Stochastic Analysis and Finance

A further development of the theory on integro-partial differential equations will be pursued in close cooperation with the PDE group. In portfolio optimization and/or in connection with option pricing when the risky assets are modelled by using jump processes, one can associate partial differential equations with integral terms (integro-PDE). There has been a great effort in analyzing integro-PDEs using viscosity solution techniques, and we aim at further developing the theory in this area. We have in mind of going in the direction of infinite-dimensional theory which is of interest in certain areas of finance. For example, in many markets there is a trade in forward contracts, which price dynamics is modelled as a stochastic process parametrized by delivery time. Further, in weather markets one may find examples where dynamics are dependent on space location. Thus, infinite-dimensional stochastic processes may be a natural framework to consider these. Questions one may study include pricing of options and stochastic control problems. These lead to the analysis of PDEs in infinite dimensions. The complexity can further be increased with jump processes, leading to integral terms.

We plan to extend our research portfolio with several new projects in the application of stochastic analysis in finance. The concept of risk or risk measure is crucial in mathematical finance and insurance, and recently there has been much research activity linked to this concept and its applications. It turns out that a dynamic risk measure can be represented by a kind of generalized conditional expectation called a G expectation, which again can be obtained as the solution of a backward stochastic differential equation (BSDE). This gives an interesting link between risk measures and BSDEs. Moreover, in view of another representation of risk measures, there is also a link to stochastic differential games. We want to study the stochastic control problems and stochastic differential games that arise in risk-related problems in finance and insurance. Moreover, the risk measure of a given random variable X can also be represented as the supremum of the Q-expectation of -X, minus a penalty term p(Q), the supremum being taken over all Q belonging to a certain family M of probability measures. Therefore the problem, say, of finding the portfolio which minimizes the risk of the terminal wealth can be viewed as a zero-sum stochastic differential game problem (an infsup problem.) This gives an interesting link between problems related to risk in finance and stochastic differential games. We will explore these links in future research. Our group has already written one paper on this, and another one is in preparation.

The group would like to extend further its activities in the modelling and analysis of electricity markets. The issue of linking the forward price dynamics with the electricity spot price is a major issue and open problem. In the electricity market this connection is not as obvious as in other commodity markets due to the non-storability of electricity. In energy markets there are several complicated crosscommodity derivatives which we want to analyse further. For instance, swing options are options which give the owner the right to exercise several times, and also to decide the volume at exercise. To study such contracts, one needs to combine methods for stochastic control with option pricing theory. The markets are incomplete. Natural models for the underlying processes will be mean-reverting, seasonal, and must include jump processes to take care of the spikes, which are distinct features of electricity markets. It seems reasonable to use time-inhomogeneous jump processes to model such spikes, which usually are more frequent in the winter period. A natural class of such models may be the independent increment processes. Our aims include both theoretical analysis and development of numerical schemes for practical solution. Infinite dimensional theory (see above) appears naturally since the forward/futures markets here are huge and derivatives written on them are complicated. Certain problems in electricity markets, like complicated bids involving, e.g., fixed prices and volumes, lead to interesting mathematical optimization problems. We want to investigate such problems, theoretically and computationally, using ideas and techniques from linear and combinatorial optimization.

The group would like to create an activity in insurance mathematics, in particular in problems related to life and pension insurance. Internationally there is already considerable activity in applying mathematical finance and stochastic analysis to problems in life and pension insurance. Our interest in entering this field is motivated by the large change in the regulation of the Norwegian life and pension industry, where in particular interest-rate guarantees need to be explicitly priced. A problem we would like to study more closely is management of pension funds, which have theirspecific regulations and compositions introducing complicated constraints on the optimization. Also, the fund managers want to reduce risk of returns below the guarantee, a desire which adds further to the complexity. An interesting area is portfolio optimization, where discretization and scenario-based modelling of the uncertainty allow for efficient numerical optimization algorithms. In this area some new measures of risk have been introduced (e.g., "conditional-value-at-risk") and wewould like to study some of these models theoretically and computationally. On the other hand, the guarantee is in effect a put option sold to the insured, and this needs to be priced and hedged. The pension funds may be controlled by the company, leading to a combined control and options pricing problem. The pension funds are usually so big that full hedging is not feasible since the market impact of such transactions affect prices. Most financial theory concentrates on small actors who do not influence market prices by their operations. Furthermore, some assets may not be possible to hedge efficiently, leading us to incomplete option pricing theory. Our focus will be both on theoretical problems and practical issues. One paper has already been written on applying indifference pricing of interest-rate guarantees. Other questions that will be investigated further by the group include optimal dividend allocation (stochastic control theory) and equivalent martingale measures and risk distribution in insurance. (A Ph.D. student is already working on the last of these problems.)