Mihail Zervos

The London School of Economics and Political Science | LSE · Department of Mathematics

We study the mean–variance hedging of an American type contingent claim that is exercised at a random time in a Markovian setting. This problem is motivated by applications in the areas of employee stock option valuation, credit risk, or equity-linked life insurance policies with an underlying risky asset value guarantee. Our analysis is based on d...

We derive the explicit solution to a singular stochastic control problem of the monotone follower type with an expected ergodic criterion as well as to its counterpart with a pathwise ergodic criterion. These problems have been motivated by the optimal sustainable exploitation of an ecosystem, such as a natural fishery. Under general assumptions on...

We consider the problem of optimally stopping a general one-dimensional stochastic differential equation (SDE) with generalised drift over an infinite time horizon. First, we derive a complete characterisation of the solution to this problem in terms of vari- ational inequalities. In particular, we prove that the problem’s value function is the dif...

We consider the problem of optimally stopping a general one-dimensional stochastic differential equation (SDE) with generalised drift over an infinite time horizon. First, we derive a complete characterisation of the solution to this problem in terms of variational inequalities. In particular, we prove that the problem's value function is the diffe...

We consider an investment project that operates within a random environment and yields a payoff rate that is a function of a stochastic economic indicator such as the price of or the demand for the project's output commodity. We assume that the investment project can operate in two modes, an "open" one and a "closed" one. The transitions from one o...

We consider the problem of ESO valuation in continuous time. In particular, we consider models that assume that an appropriate random time serves as a proxy for anything that causes the ESO's holder to exercise the option early, namely, reflects the ESO holder's job termination risk as well as early exercise behaviour. In this context, we study the...

We study managerial incentive provision under moral hazard when growth opportunities arrive stochastically and pursuing them requires a change in management. A trade-off arises between the benefit of always having the “right” manager and the cost of incentive provision. The prospect of growth-induced turnover limits the firm's ability to rely on de...

We consider an irreversible capacity expansion model in which additional investment has a strictly negative effect on the value of an underlying stochastic economic indicator. The associated optimisation problem takes the form of a singular stochastic control problem that admits an explicit solution. A special characteristic of this stochastic cont...

We consider a new family of derivatives whose payoffs become strictly positive when the price of their underlying asset falls relative to its historical maximum. We derive the solution to the discretionary stopping problems arising in the context of pricing their perpetual American versions by means of an explicit construction of their value functi...

We consider the decreasing and the increasing $r$-excessive functions $\varphi_r$ and $\psi_r$ that are associated with a one-dimensional conservative regular continuous strong Markov process $X$ with values in an interval with endpoints $\alpha < \beta$. We prove that the $r$-excessive local martingale $\bigl( e^{-r (t \wedge T_\alpha)} \varphi_r...

S. Kotani (2006) has characterised the martingale property of a one-dimensional diffusion in natural scale in terms of the classification of its boundaries. We complement this result by establishing a necessary and sufficient condition for a one-dimensional diffusion in natural scale to be a submartingale or a supermartingale. Furthermore, we study...

We consider the so-called optimal execution problem in algorithmic trading, which is the problem faced by an investor who has a large number of stock shares to sell over a given time horizon and whose actions have an impact on the stock price. In particular, we develop and study a price model that presents the stochastic dynamics of a geometric Bro...

We consider a stochastic differential equation that is controlled by means of an additive �nite-variation process. A singular stochastic controller, who is a minimiser, determines this infinite-variation process while a discretionary stopper, who is a maximiser, chooses a stopping time at which the game terminates. We consider two closely related g...

We consider a one-dimensional diffusion which solves a stochastic differential equation with Borel-measurable coefficients in an open interval. We allow for the endpoints to be inaccessible or absorbing. Given a Borel-measurable function $r$ that is uniformly bounded away from 0, we establish a new analytic representation of the $r$-potential of a...

We consider an investment project that produces a single commodity. The project’s operation yields payoff at a rate that depends on the project’s installed capacity level and on an underlying economic indicator such as the output commodity’s price or demand, which we model by an ergodic, one-dimensional Itˆo diffusion. The project’s capacity level...

We formulate and solve a problem that combines the features of the so-called monotone follower of singular stochastic control theory with optimal stopping. In particular, we consider a stochastic system whose uncontrolled state dynamics are modelled by a general one-dimensional Itô diffusion. The aim of the problem that we solve is to maximise the...

