## 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.

### 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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### References:

  American Academy of Actuaries (AAA), Final Report of the Equity Indexed Products Task Force (1997)  Black F., Journal of Political Economy 81 pp 637– (1973) · Zbl 1092.91524  Brekke K. A, Stochastic Models and Option Values: Applications to Resources, Environment and Investment Problems pp 187– (1991)  Dubins L. E., Theory of Probability and Its Applications 38 pp 226– (1993) · Zbl 04520813  Duffie J. D., Annals of Applied Probability 3 pp 641– (1993) · Zbl 0783.90009  Gerber H. U., Transactions of the XXV International Congress of Actuaries 3 pp 243– (1995)  Gerber H. U., North American Actuarial Journal 4 (2) pp 28– (2000) · Zbl 1083.91516  Gerber H. U., Bulletin of the Swiss Association of Actuaries 94 pp 143– (1994)  Gerber H. U., Mathematical Finance 6 pp 303– (1996) · Zbl 0919.90009  Gerber H. U., Insurance: Mathematics & Economics 18 pp 183– (1996) · Zbl 0896.62112  Gerber H. U., Derivatives and Financial Mathematics pp 91– (1997)  Goldman M. B., Journal of Finance 34 pp 1111– (1979)  Guo X., Recent Development in Mathematical Finance pp 39– (2002)  Guo X., Journal of Applied Probability 38 pp 647– (2001) · Zbl 1026.91048  Imai J., North American Actuarial Journal 5 (3) pp 31– (2001) · Zbl 1083.60513  Ingersoll J. E., Theory of Financial Decision Making (1987)  Kallianpur G., Stochastic Processes and Related Topics: In Memory of Stamatis Cambanis 1943–1995 pp 231– (1998)  D. O. Kramkov, Theory of Probability and Its Applications 39 pp 153– (1994) · Zbl 04525924  Kwok Y. K., Mathematical Models of Financial Derivatives (1998) · Zbl 0931.91018  Lee H., Pricing Exotic Options with Applications to Equity Indexed Annuities (2002)  Mitchell G. T., Equity-Indexed Annuities: New Territory on the Efficient Frontier (1996)  Panjer H. H., Financial Economics: With Applications to Investments, Insurance, and Pensions (1998)  Peskir G., Principles of Optimal Stopping and FreeBoundary Problem (2001)  Samuelson P. A., Industrial Management Review 6 (2) pp 13– (1965)  Shepp L., Annals of Applied Probability 3 (3) pp 631– (1993) · Zbl 0783.90011  Shepp L., Theory of Probability and Its Applications 39 pp 103– (1994) · Zbl 04525886  Shiryaev A. N., Essentials of Stochastic Finance: Facts, Models, Theory (1999)  Streiff T. F., Equity Indexed Annuities (1999)  Tiong S., Equity Indexed Annuities in the Black-Scholes Environment (2000)  Tiong S., North American Actuarial Journal 4 (4) pp 149– (2000) · Zbl 1083.62545
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