Thaleia Zariphopoulou

• Chair at University of Texas at Austin and the Oxford-Man Institute, Oxford

Motivated by optimal allocation models with relative performance criteria, we introduce a mean field game in which the terminal expected utility of the representative agent depends on her own state as well as the average of her peers. We derive the master equation, which, in view of the presence of controls in the volatility, needs to be coupled wi...

I step out of the van and straight into a real-world approximation of J.R.R. Tolkien’s Lothlórien, which was the famous Elven forest lying in a woodland valley. I am surrounded by large red and white pine trees. Although the pines are clearly dominating the landscapes, there are occasional low-lying cedar trees interspersed with tall ferns and moss...

Entropy regularization has been extensively adopted to improve the efficiency, the stability, and the convergence of algorithms in reinforcement learning. This paper analyzes both quantitatively and qualitatively the impact of entropy regularization for mean field games (MFGs) with learning in a finite time horizon. Our study provides a theoretical...

This Special Issue of Mathematical Finance celebrates the memory of Mark H. A. Davis, one of the founding editors of the journal, and his numerous contributions to mathematical finance.

Automated investment managers, or robo-advisors, have emerged as an alternative to traditional financial advisors. The viability of robo-advisors crucially depends on their ability to offer personalized financial advice. We introduce a novel framework in which a robo-advisor interacts with a client to solve an adaptive mean-variance portfolio optim...

The paper introduces and analyzes $N$-player and common-noise mean-field games in It\^{o}-diffusion environments, in the context of both complete and incomplete financial markets. The players invest in a finite horizon and in a common market by either competitive or homophilous interactions. In both kinds of market environments, the players have in...

We introduce the concept of forward rank‐dependent performance criteria, extending the original notion to forward criteria that incorporate probability distortions. A fundamental challenge is how to reconcile the time‐consistent nature of forward performance criteria with the time‐inconsistency stemming from probability distortions. For this, we fi...

In an Itô-diffusion market, two fund managers trade under relative performance concerns. For both the asset specialization and diversification settings, we analyze the passive and competitive cases. We measure the performance of the managers' strategies via forward relative performance criteria, leading to the respective notions of forward best-res...

Entropy regularization has been extensively adopted to improve the efficiency, the stability, and the convergence of algorithms in reinforcement learning. This paper analyzes both quantitatively and qualitatively the impact of entropy regularization for Mean Field Game (MFG) with learning in a finite time horizon. Our study provides a theoretical j...

We consider reinforcement learning (RL) in continuous time with continuous feature and action spaces. We motivate and devise an exploratory formulation for the feature dynamics that captures learning under exploration, with the resulting optimization problem being a revitalization of the classical relaxed stochastic control. We then study the probl...

Medicine is, in its essence, decision making under uncertainty; The decisions are made about tests to be performed and treatments to be administered. Traditionally the uncertainty in decision making was handled using expertise collected by individual providers, and more recently systematic appraisal of research in the form of evidence-based medicin...

We study the analyticity of the value function in optimal investment with expected utility from terminal wealth. We identify both a class of utilities and a class of semi-martingale models for which we establish analyticity. Specifically, these utilities have completely monotonic inverse marginals, while the market models have a maximal element in...

Automated investment managers, or robo-advisors, have emerged as an alternative to traditional financial advisors. Their viability crucially depends on timely communication of information from the clients they serve. We introduce and develop a novel human-machine interaction framework, in which the robo-advisor solves an adaptive mean-variance cont...

We introduce the concept of forward rank-dependent performance processes, extending the original notion to forward criteria that incorporate probability distortions. A fundamental challenge is how to reconcile the time-consistent nature of forward performance criteria with the time-inconsistency stemming from probability distortions. For this, we f...

Using elements from the theory of ergodic backward stochastic differential equations (BSDE), we study the behavior of forward entropic risk measures in stochastic factor models. We derive general representation results (via both BSDE and convex duality) and examine their asymptotic behavior for risk positions of large maturities. We also compare th...

We consider reinforcement learning (RL) in continuous time and study the problem of achieving the best trade-off between exploration of a black box environment and exploitation of current knowledge. We propose an entropy-regularized reward function involving the differential entropy of the distributions of actions, and motivate and devise an explor...

We combine forward investment performance processes and ambiguity-averse portfolio selection. We introduce robust forward criteria which address ambiguity in the specification of the model, the risk preferences and the investment horizon. They encode the evolution of dynamically consistent ambiguity-averse preferences. We focus on establishing dual...

