Interaction of Stochastic Systems and Financial Mathematics
by László Gerencsér
This article summarizes recent results obtained in the area of stochastic systems and financial mathematics, the focus being on the statistical theory of Hidden Markov Models, financial time series and option pricing on incomplete markets.
Hidden Markov Models (HMM) and financial mathematics have become our main technical area of research in recent years, with a strong interaction between the two areas. We have carried out research in Hidden Markov Models to derive new techniques in modelling economic time series, such as stochastic volatility processes, while research on financial mathematics has been focused on more conceptual problems, such as hedging under the presence of transaction costs. The main advance in these areas is that we have established a link between HMM-s and linear stochastic systems, enabling us to use a well-developed arsenal for the statistical analysis of HMM-s. Furthermore, in a cooperative effort we have developed new techniques for analyzing portfolio processes under transaction costs, leading to new, fundamental theorems on option pricing.
Our interest in Hidden Markov Models originates in our interaction with the group of Ian Hunter at MIT on control problems related to master-slave microrobots for eye surgery and heart surgery. To achieve high-accuracy control with low-accuracy sensors has led us to the mathematical problem of identifying a Gaussian ARMA-process under quantized observations, meaning that the actual signals are observed with finite precision. This seemingly innocent problem has turned out to be unexpectedly hard and is still unsolved: neither solid statistical theory, nor a computationally viable procedure is available today. An advance in this area is that the identifiability of quantized Gaussian ARMA processes has been established recently by a former member of our group (Ádám Szeidl et al.).
Hidden Markov Models constitute a significant extension of the classical theory of linear systems developed by Kalman. Its development was prompted by speech processing in the sixties. In a Hidden Markov Model we have a state, such as a phoneme, that follows a Markovian dynamics and we have an observation, called a read-out, which is a probabilistic function of the state, such as a digitized version of a phoneme. A fundamental problem of HMM theory is to identify the dynamics and the read-out kernel based on observed values of the process and to estimate the state. The significance of HMM-s in control theory has been recognized in the past few years. HMM-s have become a key tool for modelling not only in control but also in communication, biological systems and economics.
For the statistical analysis of HMM-s we have developed, with Gábor Molnár-Sáska, György Michaletzky and Gábor Tusnády, a fundamentally new tool establishing a link with the statistical theory of linear stochastic systems.
While the computation of the maximum-likelihood estimate of HMM parameters, such as system parameters for a quantized Gaussian ARMA-processes, are hopelessly hard, we did have significant advance in the problem of estimating a quantized linear regression. With Ildikó Kmecs we have developed a computationally feasible method for implementing the well-known EM-method, in which the unknown log-likelihood function is replaced by its conditional mean, given the observations and a tentative, a priori value for the unknown parameter. The key computational difficulty, as in all HMM estimation problems, is to compute a large number of conditional expectations. Using a Markov-Chain Monte-Carlo method we have arrived at a randomized EM-method, the theoretical justification of which has been given by a novel application of the theory of recursive estimation as developed by Benveniste, Metivier and Priouret. Extensive experimental work has proved the viability of the method.
We have cooperated in the area of HMM-s with Francois Legland, IRISA, Rennes, Lorenzo Finesso, LADSEB, Padova and Jan van Schuppen, CWI, Amsterdam, all members of the ERCIM Working Group on Systems and Control.
Our research activity in financial mathematics has been motivated by the emerging interest in option pricing within the control community. In the original theory of Black-Scholes a key condition is that of completeness, meaning the possibility of exact hedging, or synthetizing of any terminal claim starting with an initial capital which is the price of the option, and re-balancing our portfolio over time with no additional cash movement. However, real-world markets are incomplete, and thus hedging against future risk can be done at best with a prescribed probability. This so-called quantile-hedging requires tools of stochastic programming and control.
A prime example of an incomplete market is a market with friction: these are markets with positive transaction costs. A basic problem for the seller of the option is an initial endowment being sufficiently high and well-structured to superreplicate a given contingent claim at the time of maturity. This fundamental problem has been solved by a member of our group, Miklós Rásonyi, jointly with Youri Kabanov and Christophe Stricker, of the Université de Besançon.
To connect theory and practice we need to understand data. The analysis of economic time series is a fascinating area where well-established techniques of system identification can be mixed with novel ideas. With Zoltán Reppa we have analyzed the marginal distributions of the daily returns of some of the most liquid shares traded on the Budapest Stock Exchange. This study has led us to the analysis of ARMA-processes with heavy-tailed, normal inverse Gauss (NIG) innovations. We have extended basic results of system identification, implying significant reduction of secondary sources of prediction errors, and developed a computationally feasible estimation method.
A model class particular for financial mathematics is the class of stochastic volatility models, in particular GARCH models. These have been studied in cooperation with Zsuzsanna Vágó. Detection of changes in the dynamics of physical systems, such as the vibration characteristics of an off-shore oil platform is a hot area of control theory. It is natural to ask if and how the techniques of these areas are applicable to the detection of changes of financial time series. Our earlier work with Jimmy Baikovicius on real-time change-point detection of ARMA-processes has been extended to the analysis of stochastic volatility processes.