Pensions modelling

Valuable insights into the problem of how to fund defined benefitpension schemes can be obtained by analysis using the standardBlack–Scholes/Merton option pricing model, consideringthe pension fund finances jointly with those of the sponsoringcompany. The nature of the fund assets and liabilities is completelydifferent, and this lies behind current controversies aboutthe appropriate discount rate, valuation, financial accountingand preferential status for pension fund claimants in insolvency.     

具有跳跃特征的资产价格过程的期权定价模型研究

资产价格具有跳跃特征时，衍生于该资产的期权就不能利用传统的Black-Schoels公式进行定价。本主要研究基于Poisson过程和固定跳跃Merton模型的期权定价与风险对冲问题，利用e-套利定价法，得到期权的风险对冲策略所满足的偏微分方程和近似期权定价。     

Mixed Brownian–fractional Brownian model: absence of arbitrage and related topics

We consider a mixed Brownian–fractional-Brownian model of a financial market. The class of self-financing strategies is restricted to Markov-type smooth functions. It is proved that such strategies satisfy a parabolic equation that can be reduced to heat equation. Then it is proved that the mixed model is arbitrage-free. Finally, the capital of the model is presented as the limit of a sequence of semimartingales.     

Investment and consumption without commitment

In this paper, we investigate the Merton portfolio management problem in the context of non-exponential discounting. This gives rise to time-inconsistency of the decision-maker. If the decision-maker at time t = 0 can commit her successors, she can choose the policy that is optimal from her point of view, and constrain the others to abide by it, although they do not see it as optimal for them. If there is no commitment mechanism, one must seek a subgame-perfect equilibrium policy between the successive decision-makers. In the line of the earlier work by Ekeland and Lazrak (Preprint, 2006) we give a precise definition of equilibrium policies in the context of the portfolio management problem, with finite horizon. We characterize them by a system of partial differential equations, and establish their existence in the case of CRRA utility. An explicit solution is provided for the case of logarithmic utility. We also investigate the infinite-horizon case and provide two different equilibrium policies for CRRA utility (in contrast with the case of exponential discounting, where there is only one optimal policy). Some of our results are proved under the assumption that the discount function h(t) is a linear combination of two exponentials, or is the product of an exponential by a linear function. I. Ekeland was supported by PIMS under NSERC grant 298427-04.     

Default Times in a Continuous-Time Markovian Regime Switching Model

We investigate the default time of a firm when a stochastic discount factor is used so that both diffusion and regime switching risks are priced. We establish the relationship between the probability distribution of the default time and the solution of a system of coupled partial differential equations.     

Group classification of a generalized Black–Scholes–Merton equation

The complete group classification of a generalization of the Black–Scholes–Merton model is carried out by making use of the underlying equivalence and additional equivalence transformations. For each nonlinear case obtained through this classification, invariant solutions are given. To that end, two boundary conditions of financial interest are considered, the terminal and the barrier option conditions.     

Identification and control in the partially known Merton portfolio selection model

Different aspects of the Merton two-asset portfolio selection model are studied in the case where the average rate of return for the risky asset is not known to the investor. In particular, we investigate the question of estimation of under arbitrary admissible allocation policy. The question of optimality of adaptive allocation policies is also discussed.The authors would like to thank T. Duncan and B. Pasik-Duncan for stimulating discussion while preparing this paper.     

The Tradeoff between Consumption and Investment in Incomplete Financial Markets

In this paper we are concerned with the tradeoff between long term growth of the expected utility of wealth and consumption. The goal is to find a consumption policy for which the optimal rate of capital growth is zero, i.e. a policy for which balance between consumption and investment is reached. The asymptotic limit of this investment problem when the HARA parameter γ → -∝ is also studied.     

Burgers and Black–Merton–Scholes equations with real time variable and complex spatial variable

The purpose of this article is to study the Burgers and Black–Merton–Scholes equations with real time variable and complex spatial variable. The complexification of the spatial variable in these equations is made by two different methods which produce different equations: first, one complexifies the spatial variable in the corresponding (real) solution by replacing the usual sum of variables (translation) by an exponential product (rotation) and secondly, one complexifies the spatial variable in the corresponding evolution equation and then one searches for analytic and non-analytic solutions. By both methods, new kinds of evolution equations (or systems of equations) in two dimensional spatial variables are generated and their solutions are constructed.