Mihail Zervos

Dynamical pricing of weather derivatives

Abstract The dynamics of temperature can be modelled by means of a stochastic process known as fractional Brownian motion. Based on this empirical observation, we characterize temperature dynamics by a fractional Ornstein–Uhlenbeck process. This model is used to price two types of contingent claims: one based on heating and cooling degree days, and one based on cumulative temperature. We derive analytic expressions for the expected discounted payoffs of such derivatives, and discuss the dependence of the results on the fractionality of the temperature dynamics.

A Model for Reversible Investment Capacity Expansion

We consider the problem of determining the optimal investment level that a firm should maintain in the presence of random price and/or demand fluctuations. We model market uncertainty by means of a geometric Brownian motion, and we consider general running payoff functions. Our model allows for capacity expansion as well as for capacity reduction, with each of these actions being associated with proportional costs. The resulting optimization problem takes the form of a singular stochastic control problem that we solve explicitly. We illustrate our results by means of the so-called Cobb-Douglas production function. The problem that we study presents a model in which the associated Hamilton-Jacobi-Bellman equation admits a classical solution that conforms with the underlying economic intuition but does not necessarily identify with the corresponding value function, which may be identically equal to

A Problem of Sequential Entry and Exit Decisions Combined with Discretionary Stopping

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Pricing a Class of Exotic Options Via Moments and SDP Relaxations

. Thus, our model provides a situation that highlights the need for rigorous mathematical analysis when addressing stochastic optimization applications in finance and economics, as well as in other fields.

A model for investments in the natural resource industry with switching costs

We consider a stochastic control problem that has emerged in the economics literature as an investment model under uncertainty. This problem combines features of both stochastic impulse control and optimal stopping. The aim is to discover the form of the optimal strategy. It turns out that this has a priori rather unexpected features. The results that we establish are of an explicit nature. We also construct an example whose value function does not possess C1 regularity.

Finite-fuel singular control with discretionary stopping

We present a new methodology for the numerical pricing of a class of exotic derivatives such as Asian or barrier options when the underlying asset price dynamics are modeled by a geometric Brownian motion or a number of mean-reverting processes of interest. This methodology identifies derivative prices with infinite-dimensional linear programming problems involving the moments of appropriate measures, and then develops suitable finite-dimensional relaxations that take the form of semidefinite programs (SDP) indexed by the number of moments involved. By maximizing or minimizing appropriate criteria, monotone sequences of both upper and lower bounds are obtained. Numerical investigation shows that very good results are obtained with only a small number of moments. Theoretical convergence results are also established.

Buy‐Low and Sell‐High Investment Strategies

We consider a model for investment decisions in the natural resource industry with switching costs. This model gives rise to a problem combining features of both absolutely continuous and impulse stochastic control that we explicitly solve. The solution takes qualitatively different forms, depending on parameter values.

Valuation of Investments in Real Assets with Implications for the Stock Prices

We discuss the finite-fuel, singular stochastic control problem of optimally tracking the standard Brownian motion started at , by an adapted process of bounded total variation , so as to minimize the total expected discounted cost over such processes and stopping times τ. Here , and are given real numbers. In its form this problem goes back to the seminal paper of Bene[sbreve], Shepp and Witsenhausen (1980). For fixed α>0 and δ>0 we characterize explicitly the optimal policy in the case λ>αδ (of the “act-or-stop” type, since the continuation cost is relatively large), and in the case with (of the “act, stop, or wait” type, since the relative continuation cost is relatively small). In the latter case, an associated free-boundary problem is solved exactly. The case , of “moderate” relative continuation cost, is suggested as an open question

Global eradication of lymphatic filariasis: the value of chronic disease control in parasite elimination programmes.

Buy-low and sell-high investment strategies are a recurrent theme in the considerations of many investors. In this paper, we consider an investor who aims at maximizing the expected discounted cash-flow that can be generated by sequentially buying and selling one share of a given asset at fixed transaction costs. We model the underlying asset price by means of a general one-dimensional Ito diffusion X, we solve the resulting stochastic control problem in a closed analytic form, and we completely characterize the optimal strategy. In particular, we show that, if 0 is a natural boundary point of X, e.g., if X is a geometric Brownian motion, then it is never optimal to sequentially buy and sell. On the other hand, we prove that, if 0 is an entrance point of X, e.g., if X is a mean-reverting constant elasticity of variance (CEV) process, then it may be optimal to sequentially buy and sell, depending on the problem data.

The explicit solution to a sequential switching problem with non-smooth data

A general model for the valuation of natural resource investments is formulated and analyzed within a stochastic control theoretic framework. Using dynamic programming, the value of such an investment with a general payoff function is determined under the assumption that the commodity price process is given by a stochastic differential equation. The analysis results in closed form analytic solutions which can easily be computed and exhibits qualitatively different optimal behaviors, depending on parameter values. Implications for stocks and options are also considered.