In this paper, we construct a solution to the optimal contract problem for delegated portfolio management of the fist-best (risk-sharing) type. The novelty of our result is (i) in the robustness of the optimal contract with respect to perturbations of the wealth process (interpreted as capital injections), and (ii) in the more general form of princ...

In this paper, we construct a solution to the optimal contract problem for delegated portfolio management of the fist-best (risk-sharing) type. The novelty of our result is (i) in the robustness of the optimal contract with respect to perturbations of the wealth process (interpreted as capital injections), and (ii) in the more general form of princ...

We propose a new splitting algorithm to solve a class of semilinear parabolic PDEs with convex and quadratic growth gradients. By splitting the original equation into a linear parabolic equation and a Hamilton-Jacobi equation, we are able to solve both equations explicitly. In particular, we solve the associated Hamilton-Jacobi equation by the Hopf...

We analyze a nonlinear equation proposed by F. Black (1968) for the optimal portfolio function in a log-normal model. We cast it in terms of the risk tolerance function and provide, for general utility functions, existence, uniqueness and regularity results, and we also examine various monotonicity, concavity/convexity and S-shape properties. Stron...

We analyze a family of portfolio management problems under relative performance criteria, for fund managers having CARA or CRRA utilities and trading in a common time horizon in log-normal markets. We construct explicit time-independent equilibrium strategies for both the finite population games and the corresponding mean field games, which we show...

We analyze a family of portfolio management problems under relative performance criteria, for fund managers having CARA or CRRA utilities and trading in a common investment horizon in log-normal markets. We construct explicit constant equilibrium strategies for both the finite population games and the corresponding mean field games, which we show a...

We present turnpike-type results for the risk tolerance function in an incomplete market setting under time-monotone forward performance criteria. We show that, contrary to the classical case, the temporal and spatial limits do not coincide. We also show that they depend directly on the left- and right-end of the support of an underlying measure, w...

We present turnpike-type results for the risk tolerance function in an incomplete market setting under time-monotone forward performance criteria. We show that, contrary to the classical case, the temporal and spatial limits do not coincide. We also show that they depend directly on the left- and right-end of the support of an underlying measure, w...

In an incomplete market, with incompleteness stemming from stochastic factors imperfectly correlated with the underlying stocks, we derive representations of homothetic (power, exponential, and logarithmic) forward performance processes in factor-form using ergodic BSDE. We also develop a connection between the forward processes and infinite horizo...

We introduce a new class of forward performance processes that are predictable with regards to an underlying filtration and are updated in discrete time. Such performance criteria may accommodate short-term predictability of asset returns, sequential learning and other dynamically unfolding factors affecting optimal portfolio choice. We analyze in...

We introduce a new class of forward performance processes that are endogenous and predictable with regards to an underlying market information set and, furthermore, are updated at discrete times. We analyze in detail a binomial model whose parameters are random and updated dynamically as the market evolves. We show that the key step in the construc...

This paper shows that the long-time behavior of the entropic risk measure (under both forward performance process framework and classical utility framework) converges to a constant, which is independent of the initial state of the stochastic factors in a stochastic factor model. The exponential convergence rate to the long-term limit is also obtain...

We analyze the optimal portfolio choices of portfolio managers under competitive pressure when there is no clear time horizon. To this end, we introduce the relative time-monotone forward performance criteria and derive the portfolio strategies that are optimal under such criteria for two competing managers but also constitute a Nash equilibrium.

Using elements from the theory of ergodic backward stochastic differential equations, we study the behavior of forward entropic risk measures. We provide a general representation result and examine their behavior for risk positions of large maturities. We also compare them with their classical counterparts and derive a parity result.

In an incomplete market, with incompleteness stemming from stochastic factors imperfectly correlated with the underlying stocks, we derive representations of homothetic forward investment performance processes (power, exponential and logarithmic). We develop a connection with ergodic and infinite horizon quadratic BSDE, and with a risk-sensitive co...

In an incomplete market, with incompleteness stemming from stochastic factors imperfectly correlated with the underlying stocks, we derive representations of homothetic (power, exponential and logarithmic) forward performance processes in factor-form using ergodic BSDE. We also develop a connection between the forward processes and infinite horizon...

We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. The dynamics of the prices of the traded assets depend on a pair of stochastic factors, namely, a slow factor (e.g. a macroeconomic indicator) and a fast factor (e.g. stochastic volatility). We analyze the associated forward...

