Optimal Market Making in the Foreign Exchange Market

Abstract

This paper is concerned with optimal market making in the foreign exchange market. The market maker's holdings in the different currencies are modelled as stochastic processes that are influenced by both the stochastic exchange rates and the stochastic customer buy and sell orders. The market maker can control their own bid and ask price quotes and, additionally, can buy and sell at other market participants' quotes. The resulting stochastic control problem consists of a controlled diffusion problem for the optimal quotes and a singular control problem for optimal trades at other market participants' quotes. A Markov chain approximation is used to derive optimal strategies.     

Optimal portfolios of a small investor in a limit order market: a shadow price approach

We study Merton’s portfolio optimization problem in a limit order market. An investor trading in a limit order market has the choice between market orders that allow immediate transactions and limit orders that trade at more favorable prices but are executed only when another market participant places a corresponding market order. Assuming Poisson arrivals of market orders from other traders we use a shadow price approach, similar to Kallsen and Muhle-Karbe (Ann Appl Probab, forthcoming) for models with proportional transaction costs, to show that the optimal strategy consists of using market orders to keep the proportion of wealth invested in the risky asset within certain boundaries, similar to the result for proportional transaction costs, while within these boundaries limit orders are used to profit from the bid–ask spread. Although the given best-bid and best-ask price processes are geometric Brownian motions the resulting shadow price process possesses jumps.     

A model for optimal execution of atomic orders

Atomic Orders are the basic elements of any algorithm for automated trading in electronic stock exchanges. The main concern in their execution is achieving the most efficient price. We propose two optimal strategies for the execution of atomic orders based on minimization of impact and volatility costs. The first considered strategy is based on a relatively simple nonlinear optimization model while the second allows re-optimization at some time point within a given execution time. In both cases a combination of market and limit orders is used. The key innovation in our approach is the introduction of a Fill Probability function which allows a combination of market and limit orders in the two optimization models we are discussing in this paper. Under certain conditions the objective functions of both considered problems are convex and therefore standard optimization tools can be applied. The efficiency of the resulting strategies is tested against two benchmarks representing common market practice on a representative sample of real trading data.     

Smooth solutions to portfolio liquidation problems under price-sensitive market impact

We consider the stochastic control problem of a financial trader that needs to unwind a large asset portfolio within a short period of time. The trader can simultaneously submit active orders to a primary market and passive orders to a dark pool. Our framework is flexible enough to allow for price-dependent impact functions describing the trading costs in the primary market and price-dependent adverse selection costs associated with dark pool trading. We prove that the value function can be characterized in terms of the unique smooth solution to a PDE with singular terminal value, establish its explicit asymptotic behavior at the terminal time, and give the optimal trading strategy in feedback form.     

Optimal investment–reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets

In this paper, we investigate the optimal time-consistent investment–reinsurance strategies for an insurer with state dependent risk aversion and Value-at-Risk (VaR) constraints. The insurer can purchase proportional reinsurance to reduce its insurance risks and invest its wealth in a financial market consisting of one risk-free asset and one risky asset, whose price process follows a geometric Brownian motion. The surplus process of the insurer is approximated by a Brownian motion with drift. The two Brownian motions in the insurer’s surplus process and the risky asset’s price process are correlated, which describe the correlation or dependence between the insurance market and the financial market. We introduce the VaR control levels for the insurer to control its loss in investment–reinsurance strategies, which also represent the requirement of regulators on the insurer’s investment behavior. Under the mean–variance criterion, we formulate the optimal investment–reinsurance problem within a game theoretic framework. By using the technique of stochastic control theory and solving the corresponding extended Hamilton–Jacobi–Bellman (HJB) system of equations, we derive the closed-form expressions of the optimal investment–reinsurance strategies. In addition, we illustrate the optimal investment–reinsurance strategies by numerical examples and discuss the impact of the risk aversion, the correlation between the insurance market and the financial market, and the VaR control levels on the optimal strategies.     

Fast automatic differentiation as applied to the computation of second derivatives of composite functions

A technique for deriving formulas for the second derivatives of a composite function with constrained variables is proposed. The original system of constraint equations is associated with a linear system of equations, whose solution is used to determine the Hessian of the function. The resulting formulas are applied to discrete problems obtained by approximating optimal control problems with the use of Runge-Kutta methods of various orders. For a particular optimal control problem, the numerical results obtained by the gradient method and Newton’s method with the resulting formulas are described and analyzed in detail.     

Ordering and pricing of service products in an advance sales system with price-dependent demand

This investigation addresses a service inventory control problem in which a firm orders and sells a service which will be used or consumed by customers on a specific future date. The firm sells the product through an advance booking system, aiming to optimize product price to maximize the total expected profit. Considering situations in which product demand is price-dependent and customers with reservations may cancel advance orders, this work develops a continuous-time model to simultaneously determine the order quantity and selling prices. The analytical results reveal that the optimal ordering quantity and prices are derived via closed-form solutions. In addition, sensitivity analysis of the optimal prices with respect to the system parameters is also conducted to illustrate optimal decision characteristics.     

