Mihail Zervos

Bio: Mihail Zervos is an academic researcher from London School of Economics and Political Science. The author has contributed to research in topics: Stochastic control & Optimal stopping. The author has an hindex of 21, co-authored 54 publications receiving 1440 citations. Previous affiliations of Mihail Zervos include Imperial College London & King's College London.

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.

Optimal dividend and issuance of equity policies in the presence of proportional costs

Abstract: We consider three optimisation problems faced by a company that can control its liquid reserves by paying dividends and by issuing new equity. The first of these problems involves no issuance of new equity and has been considered by several authors in the literature. The second one aims at maximising the expected discounted dividend payments minus the expected discounted costs of issuing new equity over all strategies associated with positive reserves at all times. The third problem has the same objective as the second one, but with no constraints on the reserves. Assuming proportional issuance of equity costs, we derive closed form solutions and we completely characterise the optimal strategies. We also provide a relationship between the three problems.

A Model for Investment Decisions with Switching Costs

Abstract: We address the problem of determining in an optimal way the sequence of times at which a firm can enter or exit an economic activity. In particular, we consider an investment model which involves production scheduling as well as a sequence of entry and exit decisions. The pricing of an investment conforming with this model gives rise to a stochastic impulse control problem that we explicitly solve. Our solution takes qualitatively different forms, depending on the problem's data.

A Model for Reversible Investment Capacity Expansion

Abstract: 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 $\infty$. 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.

Pricing a class of exotic options via moments and sdp relaxations

Abstract: 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.

Continuous-time stochastic control and optimization with financial applications / Huyen Pham

Abstract: Stochastic optimization problems arise in decision-making problems under uncertainty, and find various applications in economics and finance. On the other hand, problems in finance have recently led to new developments in the theory of stochastic control. This volume provides a systematic treatment of stochastic optimization problems applied to finance by presenting the different existing methods: dynamic programming, viscosity solutions, backward stochastic differential equations, and martingale duality methods. The theory is discussed in the context of recent developments in this field, with complete and detailed proofs, and is illustrated by means of concrete examples from the world of finance: portfolio allocation, option hedging, real options, optimal investment, etc. This book is directed towards graduate students and researchers in mathematical finance, and will also benefit applied mathematicians interested in financial applications and practitioners wishing to know more about the use of stochastic optimization methods in finance.

On the optimal dividend problem for a spectrally negative Lévy process

Abstract: In this paper we consider the optimal dividend problem for an insurance company whose risk process evolves as a spectrally negative Levy process in the absence of dividend payments. The classical dividend problem for an insurance company consists in finding a dividend payment policy that maximizes the total expected discounted dividends. Related is the problem where we impose the restriction that ruin be prevented: the beneficiaries of the dividends must then keep the insurance company solvent by bail-out loans. Drawing on the fluctuation theory of spectrally negative Levy processes we give an explicit analytical description of the optimal strategy in the set of barrier strategies and the corresponding value function, for either of the problems. Subsequently we investigate when the dividend policy that is optimal among all admissible ones takes the form of a barrier strategy.

A general fractional white noise theory and applications to finance

Abstract: We present a new framework for fractional Brownian motion in which processes with all indices can be considered under the same probability measure. Our results extend recent contributions by Hu, Oksendal, Duncan, Pasik-Duncan, and others. As an application we develop option pricing in a fractional Black-Scholesmarket with a noise process driven by a sum of fractional Brownian motions with various Hurst indices.

Nonlinear Optimal Control via Occupation Measures and LMI-Relaxations

Abstract: We consider the class of nonlinear optimal control problems (OCPs) with polynomial data, i.e., the differential equation, state and control constraints, and cost are all described by polynomials, and more generally for OCPs with smooth data. In addition, state constraints as well as state and/or action constraints are allowed. We provide a simple hierarchy of LMI- (linear matrix inequality)-relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value. Under some convexity assumptions, the sequence converges to the optimal value of the OCP. Preliminary results show that good approximations are obtained with few moments.