Buy-low and sell-high investment strategies are a recurrent theme in the considera-tions of many investors. In this paper, we consider an investor who aims at maximising 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 pric...

World leading experts give their accounts of the modern mathematical models in the field: Markov Decision Processes, controlled diffusions, piece-wise deterministic processes etc, with a wide range of performance functionals. One of the aims is to give a general view on the state-of-the-art. The authors use Dynamic Programming, Convex Analytic Appr...

We consider a discretionary stopping problem that arises in the context of pricing a class of perpetual American-type call options, which include the perpetual American, Russian and lookback-American call options as special cases. We solve this genuinely two-dimensional optimal stopping problem by means of an explicit construction of its value func...

We consider a discretionary stopping problem that arises in the context of pricing a class of perpetual American-type call options, which include the perpetual American, Russian and lookback-American call options as special cases. We solve this genuinely two-dimensional optimal stopping problem by means of an explicit construction of its value func...

We consider the problem faced by a decision maker who can switch between two random payoff flows. Each of these payoff flows is an additive functional of a general 1D Ito diffusion. There are no bounds on the number or on the frequency of the times at which the decision maker can switch, but each switching incurs a cost, which may depend on the und...

Details of data and studies used in the analysis shown in Figure 1. (0.17 MB DOC)

Logit proportions ( = log odds) of chronic LF disease against individual study (open circles) and pentiles (closed circles) of mf prevalence (%) values. The figures shown on the graph represent the chi-square statistic and p - values obtained by applying the chi-square test described in the text for data grouped into pentiles of mf prevalence. (6.7...

Adequacy of fit of the logistic dose-response regression model with a threshold (0.03 MB DOC)

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 ultimate goal of the global programme against lymphatic filariasis is eradication through irrevocable cessation of transmission using 4 to 6 years of annual single dose mass drug administration. The costs of eradication, and logistical and managerial impediments to executing national and regional control programmes, and scientific uncertainty a...

The ultimate goal of the global programme against lymphatic filariasis is eradication through irrevocable cessation of transmission using 4 to 6 years of annual single dose mass drug administration. The costs of eradication, managerial impediments to executing national control programmes, and scientific uncertainty about transmission endpoints, are...

We consider a stochastic system whose uncontrolled state dynamics are modelled by a general one-dimensional It\^{o} diffusion. The control effort that can be applied to this system takes the form that is associated with the so-called monotone follower problem of singular stochastic control. The control problem that we address aims at maximising a p...

We address the problem of controlling a general one-dimensional Itô diffusion within an externally specified bounded interval [A, B] by means of an impulse control process. We minimise a long-term average criterion that penalises deviations of the state process from a given nominal point within this region as well as the use of impulsive control ef...

We consider the solvability of the ordinary differential equation (ODE)inside an interval , where σ, b, r are given functions and h is a locally finite measure. This ODE is associated with the Hamilton–Jacobi–Bellman (HJB) equations arising in the study of a wide range of stochastic optimisation problems. These problems are motivated by numerous ap...

We consider a problem that combines impulse control with absolutely continuous control of the drift of a general one-dimensional Ito diffusion. The objec- tive of the control problem is to minimize an ergodic or long-term average criterion that penalizes both deviations of the state process from a given nominal point and the use of control effort....

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,...

We consider the problem of determining in a dynamical way the optimal capacity level of an investment project that operates within a random economic environ- ment. In particular, we consider an investment project that yields a payoff at a rate that depends on its installed capacity level and on a random economic indicator such as, for instance, the...

We formulate an abstract mathematical model for investments in real assets from the perspective of the real options approach. We then derive an analytic expression for its fair price under a market completeness assumption. This expression is the solution of a stochastic optimisation problem. Also, we consider certain associated control theoretic as...

We consider the problem of optimally stopping a general one-dimensional Itô diffusion X. In particular, we solve the problem that aims at maximising the performance criterion E x [exp(-∫0τr(X s )ds)f(Xτ)] over all stopping times τ, where the reward function f can take only a finite number of values and has a ‘staircase’ form. This problem is partly...

