Pricing a Guaranteed Fund Payoff as a Bond and Call

Summary

The document considers a fund guarantee that pays the customer the greater of the initial investment or the fee-adjusted fund value at maturity. The answer rewrites the payoff as the initial amount plus a call option on the fund, with a strike increased to account for the fee. Under the Black–Scholes framework, the call component can then be priced using the ordinary option formula and scaled as appropriate.

This decomposition connects a floor guarantee to familiar option pricing and gives a direct way to approach valuation at inception. The provided response is brief and does not work through a numerical valuation, specify the necessary market inputs, or explain how to value the contract at an intermediate time. Its displayed time subscripts appear inconsistent between the guarantee horizon and the adjustment factor, so the fee and valuation dates should be checked before applying the expression. The argument also depends on standard Black–Scholes assumptions.

Key ideas

• A guaranteed payoff can be decomposed into a fixed initial amount and a call-like payoff.
• The corresponding call strike reflects the fund fee over the guarantee period.
• The option component can be valued with a Black–Scholes call formula under its assumptions.
• The displayed time indices should be checked for consistency before using the decomposition.

Tags

# Use of Black-Scholes Model on Guaranteed Fund Investment


# Use of Black-Scholes Model on Guaranteed Fund Investment












I am stuck with a revision question at home on Black-Scholes pricing model.

The question is on a fund manager selling one unit of the fund to a customer for $S(0)$ at time $0$ and then guaranteeing at time T to pay customer maximum of $S(0)$ or $e^{-aT} S(T)$ for $a>0$ being the fund fee. The model I read about talks about maximum of $S(T)-K$ or $0$.

How can one value the customer pay-off at time $0$ or time t?

## Answer by mbison (score 1)

https://quant.stackexchange.com/a/20650

you can just transform the payoff into a form that becomes a linear combination of the ordinary black scholes:

$max(S(0), e^{-aT} S(T))$ = $S(0) + e^{-at}max(0, S(T) - S(0)e^{aT})$ this equals $S(0) + constant * max(0, S(T)- K)$ where $K = S(0)e^{aT}$.

The option you can value with regular black scholes, then just scale it etc.

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