## Pricing lookback options and dynamic guarantees. With discussion by Griselda Deelstra.(English)Zbl 1084.91507

Summary: Pricing exotic options or guarantees in equity-indexed annuities can be problematic. The authors present closed-form formulas for pricing lookback options and dynamic guarantees that facilitate the hedging and reserving for such products. The principal tool used is a closed-form expression for $$B(u,T)$$, the Laplace-Stieltjes transform of the expected excess of the running maximum of a Wiener process above a positive constant u in a finite time interval of length T. If the aggregate net income of a company is modeled with a Wiener process, then the excess of the running maximum above $$u$$ can be interpreted as aggregate dividend payments, and the quantity $$B(u,T)$$ is the expectation of the discounted value of the dividend payments up to time $$T$$. The formula for $$B(u,T)$$ is used to price European lookback options (call and put, fixed and floating strike). It is also used to price dynamic fund protection, which is a guarantee on an investment fund: The number of units of the investment fund is increased whenever necessary, so that their total value does not fall below a guaranteed level. The guaranteed level can be stochastic, such as that given by a stock index. Some well-known results for the first passage time of the Wiener process are explained in the appendix.

### MSC:

 91G20 Derivative securities (option pricing, hedging, etc.) 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Full Text:

### References:

 [1] American Academy Of Actuaries (AAA), Final Report of the Equity Indexed Products Task Force,” American Academy of Actuaries, Washington, D.C. Reprinted in part as ”Society of Actuaries Study Note 8V-312-01 (1997) [2] Bermin H.-P., Applied Mathematical Finance 7 pp 75– (2000) · Zbl 1020.91019 [3] Björk T., Arbitrage Theory in Continuous Time (1998) · Zbl 1140.91038 [4] Black F., Journal of Political Economy 81 pp 637– (1973) · Zbl 1092.91524 [5] Briys E., Options, Futures and Exotic Derivatives (1998) [6] Chhikara R. S., The Inverse Gaussian Distribution: Theory, Methodology, and Applications (1989) · Zbl 0701.62009 [7] Clewlow L., Exotic Options: The State of the Art (1997) [8] Conze A., Journal of Finance 46 pp 1893– (1991) [9] Feller W., An Introduction to Probability Theory and Its Applications, 2. ed. (1971) · Zbl 0219.60003 [10] Gerber H. U., Rivista di Matematica per le Scienze Economiche e Sociali 21 pp 125– (1998) [11] Gerber H. U., North American Actuarial Journal 4 (2) pp 28– (2000) · Zbl 1083.91516 [12] Gerber H. U., Transactions of the Society of Actuaries 36 pp 99– (1994) [13] Gerber H. U., Insurance: Mathematics & Economics 18 pp 183– (1996) · Zbl 0896.62112 [14] Gerber H. U., North American Actuarial Journal 4 (4) pp 164– (2000) [15] Gerber H. U., Metodi Statistici per la Finanza e le Assicurazioni (2002) [16] Gerber H. U., North American Actuarial Journal 7 (2) (2003) · Zbl 1084.60512 [17] Goldman M. B., Journal of Finance 34 pp 1111– (1979) [18] Haug E. P., The Complete Guide to Option Pricing Formulas (1998) [19] Huang Y.-C., North American Actuarial Journal 5 (1) pp 153– (2001) · Zbl 1083.91542 [20] Imai J., North American Actuarial Journal 5 (3) pp 31– (2001) · Zbl 1083.60513 [21] Kwok Y.-K., Mathematical Models of Financial Derivatives (1998) · Zbl 0931.91018 [22] Lee H., Pricing Exotic Options with Applications to Equity-Indexed Annuities (2002) [23] Lin X. S., Insurance: Mathematics & Economics 23 pp 45– (1998) · Zbl 0942.60066 [24] Lundberg F., Skandinavisk Aktuarietidskrift 15 pp 137– (1932) [25] Margrabe W., Journal of Finance 33 pp 177– (1978) [26] Mitchell G. T., Equity-Indexed Annuities: New Territory on the Efficient Frontier (1996) [27] Nelken I., The Handbook of Exotic Options: Instruments, Analysis, and Applications (1996) [28] Panjer H. H., Financial Economics: With Applications to Investments, Insurance, and Pensions (1998) [29] Seshadri V., The Inverse Gaussian Distribution: A Case Study in Exponential Family (1993) [30] Streiff T. F., Equity-Indexed Annuities (1999) [31] Tiong S., Equity-Indexed Annuities in the Black-Scholes Environment (2000) · Zbl 1083.62545 [32] Tiong S., North American Actuarial Journal 4 (4) pp 149– (2000) · Zbl 1083.62545 [33] Wilmott P., Derivatives: The Theory and Practice of Financial Engineering (1998) [34] Zhang P. G., Exotic Options: A Guide to Second Generation Options, 2. ed. (1998) · Zbl 0934.91030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.