Utility Indifference Pricing: A Time Consistent Approach

arXiv:1102.5075v1 [math.OC] 24 Feb 2011 Utility Indifference Pricing: A Time Consistent Approach ∗ Traian A. Pirvu Dept of Mathematics & Statistics McMaster University 1280 Main Street West Hamilton, ON, L8S 4K1 tpirvu@math.mcmaster.ca Huayue Zhang Dept of Finance Nankai University 94 Weijin Road Tianjin, China, 300071 hyzhang69@nankai.edu.cn August 17, 2021 Abstract This paper considers the optimal portfolio selection problem in a dynamic multi-period stochastic framework with regime switching. The risk preferences are of exponential (CARA) type with an absolute coefficient of risk aversion which changes with the regime. The market model is incomplete and there are two risky assets: one tradable and one non-tradable. In this context, the optimal investment strategies are time inconsistent. Consequently, the subgame perfect equilibrium strategies are considered. The utility indifference prices of a contingent claim written on the risky assets are computed via an indifference valuation algorithm. By running numerical experiments, we examine how these prices vary in response to changes in model parameters. Keywords: Time consistency, time inconsistent control, incomplete market, utility indiffer- ence price. 1 Introduction One of the most important problems in mathematical finance is the valuation of contingent claims in incomplete financial markets. This paper studies the indifference valuation of contingent claims in a multi-period stochastic model under regime switching. The risk preferences are of exponential type and they are allowed to change with the regime. ∗Work supported by NSERC grant 371653-09, MITACS grant 5-26761 and the NSFC grant 10901086. The problem of pricing contingent claims by utility indifference in an incomplete binomial model was studied in [7] and [8]. The work of [7] constructs a probabilistic iterative algorithm to obtain utility indifference prices of contingent claims. This algorithm at each step consists of a nonlinear pricing functional which is applied to prices obtained at the earlier steps. This functional is represented in terms of risk aversion and a special martingale measure. In [8], a more general model is considered, with an stochastic factor which may affect the transition probabilities and the contingent claim’s payoff. Two pricing algorithms are proposed in this paper to produce the utility indifference prices. They employ two martingale measures: the minimal martingale measure and the minimal entropy measure. This paper also analyses the dependence of the utility indifference prices on the choice of the trading horizon. Our paper proposes a model with regime switching. Recently, many papers considered the pricing of contingent claims on regime switching market. Here we recall only two such works, [6] and [5]. In [6], the author considers a stock price model which allows for the drift and the volatility coefficients to switch according to two-states. This market is incomplete, but it is completed with new securities. In [5] the problem of option pricing is considered in a model where the risky underlying assets are driven by Markov-modulated Geometric Brownian motions. A regime switching Esscher transform is used to find a martingale pricing measure. The novelty our model brings is the change in risk preference during the investemnt horizon. The issue of loss aversion changing with time was addressed in financial economic literature. For instance, [1] considers a model in which the loss aversion depends on prior gains and losses, so it may change through time. We choose to model this effect by allowing the risk aversion to change between two exponential type utilities according to the two states of the market (bull and bear). In a bull market we expect investors to be willing to take more risk and this is modeled by a lower coefficient of relative risk aversion as compared with the bear market. This type of risk preferences lead to time inconsistent investemnt strategies. That is, an investor may have an incentive to deviate from the optimal strategies that he/she computed at some past time. To deal with this issue, [2] developed a theory for stochastic control problems which are time inconsistent in the sense that they do not admit a Bellman optimality principle. Inspired by [3] and [4], the work of [2] introduced the subgame perfect Nash equilibrium strategies in a discrete time model. These strategies are optimal to be implemented in the next time interval given that they are optimal in the future. In dealing with the problem of time consistency we choose the approach proposed by [2]. Our paper proposes an indifference valuation algorithm for pricing contingent claims in a discrete time incomplete market with regime switching. At each step, the pricing functional depends on the risk aversion, a martingale measure (the minimal martingale measure) and a process which keeps track of the previous optimal wealth levels. In the special case of non- switching preferences our results recover

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