Jukka Lempa

On the Optimal Stochastic Impulse Control of Linear Diffusions

We consider a class of stochastic impulse control problems of linear diffusions arising in studies considering the determination of optimal dividend policies. This class of problems appears also in studies analyzing the optimal management of renewable resources. We state a set of weak conditions guaranteeing both existence and uniqueness of the boundary characterizing the optimal policy and its value. We also analyze two associated stochastic control problems and establish a general ordering for both the values and the marginal values of the considered stochastic control problems. In this way we extend previous findings obtained by relying on linear payoff characterizations.

On the optimal exercise of swing options in electricity markets

We study the optimal exercise of a swing option in electricity markets. To this end, we set up a model in terms of a stochastic control problem. In this model, the option can be exercised in continuous time and is subject to a total volume constraint. We analyze some fundamental properties of the model and carry out a numerical analysis. Finally, the results are illustrated numerically.

A note on optimal stopping of diffusions with a two-sided optimal rule

We consider a class of optimal stopping problems of diffusions with a two-sided optimal rule. We propose an approach for finding and characterizing the solution. We establish that the optimal stopping rule can be associated with the unique fixed point of an auxiliary function. The results are illustrated with an explicit example.

A DYNKIN GAME WITH ASYMMETRIC INFORMATION

We study a Dynkin game with asymmetric information. The game has a random expiry time, which is exponentially distributed and independent of the underlying process. The players have asymmetric information on the expiry time, namely only one of the players is able to observe its occurrence. We propose a set of conditions under which we solve the saddle point equilibrium and study the implications of the information asymmetry. Results are illustrated with an explicit example.

Swing options in commodity markets: a multidimensional Lévy diffusion model

We study valuation of swing options on commodity markets when the commodity prices are driven by multiple factors. The factors are modeled as diffusion processes driven by a multidimensional Lévy process. We set up a valuation model in terms of a dynamic programming problem where the option can be exercised continuously in time. Here, the number of swing rights is given by a total volume constraint. We analyze some general properties of the model and study the solution by analyzing the associated HJB-equation. Furthermore, we discuss the issues caused by the multi-dimensionality of the commodity price model. The results are illustrated numerically with three explicit examples.

Optimal portfolios in commodity futures markets

We develop a general approach to portfolio optimization in futures markets. Following the Heath–Jarrow–Morton (HJM) approach, we model the entire futures price curve at once as a solution of a stochastic partial differential equation. We also develop a general formalism to handle portfolios of futures contracts. In the portfolio optimization problem, the agent invests in futures contracts and a risk-free asset, and her objective is to maximize the utility from final wealth. In order to capture self-consistent futures price dynamics, we study a class of futures price curve models which admit a finite-dimensional realization. More precisely, we establish conditions under which the futures price dynamics can be realized in finite dimensions. Using the finite-dimensional realization, we derive a finite-dimensional form of the portfolio optimization problem and study its solution. We also give an economic interpretation of the coordinate process driving the finite-dimensional realization.

Optimal stopping with random exercise lag

We study optimal stopping with exponentially distributed exercise lag. We formalize the problem first in a general Markovian setting and derive a set of conditions under which the solution exists. In particular, no semicontinuity assumptions of the payoff function are needed. We analyze also some specific classes of lagged optimal stopping problems with one-dimensional diffusion dynamics where the solution can be characterized in closed form. Finally, the results are illustrated with an explicit example.

Mathematics of Swing Options: A Survey

This paper is a survey article on mathematical theories and techniques used in the study of swing options. In financial terms, swing options can be regarded as multiple-strike American or Bermudan options with specific constraints on the exerciseability. We focus on two categories of approaches: martingale and Markovian methods. Martingale methods build on purely probabilistic properties of the models whereas Markovian methods draw on the interplay between stochastic control and partial differential equations. We also review other techniques available in the literature.

On Infinite Horizon Optimal Stopping of General Random Walk

The objective of this study is to provide an alternative characterization of the optimal value function of a certain Black–Scholes-type optimal stopping problem where the underlying stochastic process is a general random walk, i.e. the process constituted by partial sums of an IID sequence of random variables. Furthermore, the pasting principle of this optimal stopping problem is studied.

A Class of Solvable Multiple Entry Problems with Forced Exits

We study an optimal investment problem with multiple entries and forced exits. A closed form solution of the optimisation problem is presented for general underlying diffusion dynamics and a general running payoff function in the case when forced exits occur on the jump times of a Poisson process. Furthermore, we allow the investment opportunity to be subject to the risk of a catastrophe that can occur at the jumps of the Poisson process. More precisely, we attach IID Bernoulli trials to the jump times and if the trial fails, no further re-entries are allowed. Interestingly, we find in the general case that the optimal investment threshold is independent of the success probability is the Bernoulli trials. The results are illustrated with explicit examples.