Using a stochastic representation of the optimal wealth process in the classical Merton problem, we calculate its cumulative distribution and density functions and provide bounds and monotonicity results for these quantities under general risk preferences. We also show that the optimal wealth and portfolio processes for different utility functions...

The present volume is dedicated to Marek Musiela, an eminent scholar and practitioner who is perhaps best-known for his important contributions to problems of derivative pricing, theory of term structure of interest rates, theory of defaultable securities and other topics in modern mathematical finance. It includes 25 research papers by 47 authors,...

Articles from many of the main contributors to recent progress in stochastic analysis are included in this volume, which provides a snapshot of the current state of the area and its ongoing developments. It constitutes the proceedings of the conference on "Stochastic Analysis and Applications" held at the University of Oxford and the Oxford-Man Ins...

We provide an approximation scheme for the maximal expected utility and optimal investment policies for the portfolio choice problem in an incomplete market. Incompleteness stems from the presence of a stochastic factor which affects the dynamics of the correlated stock price. The scheme is built on the Trotter-Kato approximation and is based on an...

We study the Merton portfolio optimization problem in the presence of stochastic volatility using asymptotic approximations when the volatility process is characterized by its time scales of fluctuation. This approach is tractable because it treats the incomplete markets problem as a perturbation around the complete market constant volatility probl...

We analyze the American option valuation problem with the forward performance criterion introduced by Musiela and Zariphopoulou (2008). In this framework, utility evolves forward in time without reference to a specific future time horizon. Moreover, risk preferences change with stochastic market conditions, which is natural as investors are clearly...

We study forward investment performance processes with non-zero forward volatility. We focus on the class of homothetic preferences in a single stochastic factor model. The forward performance process is represented in a closed-form via a deterministic function of the wealth and the stochastic factor. This function is, in turn, given as a distortio...

The paper offers a new perspective on optimal portfolio choice by investigating how and to what extent knowledge of an investor's desirable initial investment choice can be used to determine his future optimal portfolio allocations. Optimality of investment decisions is built on the so-called forward investment performance criteria and, in particul...

We study the impact of risk-aversion on the valuation of credit derivatives. Using the technology of utility-indifference pricing in intensity-based models of default risk, we analyse resulting yield spreads in multi-name credit derivatives, particularly CDOs. We study first the idealized problem with constant intensities where solutions are essent...

We introduce a stochastic partial differential equation which describes the evolution of the investment performance process in portfolio choice models. The equation is derived for two formulations of the investment problem, namely, the traditional one (based on maximal expected utility of terminal wealth) and the recently developed forward formulat...

The class of time-decreasing forward performance processes is analyzed in a portfolio choice model of Itô-type asset dynamics. The associated optimal wealth and portfolio processes are explicitly constructed and their probabilistic properties are discussed. These formulae are, in turn, used in analyzing how the investor's preferences can be calibra...

The indierence valuation problem in incomplete binomial models is analyzed. The model is more general than the ones studied so far, because the stochastic factor, which generates the market incompleteness, may aect the transition propabilities and/or the values of the traded asset as well as the claim’s payo. Two pricing algorithms are constructed...

Using forward optimality criteria, we analyze a portfolio choice problem when the local risk tolerance is time dependent and asymptotically linear in wealth. This class corresponds to a dynamic extension of the traditional (static) risk tolerances associated with the power, logarithmic, and exponential utilities. We provide explicit solutions for t...

A new dynamic criterion for measuring the performance of self-financing investment strategies is introduced. To this aim, a family of stochastic processes defined on [0, ∞) and indexed by a wealth argument is used. Optimality is associated with their martingale property along the optimal wealth trajectory. The optimal portfolios are constructed via...

This paper deals with the problem one faces when the maturity (horizon, expiration date, etc.) associated with a particular risky position is not fixed. We take the view that the mechanism used to measure the risk content of a certain random variable should not depend on any a priori choice of the measurement horizon. The case in complete financial...

This paper provides an overview of the optimal investment problem in a market in which the dynamics of the risky security are aected by a correlated stochastic factor. The performance of investment strategies is measured using two criteria. The …rst criterion is the traditional one, formulated in terms of expected utility from terminal wealth while...

This article provides an overview of risk-neutral valuation methodology and presents historical milestones in the development of quantitative finance. It also discusses current challenges and new perspectives in model choice, pricing and hedging.

