuous time notation, is $max(S_T - K_T,0)$, where the floating strike is $$K_T = K_0 \exp \left\{ gT + \int_0^T max \left(\frac{ d\Pi_t}{\Pi_t} - g \,dt, 0 \right) \right\}$$ where $K_0$ is the initial guarantee level and $g$ the roll up rate. $\endgroup$
    – Frido
    Mar 28, 2023 at 7:56
  • $\begingroup$ Thanks for your reply! The stock is not involved in the final payoff. Essentially, this is an insurance liability with a guaranteed rate $g$ but any surplus of the hedging portfolio is shared between the insurer (investor) and the policyholder (client) so that the final payoff is $$ C_T=C_0 \exp \left\{g T+\int_0^T \max \left(\alpha \frac{d \Pi_t}{\Pi_t}-g d t, 0\right)\right\} $$ where $C_0$ is the initial guarantee level and $g$ the roll up rate and $0<\alpha<1$ is the proportion of the surplus that is redistributed. $\endgroup$
    – Wiles01
    Mar 28, 2023 at 16:27
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    $\begingroup$ You should probably google about the passport option. Quite a few famous quants have worked on this years ago. $\endgroup$
    – Kurt G.
    Mar 28, 2023 at 17:00
  • $\begingroup$ I don't understand: if the stock / funds is not involved, then what exactly are you hedging, ie what is the hedging portfolio? If you are talking about a with profits product then also I do not see what the hedge portfolio is. Initially I thought you were looking at some exotic type of VA. $\endgroup$
    – Frido
    Mar 28, 2023 at 18:31
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    $\begingroup$ @Wiles01 It's been a while for me, and i've read I believe Kleinow and others on participating products. Will take another look and get back to you if I get new ideas. But I think / I'm afraid the only way to value these liabilities is by brute force methods, even under `simple' Black-Scholes assumptions. $\endgroup$
    – Frido
    Mar 28, 2023 at 21:09

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