Carl Lindberg

Optimal Liquidation of a Pairs Trade

Pairs trading is a common strategy used by hedge funds. When the spread between two highly correlated assets is observed to deviate from historical observations, a long position is taken in the underpriced asset, and a short position in the overpriced one. If the spread narrows, both positions are closed, thus generating a profit. We study when to optimally liquidate a pairs trading strategy when the difference between the two assets is modeled by an Ornstein–Uhlenbeck process. We also provide a sensitivity analysis in the model parameters.

News-Generated Dependence and Optimal Portfolios for N Stocks in a Market of Barndorff-Nielsen and Shephard Type

We consider Mertons portfolio optimization problem in a Black and Scholes market with non-Gaussian stochastic volatility of Ornstein-Uhlenbeck type. The investor can trade in n stocks and a risk-free bond. We assume that the dependence between stocks lies in that they partly share the Ornstein-Uhlenbeck processes of the volatility. We refer to these as news processes, and interpret this as that dependence between stocks lies solely in their reactions to the same news. The model is primarily intended for assets that are dependent, but not too dependent, such as stocks from different branches of industry. We show that this dependence generates covariance, and give statistical methods for both the fitting and verification of the model to data. Using dynamic programming, we derive and verify explicit trading strategies and Feynman-Kac representations of the value function for power utility.

Portfolio Optimization and a Factor Model in a Stochastic Volatility Market

The aim of this paper is to find optimal portfolio strategies for an n-stock extension of the stochastic volatility model proposed in Ref. [2]. It is a modification of [19], and characterizes the dependence by the use of a factor structure. The idea of a factor structure is that the diffusion components of the stocks contain one Brownian motion that is unique for each stock, and a few Brownian motions that all stocks share. Our model can obtain strong correlations between the returns for different stocks without affecting their marginal distributions. This was not possible in Ref. [19]. Further, the number of model parameters does not grow too fast as the number of stocks n grows. We use dynamic programming to solve Merton’s optimization problem for power utility, with utility drawn from terminal wealth. Optimal strategies for n stocks are obtained. We also discuss how to estimate the model from data. This is illustrated with an example.

Game Intelligence in Team Sports

We set up a game theoretic framework to analyze a wide range of situations from team sports. A fundamental idea is the concept of potential; the probability of the offense scoring the next goal minus the probability that the next goal is made by the defense. We develop categorical as well as continuous models, and obtain optimal strategies for both offense and defense. A main result is that the optimal defensive strategy is to minimize the maximum potential of all offensive strategies.

Optimal Liquidation of a Call Spread

We study the optimal liquidation strategy for a call spread in the case when an investor, who does not hedge, believes in a volatility that differs from the implied volatility. The liquidation problem is formulated as an optimal stopping problem, which we solve explicitly. We also provide a sensitivity analysis with respect to the model parameters.

Optimal closing of a pair trade with a model containing jumps

A pair trade is a portfolio consisting of a long position in one asset and a short position in another, and it is a widely used investment strategy in the financial industry. Recently, Ekström, Lindberg, and Tysk studied the problem of optimally closing a pair trading strategy when the difference of the two assets is modelled by an Ornstein-Uhlenbeck process. In the present work the model is generalized to also include jumps. More precisely, we assume that the difference between the assets is an Ornstein-Uhlenbeck type process, driven by a Lévy process of finite activity. We prove a necessary condition for optimality (a so-called verification theorem), which takes the form of a free boundary problem for an integro-differential equation. We analyze a finite element method for this problem and prove rigorous error estimates, which are used to draw conclusions from numerical simulations. In particular, we present strong evidence for the existence and uniqueness of an optimal solution.

Merton’s problem for an investor with a benchmark in a Barndorff-Nielsen and Shephard market

To try to outperform an externally given benchmark with known weights is the most common equity mandate in the financial industry. For quantitative investors, this task is predominantly approached by optimizing their portfolios consecutively over short time horizons with one-period models. We seek in this paper to provide a theoretical justification to this practice when the underlying market is of Barndorff-Nielsen and Shephard type. This is done by verifying that an investor who seeks to maximize her expected terminal exponential utility of wealth in excess of her benchmark will in fact use an optimal portfolio equivalent to the one-period Markowitz mean-variance problem in continuum under the corresponding Black-Scholes market. Further, we can represent the solution to the optimization problem as in Feynman-Kac form. Hence, the problem, and its solution, is analogous to Merton’s classical portfolio problem, with the main difference that Merton maximizes expected utility of terminal wealth, not wealth in excess of a benchmark.

A note on contracts on quadratic variation

Given a Black stochastic volatility model for a future F, and a function g, we show that the price of 12∫0Tg(t,F(t))F2(t)σ2(t)dt can be represented by portfolios of put and call options. This generalizes the classical representation result for the variance swap. Further, in a local volatility model, we give an example based on Dupire’s formula which shows how the theorem can be used to design variance related contracts with desirable characteristics.