The solution and sensitivity of a general optimal control problem

A solution method for the general optimal control problem is presented. This can be used to solve optimal control problems for which the system dynamics are not necessarily described by differential or difference equations. Having obtained the solution it is of theoretical and practical interest to investigate the sensitivity of the optimal trajectory to perturbations. Indicators of this sensitivity are the adjoint variables derived in the maximum principle. A method of deriving the adjoint variables from the solution is described. To illustrate the solution method and the determination of the adjoint variables a problem in the urban housing market is used.     

Pricing general insurance in a reactive and competitive market

A simple parameterisation is introduced which represents the insurance market’s response to an insurer adopting a pricing strategy determined via optimal control theory. Claims are modelled using a lognormally distributed mean claim size rate, and the market average premium is determined via the expected value principle. If the insurer maximises its expected wealth then the resulting Bellman equation has a moving boundary in state space that determines when it is optimal to stop selling insurance. This stochastic optimisation problem is simplified by the introduction of a stopping time that prevents an insurer leaving and then re-entering the insurance market. Three finite difference schemes are used to verify the existence of a solution to the resulting Bellman equation when there is market reaction. All of the schemes use a front-fixing transformation. If the market reacts, then it is found that the optimal strategy is altered, in that premiums are raised if the strategy is of loss-leading type and lowered if it is optimal for the insurer to set a relatively high premium and sell little insurance.     

The determination of optimal partial fill policy for an inventory system with lumpy demand items

This paper presents a mathematical model developed for the synthesis of optimal replenishment policies for items that experience lumpy demands. In order to avoid disrupting the inventory system, a cutoff point of w units is introduced such that the system would only satisfy routinely customer orders with transaction sizes less than or equal to w units. For customer orders with transaction sizes larger than w units, the system would only supply the cutoff amount (w units). The excess units would be refused. The control discipline is the (s, S) inventory policy with continuous review, and the nature of the customer orders is approximated by a discrete stuttering Poisson distribution. The optimal values of the control parameters, w, s and S, are determined. The theoretical results obtained are illustrated with a numerical example.     

Customer Enquiry Management: Part of a Hierarchical System to Control Lead Times in Make-to-Order Companies

The authors are currently developing a hierarchical production planning system specifically designed for the ‘make-to-order’ sector of industry. Its aim is to control the delivery and manufacturing lead times of all orders processed by a firm. Two major decision levels are identified - the customer enquiry stage and the order release stage. Input/output control is exercised at both stages. This paper is concerned with the control mechanisms used at the customer enquiry stage. Two important backlogs of work, along with their associated backlog lengths, are identified. The system aims to maintain these backlogs between predetermined minimum and maximum lengths in order to process all orders within an acceptable length of time. It is shown that the two backlogs are linked in a manner that enables them to be controlled simultaneously. Hence, a procedure for dealing with incoming customer orders is presented.     

Pricing general insurance in a reactive and competitive market

A simple parameterisation is introduced which represents the insurance market’s response to an insurer adopting a pricing strategy determined via optimal control theory. Claims are modelled using a lognormally distributed mean claim size rate, and the market average premium is determined via the expected value principle. If the insurer maximises its expected wealth then the resulting Bellman equation has a moving boundary in state space that determines when it is optimal to stop selling insurance. This stochastic optimisation problem is simplified by the introduction of a stopping time that prevents an insurer leaving and then re-entering the insurance market. Three finite difference schemes are used to verify the existence of a solution to the resulting Bellman equation when there is market reaction. All of the schemes use a front-fixing transformation. If the market reacts, then it is found that the optimal strategy is altered, in that premiums are raised if the strategy is of loss-leading type and lowered if it is optimal for the insurer to set a relatively high premium and sell little insurance.     

Customer selection problem with profit from a sideline

This paper deals with the problem of selecting profitable orders to accept out of customers sequentially arriving at companies operating in service industries which provide specialized services designed to meet the various needs of their customers. When all the orders accepted up to a point of time are completed and delivered, the companies provide subsidiary services as a sideline in order to prevent their system from being idle, and to yield extra income, referred to as the profit from a sideline. Further, a cost is paid to search for customers, called the search cost. We discuss the admission control problem and pricing control problem in an identical framework. Properties of the optimal decision rule maximizing the total expected present discounted net profit gained over an infinite planning horizon are examined and clarified. It is shown that when the profit from the sideline is large, the optimal policies may not be monotone in the number of orders in the system.     

Incorporating order-flow into optimal execution

We provide an explicit closed-form strategy for an investor who executes a large order when market order-flow from all agents, including the investor’s own trades, has a permanent price impact. The strategy is found in closed-form when the permanent and temporary price impacts are linear in the market’s and investor’s rates of trading. We do this under very general assumptions about the stochastic process followed by the order-flow of the market. The optimal strategy consists of an Almgren–Chriss execution strategy adjusted by a weighted-average of the future expected net order-flow (given by the difference of the market’s rate of buy and sell market orders) over the execution trading horizon and proportional to the ratio of permanent to temporary linear impacts. We use historical data to calibrate the model to Nasdaq traded stocks and use simulations to show how the strategy performs.     

