## Theses and Dissertations

#### DOI

10.17077/etd.oi8e0vlq

Dissertation

Spring 2017

#### Degree Name

PhD (Doctor of Philosophy)

Statistics

Shiu, Elias S.W.

Jorgensen, Palle

Lo, Ambrose

Tang, Qihe

#### Abstract

Motivated by the Guaranteed Minimum Death Benefits (GMDB) in variable annuities, we are interested in valuing equity-linked options whose expiry date is the time of the death of the policyholder. Because the time-until-death distribution can be approximated by linear combinations of exponential distributions or mixtures of Erlang distributions, the analysis can be reduced to the case where the time-until-death distribution is exponential or Erlang.

We present two probability methods to price American options with an exponential expiry date. Both methods give the same results. An American option with Erlang expiry date can be seen as an extension of the exponential expiry date case. We calculate its price as the sum of the price of the corresponding European option and the early exercise premium. Because the optimal exercise boundary takes the form of a staircase, the pricing formula is a triple sum. We determine the optimal exercise boundary recursively by imposing the “smooth pasting” condition. The examples of the put option, the exchange option, and the maximum option are provided to illustrate how the methods work.

Another issue related to variable annuities is the surrender behavior of the policyholders. To model this behavior, we suggest using barrier options. We generalize the reflection principle and use it to derive explicit formulas for outside barrier options, double barrier options with constant barriers, and double barrier options with time varying exponential barriers.

Finally, we provide a method to approximate the distribution of the time-until-death random variable by combinations of exponential distributions or mixtures of Erlang distributions. Compared to directly fitting the distributions, my method has two advantages: 1) It is more robust to the initial guess. 2) It is more likely to obtain the global minimizer.

#### Public Abstract

Most variable annuities are essentially an equity investment fund embedded with options or guarantees. These options or guarantees provide the policyholders downside protection plus some chance of upside gains. To provide the protection, insurance companies may buy put options. However, due to the uncertainty of the time of payment, options with a random expiration date need to be considered.

In this thesis, we consider the valuation problem of American options with exponentially distributed or Erlang distributed expiration date. With such expiration dates, analytic pricing formulas can be obtained. Compared to the European option, an American option allows its owner to exercise the option at any time prior to the expiration date. Therefore, we can calculate the price of an American option as the sum of the price of the corresponding European option and the early exercise premium. To determine the optimal exercise boundary, we equate the exercise value with the option price at the exercise boundary.

The surrender behavior of the policyholders is another issue related to variable annuities. If policyholders choose to surrender the contract, they give up the protection, stop paying fees to the insurance company and receive a surrender value. Hence, the insurance company may choose to buy up-and-out put options instead of regular put options to provide the protection, because barrier options are cheaper. We derive explicit formulas for valuing various barrier options, including outside barrier options and double barrier options.

#### Keywords

american option, fit distribution, random expiry date, variable annuity

vii, 110 pages

#### Bibliography

Includes bibliographical references (pages 107-110).