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**Pricing perpetual fund protection with withdrawal option (With discussion and a reply by the authors).**
*(English)*
Zbl 1084.60512

Summary: Consider an American option that provides the amount
\[
F(t)=S_2(t)\max\left\{1, \max_{0\leq \tau\leq t} (S_1(\tau)/S_2(\tau))\right\},
\]
if it is exercised at time \(t\), \(t\geq 0\). For simplicity of language, we interpret \(S_1(t)\) and \(S_2(t)\) as the prices of two stocks. The option payoff is guaranteed not to fall below the price of stock 1 and is indexed by the price of stock 2 in the sense that, if \(F(t) > S_1(t)\), the instantaneous growth rate of \(F(t)\) is that of \(S_2(t)\). We call this option the dynamic fund protection option. For the two stock prices, the bivariate Black-Scholes model with constant dividend-yield rates is assumed. In the case of a perpetual option, closed-form expressions for the optimal exercise strategy and the price of the option are given. Furthermore, this price is compared with the price of the perpetual maximum option, and it is shown that the optimal exercise of the maximum option occurs before that of the dynamic fund protection option.

Two general concepts in the theory of option pricing are illustrated: the smooth pasting condition and the construction of the replicating portfolio. The general result can be applied to two special cases. One is where the guaranteed level \(S_1(t)\) is a deterministic exponential or constant function. The other is where \(S_2(t)\) is an exponential or constant function; in this case, known results concerning the pricing of Russian options are retrieved. Finally, we consider a generalization of the perpetual lookback put option that has payoff \([F(t)- \kappa S_1(t)]\), if it is exercised at time \(t\). This option can be priced with the same technique.

Two general concepts in the theory of option pricing are illustrated: the smooth pasting condition and the construction of the replicating portfolio. The general result can be applied to two special cases. One is where the guaranteed level \(S_1(t)\) is a deterministic exponential or constant function. The other is where \(S_2(t)\) is an exponential or constant function; in this case, known results concerning the pricing of Russian options are retrieved. Finally, we consider a generalization of the perpetual lookback put option that has payoff \([F(t)- \kappa S_1(t)]\), if it is exercised at time \(t\). This option can be priced with the same technique.

### MSC:

60G35 | Signal detection and filtering (aspects of stochastic processes) |

60G17 | Sample path properties |

91G20 | Derivative securities (option pricing, hedging, etc.) |

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\textit{H. U. Gerber} and \textit{E. S. W. Shiu}, N. Am. Actuar. J. 7, No. 2, 60--92 (2003; Zbl 1084.60512)

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