Jörn Sass

Optimizing the terminal wealth under partial information: The drift process as a continuous time Markov chain

Abstract.We consider a multi-stock market model where prices satisfy a stochastic differential equation with instantaneous rates of return modeled as a continuous time Markov chain with finitely many states. Partial observation means that only the prices are observable. For the investor’s objective of maximizing the expected utility of the terminal wealth we derive an explicit representation of the optimal trading strategy in terms of the unnormalized filter of the drift process, using HMM filtering results and Malliavin calculus. The optimal strategy can be determined numerically and parameters can be estimated using the EM algorithm. The results are applied to historical prices.

Utility Maximization Under Bounded Expected Loss

We consider optimal selection of portfolios for utility maximizing investors under joint budget and shortfall risk constraints. The shortfall risk is measured in terms of the expected loss. Depending on the parameters of the risk constraint, we show the existence of an optimal solution and uniqueness of the corresponding Lagrange multipliers. Using Malliavin calculus we also provide the optimal trading strategy.

Optimal portfolio policies under bounded expected loss and partial information

In a market with partial information we consider the optimal selection of portfolios for utility maximizing investors under joint budget and shortfall risk constraints. The shortfall risk is measured in terms of expected loss. Stock returns satisfy a stochastic differential equation. Under general conditions on the corresponding drift process we provide the optimal trading strategy using Malliavin calculus. We give extensive numerical results in the case that the drift is modeled as a continuous-time Markov chain with finitely many states. To deal with the problem of time-discretization when applying the results to market data, we propose a method to detect and correct possible tracking errors.

Good Portfolio Strategies under Transaction Costs: A Renewal Theoretic Approach

This paper looks at portfolio theory in the presence of costs of transactions. A fundamental result is given in Morton and Pliska (1995) ([15]) where renewal theoretic arguments and the theory of optimal stopping are used to derive optimal strategies for maximizing the asymptotic growth rate under purely fixed costs which are proportional to the portfolio value. Our paper is also devoted to maximizing the asymptotic growth rate but here we consider a combination of fixed and proportional costs. Motivated by various structural results in the work on optimal portfolio theory we introduce a class of natural trading strategies which can be described by four parameters, two for the stopping boundaries and two for the new risky fractions (fraction of the wealth invested in the stock). In this class the problem can be simplified by renewal theoretic arguments to treating one period between two trading times, where we then have to start the new risky fraction process according to the invariant distribution. This yields an explicit form for the asymptotic growth rate that can be maximized in these four parameters. The computation of best strategies in this class thus is simple, and we provide various examples. Preliminary considerations based on the fundamental results of Bielecki and Pliska (2000)([5]) and the results of this paper point out that in fact an allover optimal impulse control strategy can be found within this class.

Moment Based Regression Algorithms for Drift and Volatility Estimation in Continuous-Time Markov Switching Models

We consider a continuous time Markov switching model (MSM) which is widely used in mathematical finance. The aim is to estimate the parameters given observations in discrete time. Since there is no finite dimensional filter for estimating the underlying state of the MSM, it is not possible to compute numerically the maximum likelihood parameter estimate via the well known expectation maximization (EM) algorithm. Therefore in this paper, we propose a method of moments based parameter estimator. The moments of the observed process are computed explicitly as a function of the time discretization interval of the discrete time observation process. We then propose two algorithms for parameter estimation of the MSM. The first algorithm is based on a least-squares fit to the exact moments over different time lags, while the second algorithm is based on estimating the coefficients of the expansion (with respect to time) of the moments. Extensive numerical results comparing the algorithm with the EM algorithm for the discretized model are presented. Copyright

Portfolio optimization under transaction costs in the CRR model

In the CRR model we introduce a transaction cost structure which covers piecewise proportional, fixed and constant costs. For a general utility function we formulate the problem of maximizing the expected utility of terminal wealth as a Markov control problem. An existence result is given and optimal strategies can be described by solutions of the dynamic programming equation. For logarithmic utility we provide detailed solutions in the one-period case and provide examples for the multi-dimensional case and for complex cost structures. For a combination of fixed and proportional costs a fast multi-period algorithm is introduced.

Insurance markets and unisex tariffs: is the European Court of Justice improving or destroying welfare?

We analyze the effects of mandatory unisex tariffs in insurance contracts, such as those required by a recent ruling of the European Court of Justice, on equilibrium insurance premia and equilibrium welfare. In a unified framework, we provide a quantitative analysis of the associated insurance market equilibria in both monopolistic and competitive insurance markets. We investigate the welfare loss caused by regulatory adverse selection and show that unisex tariffs may cause market distortions that significantly reduce overall social welfare.

Optimal investment under dynamic risk constraints and partial information

We consider an investor who wants to maximize expected utility of terminal wealth. Stock returns are modelled by a stochastic differential equation with non-constant coefficients. If the drift of the stock returns depends on some process independent of the driving Brownian motion, it may not be adapted to the filtration generated by the stock prices. In such a model with partial information, due to the non-constant drift, the position in the stocks varies between extreme long and short positions making these strategies very risky when trading on a daily basis. To reduce the corresponding shortfall risk, motivated by Cuoco, He and Issaenko [Operations Research, 2008, 56, pp. 358–368.] we impose a class of risk constraints on the strategy, computed on a short horizon, and then find the optimal policy in this class. This leads to much more stable strategies that can be computed for both classical drift models, a mean reverting Ornstein–Uhlenbeck process and a continuous-time Markov chain with finitely many states. The risk constraints also reduce the influence of certain parameters that may be difficult to estimate. We provide a sensitivity analysis for the trading strategy with respect to the model parameters in the constrained and unconstrained case. The results are applied to historical stock prices.

Primal-dual methods for the computation of trading regions under proportional transaction costs

Portfolio optimization problems on a finite time horizon under proportional transaction costs are considered. The objective is to maximize the expected utility of the terminal wealth. The ensuing non-smooth time-dependent Hamilton–Jacobi–Bellman equation is solved by regularization and the application of a semi-smooth Newton method. Discretization in space is carried out by finite differences or finite elements. Computational results for one and two risky assets are provided.

Trading Regions Under Proportional Transaction Costs

In the Black-Scholes model optimal trading for maximizing expected power utility under proportional transaction costs can be described by three intervals B, NT, S: If the proportion of wealth invested in the stocks lies in B, NT, S, then buying, not trading and selling, respectively, are optimal. For a finite time horizon, the boundaries of these trading regions depend on time and on the terminal condition (liquidation or not). Following a stochastic control approach, one can derive parabolic variational inequalities whose solution is the value function of the problem. The boundaries of the active sets for the different inequalities then provide the boundaries of the trading regions. We use a duality based semi-smooth Newton method to derive an efficient algorithm to find the boundaries numerically.