We discuss the valuation of credit derivatives in extreme regimes such as when the time-to-maturity is short, or when payoff is contingent upon a large number of defaults, as with senior tranches of collateralized debt obligations. In these cases, risk aversion may play an important role, especially when there is little liquidity, and utility indif...

The new notion of maturity-independent risk measures is introduced and contrasted with the existing risk measurement concepts. It is shown, by means of two examples, one set on a finite probability space and the other in a diffusion framework, that, surprisingly, some of the widely utilized risk measures cannot be used to build maturity-independent...

The new notion of maturity-independent risk measures is introduced and contrasted with the existing risk measurement concepts. It is shown, by means of two examples, one set on a finite probability space and the other in a diffusion framework, that, surprisingly, some of the widely utilized risk measures cannot be used to build maturity-independent...

We introduce a new class of dynamic utilities that are generated forward in time. We discuss the associated value functions, optimal investments, and indifference prices and we compare them with their traditional counterparts, implied by backward dynamic utilities.

We study the effect of risk aversion on the valuation of credit derivatives. Using the technology of utility-indiffierence valuation in intensity-based models of default risk, we analyze resulting yield spreads for single-name defaultable bonds and a simple representative two-name credit derivative. The impact of risk averse valuation on prices and...

We study the impact of risk-aversion on the valuation of credit derivatives. Using the technology of utility-indifference pricing in intensity-based models of default risk, we analyze resulting yield spreads in both simple single-name credit derivatives, and complex multi-name securities, particularly CDOs. We introduce the diversity coeffi-cient t...

This work presents a novel concept in stochastic optimization, namely, the notion of forward performance. As an application, we analyze a portfolio management problem with exponential criteria. Under minimal model assumptions we explicitly construct the forward performance process and the associated optimal wealth and asset allocations. For various...

We study the optimal investment and consumption problem of a CRRA investor when the drift and volatility of the stock are driven by a correlated factor. The myopic and non-myopic components of the optimal portfolio process are characterised in terms of the market price of traded and non-traded risk of the minimax martingale measure. We find that th...

A probabilistic iterative algorithm is constructed for indifference prices of claims in a multiperiod incomplete model. At each time step, a nonlinear pricing functional is applied that isolates and prices separately the two types of risk. It is represented solely in terms of risk aversion and the pricing measure, a martingale measure that preserve...

The aim herein is to analyze utility-based prices and hedging strategies. The analysis is based on an explicitly solved example of a European claim written on a nontraded asset, in a model where risk preferences are exponential, and the traded and nontraded asset are diffusion processes with, respectively, lognormal and arbitrary dynamics. Our resu...

This paper is a contribution to the valuation of derivative securities in a stochastic volatility framework, which is a central problem in financial mathematics. The derivatives to be priced are of European type with the payoff depending on both the stock and the volatility. The valuation approach uses utility-based criteria under the assumption of...

We provide a structural charaterization result for the non-myopic op-timal portfolio of a CARA agent who invests in an incomplete market environment. The excess risky demand turns out to be the indifference risk monitoring strategy of an emerging claim written on the traded asset's Sharpe ratio and the risk tolerance of the investor. Sensitivity re...

The valuation of early exercise claims written on traded and nontraded assets via the utility based approach is studied. It is shown that the indifference price satisfies a quasilinear variational inequality with an obstacle term. It is also established that it is the solution to an optimal stopping problem of a nonlinear criterion related to the E...

We present a utility-based methodology for the valuation of early exercise contracts in incomplete markets. Incompleteness stems from nontraded assets on which the contracts are written. This methodology takes into account the individual’s attitude towards risk and yields nonlinear pricing rules. The early exercise indifference prices solve a quasi...

We consider a consumption and investment problem where an investor¡¯s investment opportunity gets enlarged when she becomes rich enough, i.e., when her wealth touches a critical level. We derive optimal consumption and investment rules assuming that the investor has a time-separable von Neumann-Morgenstern utility function. An interesting feature o...

We introduce an expected utility approach to price insurance risks in a dynamic financial market setting. The valuation method is based on comparing the maximal expected utility functions with and without incorporating the insurance product, as in the classical principle of equivalent utility. The pricing mechanism relies heavily on risk preference...

The traditional approach towards derivative pricing consists of dy-namically replicating a future liability by trading the assets on which that lia-bility is written. However, the assumption that one can trade the assets is often rather restrictive. In some cases, say of options on commodities or funds, one can at best trade another correlated asse...

