An Introduction to Financial Markets

Chapter Eleven Modeling Dynamic Uncertainty

From Chapter 8 on, we have considered static portfolio management models, where we make a decision at time t = 0 and observe the result at time t = T, the end of a predefined holding period. Representing uncertainty in this context requires the characterization of a multivariate distribution of risk factors, which is not quite trivial. However, there are even more complicated problems, calling for a dynamic characterization of uncertainty.

• A first example is asset–liability management (ALM) problems, where we consider a sequence of time instants t_i ε [0, T], i = 1, …, m, at which we must meet a possibly uncertain liability L_i. We have considered simple approaches to interest rate risk management for ALM problems in Section 6.3. In these limited approaches, we actually solve a static decision problem. It may be the case that a better plan is obtained bHy a multistage decision model, but even if we do not want to pay the price of such a challenging optimization model,¹ we may need to check the performance of whatever plan on a set of random scenarios for both the assets and the liabilities. Thus, we must characterize the uncertain evolution of the underlying risk factors over time, in order to generate a rich and reliable set of scenarios.
• Another quite relevant example is provided by the need to hedge an option dynamically. In Section 2.3.4, we have considered a single-step binomial model for option pricing. Quite clearly, we need ...