Optimal strategies for water quality control

In this paper, the use of optimal control theory to obtain optimal strategies for the control of aquatic models is illustrated. Several types of control variables are used including the rate of nutrient application and the rates of change of nutrient concentration in both the phytoplankton and zooplankton populations. Techniques are given to show how optimal control theory can be applied to several models with different states and control variables constraints. Explicit expressions and optimality conditions are given for singular controls whenever they exist. Some numerical techniques are suggested to couple the optimal control parts in the proper sequence.     

Horizon-unbiased utility functions

In this paper we consider a class of mixed optimal control/optimal stopping problems related to the choice of the best time to sell a single unit of an indivisible asset. We assume that in addition to the indivisible asset, the agent has access to a financial market. Investments in the financial market can be used for hedging, but the financial assets are only partially correlated with the indivisible asset, so that the agent faces an incomplete markets problem.     

Optimal scheduling of batch industrial facilities

An increasing interest in batch processing has been evident in recent years. This renewed interest is explained by the inherent flexibility of such plants that permits a high level of response to uncertain market conditions and requirements. This level of response does require the use of efficient tools to help the decision-making process at the design and operational level. This paper presents a Mixed Integer Linear Program (MILP) model to optimise the scheduling of batch facilities subject to changeovers and distribution constraints so as to guarantee a pre-defined objective. Such an objective can be defined as the minimum orders' total lateness or the maximum distribution units loading capacity, among others. A continuous-time representation is used as well as the concept of job predecessor and successor to effectively handle changeovers. Facilities having non-identical parallel units/lines, sequence-dependent orders, finite release times for units and orders, restrictions on the suitability of jobs to lines/units and different possible destinations to available distribution units are also considered. Based on these characteristics the proposed model is able to determine the optimal allocation of jobs to production lines/units, the sequence of jobs on every line/unit and the starting and completion production times of each order. Also, the usage and allocation of the distribution resources (eg trucks) to orders and destinations are obtained based on their availability and suitability to the orders. The model led to the development of a prototype information system that can be used as a tool to help the decision-making process at the operational plant level.Finally, the applicability of the proposed system/formulation is shown through the resolution of an industrial real case where the production of polymers is performed.     

Optimal admission and pricing control problems in service industries with multiple servers and sideline profit

This paper deals with the problem of selecting profitable customer orders sequentially arriving at a company operating in service industries with multiple servers in which two classes of services are provided. The first class of service is designed to meet the particular needs of customers; and the company (1) makes a decision on whether to accept or to reject the order for this service (admission control) and (2) decides a price of the order and offers it to an arriving customer (pricing control). The second class of service is provided as a sideline, which prevents servers from being idle when the number of customer orders for the first class is less than the number of servers. This yields the sideline profit. A cost is paid to search for customer orders, which is called the search cost. In the context of search cost, the company has an option whether to conduct the search or not. In this paper, we discuss both admission control and pricing control problems within an identical framework as well as examine the structure of the optimal policies to maximize the total expected net profit gained over an infinite planning horizon. We show that when the sideline profit is large, the optimal policies may not be monotone in the number of customer orders in the system. Copyright © 2008 John Wiley & Sons, Ltd.     

Multiperiod portfolio optimization models in stochastic markets using the mean–variance approach

We consider several multiperiod portfolio optimization models where the market consists of a riskless asset and several risky assets. The returns in any period are random with a mean vector and a covariance matrix that depend on the prevailing economic conditions in the market during that period. An important feature of our model is that the stochastic evolution of the market is described by a Markov chain with perfectly observable states. Various models involving the safety-first approach, coefficient of variation and quadratic utility functions are considered where the objective functions depend only on the mean and the variance of the final wealth. An auxiliary problem that generates the same efficient frontier as our formulations is solved using dynamic programming to identify optimal portfolio management policies for each problem. Illustrative cases are presented to demonstrate the solution procedure with an interpretation of the optimal policies.     

Optimal investment strategies and risk-sharing arrangements for a hybrid pension plan

A continuous time stochastic model is used to study a hybrid pension plan, where both the contribution and benefit levels are adjusted depending on the performance of the plan, with risk sharing between different generations. The pension fund is invested in a risk-free asset and multiple risky assets. The objective is to seek an optimal investment strategy and optimal risk-sharing arrangements for plan trustees and participants so that this proposed hybrid pension system provides adequate and stable income to retirees while adjusting contributions effectively, as well as keeping its sustainability in the long run. These goals are achieved by minimizing the expected discount disutility of intermediate adjustment for both benefits and contributions and that of terminal wealth in finite time horizon. Using the stochastic optimal control approach, closed-form solutions are derived under quadratic loss function and exponential loss function. Numerical analysis is presented to illustrate the sensitivity of the optimal strategies to parameters of the financial market and how the optimal benefit changes with respect to different risk aversions. Through numerical analysis, we find that the optimal strategies do adjust the contributions and retirement benefits according to fund performance and model objectives so the intergenerational risk sharing seem effectively achieved for this collective hybrid pension plan.