We study a class of stochastic optimization models of expected utility in markets with stochastically changing investment opportunities. The prices of the primitive assets are modelled as diffusion processes whose coefficients evolve according to correlated diffusion factors. Under certain assumptions on the individual preferences, we are able to p...

The observed discrepancies of derivative prices from their theoretical, arbitrage-free values are examined in the presence of proportional transaction costs. Analytic upper and lower bounds on the reservation write and purchase prices, respectively, are obtained when an investor's preferences exhibit constant relative risk aversion between zero and...

This paper introduces a valuation model of international pricing in the presence of political risk. Shipments between countries are charged with shipping costs and the country specific production processes are modelled as diffusion processes. The political risk is modelled as a continuous time jump process that affects the drift of the returns in t...

We consider an environment of a fixed size that can be converted to another use. This conversion can be made in steps, but it is irreversible. The future benefits (per unit) from the original use, and from the alternative use, follow a diffusion process. For a fairly general case, we show that the value function must be the unique (viscosity) solut...

This presentation provides an overview of a class of free boundary problems that arise in valuation models in markets with transaction costs. Transaction costs are a realistic feature in numerous financial transactions and their presence affects considerably the theoretical asset and derivative prices. In the area of optimal portfolio management, t...

We study distorted survival probabilities related to risks in incomplete markets. The risks are modeled as diffusion processes, and the distortions are of general type. We establish a connection between distorted survival probabilities of the original risk process and distortion-free survival probabilities of new pseudo risk diffusions; the latter...

We study the level sets of value functions in expected utility stochastic optimization models. We consider optimal portfolio management models in complete markets with lognormally distributed prices as well as asset prices modeled as diffusion processes with nonlinear dynamics. Besides the complete market cases, we analyze models in markets with fr...

The pricing of derivative securities in the presence of market frictions has always been a question of fundamental importance. The reason is twofold: market frictions are present in numerous practical applications and, in such settings, the classical valuation theories break down entirely. Examples of market frictions include among others, transact...

We study a generalization of the Merton's original problem of optimal consumption and portfolio choice for a single investor in an intertemporal economy. The agent trades between a bond and a stock account and he may consume out of his bond holdings. The price of the bond is deterministic as opposed to the stock price which is modelled as a diffusi...

Analytic bounds on the reservation write price of European-style contingent claims are derived in the presence of proportional transaction costs in a model which allows for intermediate trading. The option prices are obtained via a utility maximization approach by comparing the maximized utilities with and without the contingent claim. The mathemat...

We study a model of optimal portfolio choice for a single agent where investments take place between a bond and a stock account. The price of the bond is riskless as opposed to the price of the stock which is a diffusion process. The coefficients of the latter depend on another diffusion process which is driven by a Brownian motion correlated with...

We study the behavior of the optimal portfolio policy of a long-run investor in markets with stationary investment opportunity sets. We provide conditions on the utility function, for large wealth levels, which are sufficient for the optimal portfolio policy to approximate, as the trading horizon becomes very long, the policy of investing a constan...

We discuss existing and new results for mathematical finance models with transaction costs, namely models of portfolio management and international asset pricing in the presence of political risk. The main analysis comes from the theories of singular stochastic control and viscosity solutions for first- and second-order nonlinear partial differenti...

this paper, we have developed a model of international asset pricing in the presence of political risk. Although the model is considerably simplified, it represents a substantial step towards understanding how uncertainty about future government actions can affect the prices of tradeable assets. The recent turmoil in asset prices for several Southe...

In this paper, we use stochastic dynamic programming to study the intertemporal consumption and portfolio choice of an infinitely lived agent who faces a constant opportunity set and a borrowing constraint. We show that, under general assumptions on the agent's utility function, optimal policies exist and can be expressed as feedback functions of c...

Numerical Methods in Finance has emerged as a discipline at the intersection of probability theory, finance and numerical analysis. This book, based on lectures given at the Newton Institute as part of a broader programme, describes a wide variety of numerical methods used in financial analysis: computation of option prices, especially of American...

In the context of Merton's original problem of optimal consumption and portfolio choice in continuous time, this paper solves an extension in which the investor is endowed with a stochastic income that cannot be replicated by trading the available securities. The problem is treated by demonstrating, using analytic and, in particular, ‘viscosity sol...

We consider the problem of pricing American options in a market model similar to the Black-Scholes one except that proportional transaction charges are levied on all sales and purchases of stock. “Perfect replication” is no longer possible and holding an option involves an essential element of risk. We obtain a definition of the option price by com...