This spring's lunch seminar from the CAS group Stochastics in Environmental and Financial Economics – SEFE is held by CAS fellow Professor Bernt Øksendal (University of Oslo).

Abstract

What is the optimal portfolio for an agent in a financial market? This is a classical problem of great interest both to researchers in mathematical finance and (of course) to practitioners in finance. The solution depends not only on the financial market and on the personal utility function of the trader, but also on the information flow that is available to the agent. We consider a Black-Scholes market with logarithmic utility in the following 3 information flow cases:

(i) The classical, complete information case, where we assume that all the information that can be obtained from the market is available to the agent at all times t. This problem can be solved easily using Itô stochastic calculus.

(ii) The partial information case, in which the agent has to base her portfolio choices on less information than in (i). For example, there could be a delay in the information flow compared to the classical information flow in (i).This problem is more difficult than (i), but can still be solved within the machinery of Itô calculus.

(iii) The insider case, in which the agent has some inside information about the future of the market. This information could for example be the price of one of the assets at some future time. Mathematically this is more challenging than the cases above, because it means that the corresponding stochastic integrals describing the wealth process become anticipating, and this is beyond the framework of classical Itô calculus. To approach this problem we use the machinery of anticipating white noise calculus and Malliavin calculus.

We solve the problem in all 3 cases (i), (ii), (iii) and we compute the corresponding optimal expected logarithmic utility of the terminal wealth, denoted by V_0, V_1 and V_2, respectively. Then V_1 - V_0 represents the loss of value due to to the loss of information in (ii) compared to (i), while V_2 - V_0 represents the value gained by the inside information in (iii) compared to (i).