Modeling exotic options with maturity extensions by stochastic dynamic programming

Abstract

The exotic options that are examined in this thesis have a combination of non-standard characteristics which can be found in shout, multi-callable, pathdependent and Bermudan options. These options are called reset options. A reset option is an option which allows the holder to reset, one or more times, certain terms of the contract based on pre-specified rules during the life of the option. Overall in this thesis, an attempt has been made to tackle the modeling challenges that arise from the exotic properties of the reset option embedded in segregated funds. Initially, the relevant literature was reviewed and the lack of published work, advanced enough to deal with the complexities of the reset option, was identified. Hence, there appears to be a clear and urgent need to have more sophisticated approaches which will model the reset option. The reset option on the maturity guarantee of segregated funds is formulated as a non-stationary finite horizon Markov Decision Process. The returns from the underlying asset are modeled using a discrete time approximation of the lognormal model. An Optimal Exercise Boundary of the reset option is derived where a threshold value is depicted such that if the value of the underlying asset price exceeds it then it is optimal for the policyholder to reset his maturity guarantee. Otherwise, it is optimal for the policyholder to rollover his maturity guarantee. It is noteworthy that the model is able to depict the Optimal Exercise Boundary of not just the first but of all the segregated fund contracts which can be issued throughout the planning horizon of the policyholder. The main finding of the model is that as the segregated fund contract approaches its maturity, the threshold value in the Optimal Exercise Boundary increases. However, in the last period before the maturity of the segregated fund, the threshold value decreases. The reason for this is that if the reset option is not exercised it will expire worthless. The model is then extended to re ect on the characteristics of the range of products which are traded in the market. Firstly, the issuer of the segregated fund contract is allowed to charge a management fee to the policyholder. The effect from incorporating this fee is that the policyholder requires a higher return in order to optimally reset his maturity guarantee while the total value of the segregated fund is diminished. Secondly, the maturity guarantee becomes a function of the number of times that the reset option has been exercised. The effect is that the policyholder requires a higher return in order to choose to reset his maturity guarantee while the total value of the segregated fund is diminished. Thirdly, the policyholder is allowed to reset the maturity guarantee at any point in time within each year from the start of the planning horizon, but only once. The effect is that the total value of the segregated fund is increased since the policyholder may lock in higher market gains as he has more reset decision points. In response to the well documented deficiencies of the lognormal model to capture the jumps experienced by stock markets, extensions were built which incorporate such jumps in the original model. The effect from incorporating such jumps is that the policyholder requires a higher return in order to choose to reset his maturity guarantee while the total value of the segregated fund is diminished due to the adverse effect of the negative jumps on the value of the underlying asset.