| The purpose of quantile hedging is to look for the optimal probability of a successful hedge, and efficient hedging has the same purpose but with different structure. They are active topics of research in mathematical finance, and play a role in incomplete markets, or when perfect hedging is not possible.;My thesis works on the quantile hedging and efficient hedging models for guaranteed minimum death benefits within variable annuities.;A variable annuity is a contract between an individual and an insurance company, under which the individual make a lump-sum payment or series of payments. In return, the insurer agrees to make periodic payments to the individual beginning immediately or at some future date. A common feature of variable annuities is the death benefit. The variable annuity contract guarantee insurance company gives a heirs of the account holder or client either the account value upon death, or the value of the initial investment, whichever is greater. So the contract includes risk related to the future development of the stock index as well as uncertainty about whether or not the policyholder will survive.;The main contribution of my thesis is to the mathematical analysis of techniques for optimal quantile and efficient hedging of guaranteed minimum death benefits. And the main results come from the three parts.;In the first part, we hedge a mutual fund using the quantile hedging model and efficient hedging model respectively. In the quantile hedging case, we provide a closed form solution and a complete verification theory. In efficient hedging case, we figure out a closed form solution for a special case, and we have the verification theory under some reasonable range of parameters. Then we change the problem to resemble a finite horizon problem, and set up the theory from a relatively mathematical point of view. Also we solve the general model numerically and provide the verification theorem under some assumptions.;In the second part, we hedge guaranteed minimum death benefits, so we have no closed form solution, and we face a complex numerical problem---a free boundary problem.;In the third part, we not only hedge guaranteed minimum death benefits, but also work in an incomplete financial market. So we need to solve three variable PDEs. |
