Wolfgang J. Runggaldier

Bond Market Structure in the Presence of Marked Point Processes

We investigate the term structure of zero coupon bonds when interest rates are driven by a general marked point process as well as by a Wiener process. Developing a theory that allows for measure-valued trading portfolios, we study existence and uniqueness of a martingale measure. We also study completeness and its relation to the uniqueness of a martingale measure. For the case of a finite jump spectrum we give a fairly general completeness result and for a Wiener-Poisson model we prove the existence of a time-independent set of basic bonds. We also give sufficient conditions for the existence of an affine term structure. Copyright Blackwell Publishers Inc. 1997.

Towards a General Theory of Bond Markets

Abstract.The main purpose of the paper is to provide a mathematical background for the theory of bond markets similar to that available for stock markets. We suggest two constructions of stochastic integrals with respect to processes taking values in a space of continuous functions. Such integrals are used to define the evolution of the value of a portfolio of bonds corresponding to a trading strategy which is a measure-valued predictable process. The existence of an equivalent martingale measure is discussed and HJM-type conditions are derived for a jump-diffusion model. The question of market completeness is considered as a problem of the range of a certain integral operator. We introduce a concept of approximate market completeness and show that a market is approximately complete iff an equivalent martingale measure is unique.

Mean-variance hedging of options on stocks with Markov volatilities

We consider the problem of hedging an European call option for a diffusion model where drift and volatility are functions of a Markov jump process. The market is thus incomplete implying that perfect hedging is not possible. To derive a hedging strategy, we follow the approach based on the idea of hedging under a mean-variance criterion as suggested by Follmer, Sondermann, and Schweizer. This also leads to a generalization of the Black–Scholes formula for the corresponding option price which, for the simplest case when the jump process has only two states, is given by an explicit expression involving the distribution of the integrated telegraph signal (known also as the Kac process). In the Appendix we derive this distribution by simple considerations based on properties of the order statistics.

Connections between stochastic control and dynamic games

We consider duality relations between risk-sensitive stochastic control problems and dynamic games. They are derived from two basic duality results, the first involving free energy and relative entropy and resulting from a Legendre-type transformation, the second involving power functions. Our approach allows us to treat, in essentially the same way, continuous- and discrete-time problems, with complete and partial state observation, and leads to a very natural formal justification of the structure of the cost functional of the dual. It also allows us to obtain the solution of a stochastic game problem by solving a risk-sensitive control problem.

A NONLINEAR FILTERING APPROACH TO VOLATILITY ESTIMATION WITH A VIEW TOWARDS HIGH FREQUENCY DATA

In this paper we consider a nonlinear filtering approach to the estimation of asset price volatility. We are particularly interested in models which are suitable for high frequency data. In order to describe some of the typical features of high frequency data we consider marked point process models for the asset price dynamics. Both jump-intensity and jump-size distribution of this marked point process depend on a hidden state variable which is closely related to asset price volatility. In our setup volatility estimation can therefore be viewed as a nonlinear filtering problem with marked point process observations. We develop efficient recursive methods to compute approximations to the conditional distribution of this state variable using the so-called reference probability approach to nonlinear filtering.

Jump-Diffusion Models

We discuss jump-diffusion type models for financial market as well as methods for pricing and hedging of contingent claims in such markets. We consider both, asset price and term structure models, and deal also with situations when there is a stochastic volatility correlated with the jumps and when one has very small time scales, i.e., high frequency data. To make the presentation possibly self-contained, in a preliminary section we recall some basic notions from stochastic analysis for jump-diffusions.

Nearly optimal state feedback controls for stochastic systems with wideband noise disturbances

Much of optimal stochastic control theory is concerned with diffusion models. Such models are often only idealizations (or limits in an appropriate sense) of the actual physical process, which might be driven by a wide bandwidth (not white) process or be a discrete parameter system with correlated driving noises. Optimal or nearly optimal controls, derived for the diffusion models, would not normally be useful or even of much interest, if they were not also ‘nearly optimal’ for the physical system which the diffusion approximates. Under quite broad conditions, the ‘nearly optimal’ control for the physical process do have this robustness property, even when compared to controls which can depend on all the (past) driving noise. We treat the problem over a finite time interval, as well as the average cost per unit time problem. Weak convergence methods provide the appropriate analytical tools.

Kalman filtering for linear systems with coefficients driven by a hidden Markov jump process

Abstract We present an explicit algorithm for the solution of the nonlinear filtering problem concerning a linear, partially observed diffusion-type model, where the coefficients are driven by a hidden Markov jump process, of which one can observe the occurrence of a jump. The (recursive) filter is of a “branching” type with its (finite) dimension growing for each jump of the hidden Markov process.

Optimization of Observations: a Stochastic Control Approach

We study a stochastic control problem for the optimization of observations in a partially observable stochastic system. Using a method of discontinuous time transformation, we associate with the original problem with unbounded controls a problem that has bounded controls. This latter problem allows us to construct nearly optimal nonanticipative Lipschitz Markov controls with finite observation power for the original problem. Since the controlled observation equation may degenerate, we also derive a corresponding filtering result and show a separation property of the optimal controls.

On measure transformations for combined filtering and parameter estimation in discrete time

Combined filtering and parameter estimation in discrete time is studied as a nonlinear filtering problem. The measure transformation approach is used and the Zakai equation derived. Some applications are discussed.