We consider the problem of optimally stopping a general one-dimensional Itô diffusion X. In particular, we solve the problem that aims at maximising the performance criterion E_x[exp(-∫₀^τr(X_s)ds)f(X_τ)] over all stopping times τ, where the reward function f can take only a finite number of values...

We consider the problem of controlling a general one-dimensional Itô diffusion by means of a finite-variation process. The objective is to minimise a long-term average expected criterion as well as a long-term pathwise criterion that penalise deviations of the underlying state process from a given nominal point as well as the expenditure of control...

We consider the problem of controlling a general one-dimensional Ito diffusion by means of an impulse control process. The objective is to minimise a long-term expected criterion as well as a long-term pathwise criterion that penalise both deviations of the state process from a given nominal point and the use of impulsive control effort. In particu...

We consider the problem of optimally stopping a general one-dimensional Itô dif-fusion X that aims at maximising the performance index E x exp − τ 0 r(X t) dt f (X τ)1 {τ <∞} over all stopping times τ . We assume that the reward function f is Borel measurable and we allow for the possibility that it is unbounded. We show that the value function v o...

We consider a variant of an optimisation problem involving sequential entry and exit decisions that has emerged in the economics literature as a real option model. The problem that we solve aims at maximising an ergodic, or long-term average, performance criterion in a pathwise as well as in an expected sense. Such a performance index is probably b...

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 p...

We consider the discretionary stopping problem that aims at maximising the per-formance criterion E x e − Ê τ 0 r(Xs)ds g(X τ)1 {τ <∞} over all stopping times τ , where X is a general one-dimensional positive Itô diffusion, r is a strictly positive function and g is a given payoff function. This optimal stopping problem has several applications in...

We consider the problem of pricing weather derivatives based on linear tempera-ture indices. Anticipating the development of a liquid weather swap market, we address the issue of pricing weather derivative options using weather swaps as hedging instru-ments. Our analysis starts by considering stochastic dynamics that are appropriate for the modelli...

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 t...

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 base...

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.

We formulate a general mathematical model for investments in real assets from the perspective of the real options approach. We then derive an analytic expression for its price under a market completeness assumption. This expression is the solution of a stochastic optimization problem. The generality of the model is such that it can also provide a f...

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 stochasti...

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...

We consider an investment model which generalizes a number of models that have been studied in the literature. The model involves entry and exit decisions as well as decisions relating to production scheduling. We then address the problem of its valuation from the standpoint of the dynamic programming approach. Our analysis results in a closed form...

We consider an investment model which generalizes a number of models that have been studied in the literature. The model involves entry and exit decisions as well as decisions relating to production scheduling. We then address the problem of its valuation from the standpoint of the dynamic programming approach. Our analysis results in a closed form...

We consider a stochastic control problem that has emerged in the economics literature as an investment model under uncertainty. This problem combines some of the features of stochastic impulse control with optimal stopping. The aim is to discover the form of the optimal strategy. The results that we establish are of an explicit nature

We consider a general model for an investment producing a single commodity, and, assuming that there exists a traded asset spanning the corresponding market, we prove a "verification theorem" which relates the solution of an appropriate differential equation with the investment's contingent claim price. In this way, we show in a mathematically rigo...

The problem of strong consistency of sequences of optimal solutions to stochastic optimization problems is considered. This problem is related to a large number of applications including Bayesian decision problems and Monte Carlo simulations, as well as a number of statistical methodologies such as maximum likelihood estimation. The theory of epico...

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. Th...

We consider a general model of singular stochastic control with infinite time horizon and we prove a ``verification theorem'' under the assumption that the Hamilton—Jacobi—Bellman (HJB) equation has a C 2 solution. In the one-dimensional case, under the assumption that the HJB equation has a solution in W loc2,p(R) with \(p \geq 1\) , we prove a ve...

A general model for the valuation of an investment producing a single commodity is formulated and analysed 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 differentia...

Presents a unifying new proof for the three discrete-time linear quadratic Gaussian problems (deterministic, stochastic full information, and stochastic partial information) based on pathwise (deterministic) optimization. The essential difference between the control aspect of the three cases is that the controls should lie in different classes of “...

In this paper a simple problem of combined singular stochastic control and optimal stopping is formulated and solved. We find that the optimal strategies can take qualitatively different forms, depending on parameter values. We also study a variant on the problem in which the value function is inherently nonconvex. The proofs employ the generalised...