Continuous time trading of a small investor in a limit order market

We provide a mathematical framework to model continuous time trading of a small investor in limit order markets. We show how elementary strategies can be extended in a suitable way to general continuous time strategies containing orders with infinitely many different limit prices. The general limit buy order strategies are predictable processes with values in the set of nonincreasing demand functions. It turns out that our strategy set of limit and market orders is closed, but limit orders can turn into market orders when passing to the limit, and any element can be approximated by a sequence of elementary strategies.     

Optimal portfolio for a small investor in a market model with discontinuous prices

A consumption-investment problem is considered for a small investor in the case of a market model in which prices evolve according to a stochastic equation with a jump-process component. The techniques we use include the martingale representation theorem, Lagrange multiplier methods, and Markovian methods for the resolution of stochastic differential equations. We establish a Black-Scholes formula.     

Stylized facts of price gaps in limit order books

Price gap, defined as the logarithmic price difference between the first two occupied price levels on the same side of a limit order book (LOB), is a key determinant of market depth, which is one of the dimensions of liquidity. However, the properties of price gaps have not been thoroughly studied due to the less availability of ultrahigh frequency data. In the paper, we rebuild the LOB dynamics based on the order flow data of 26 A-share stocks traded on the Shenzhen Stock Exchange in 2003. Three key empirical statistical properties of price gaps are investigated. We find that the distribution of price gaps has a power-law tail for all stocks with an average tail exponent close to 3.2. Applying modern statistical methods, we confirm that the gap time series are long-range correlated and possess multifractal nature. These three features appear to be different in the measures across stocks, but they are similar for the buy and sell LOBs within each stock. Furthermore, we also unveil buy–sell asymmetry phenomena in the properties of price gaps on the buy and sell sides of the LOBs for individual stocks. These findings deepen our understanding of the dynamics of liquidity of common stocks and can be used to calibrate agent-based computational financial models.     

Optimal posting price of limit orders: learning by trading

We model a trader interacting with a continuous market as an iterative algorithm that adjusts limit prices at a given rhythm and propose a procedure to minimize trading costs. We prove the $a.s.$ convergence of the algorithm under assumptions on the cost function and give some practical criteria on model parameters to ensure that the conditions to use the algorithm are met (notably, using the co-monotony principle). We illustrate our results with numerical experiments on both simulated and market data.     

Optimal consumption, investment and insurance with insurable risk for an investor in a Lévy market

Numerous researchers have applied the martingale approach for models driven by Lévy processes to study optimal investment problems. The aim of this paper is to apply the martingale approach to obtain a closed form solution for the optimal investment, consumption and insurance strategies of an individual in the presence of an insurable risk when the insurable risk and risky asset returns are described by Lévy processes and the utility is a constant absolute risk aversion (CARA). The model developed in this paper can potentially be applied to absorb large insurable losses in the absence of insurance protection and to examine the level of diminishing current utility and consumption.     

Optimal spot market inventory strategies in the presence of cost and price risk

We consider a firm facing random demand at the end of a single period of random length. At any time during the period, the firm can either increase or decrease inventory by buying or selling on a spot market where price fluctuates randomly over time. The firm’s goal is to maximize expected discounted profit over the period, where profit consists of the revenue from selling goods to meet demand, on the spot market, or in salvage, minus the cost of buying goods, and transaction, penalty, and holding costs. We first show that this optimization problem is equivalent to a two-dimensional singular control problem. We then use a recently developed control-theoretic approach to show that the optimal policy is completely characterized by a simple price-dependent two-threshold policy. In a series of computational experiments, we explore the value of actively managing inventory during the period rather than making a purchase decision at the start of the period, and then passively waiting for demand. In these experiments, we observe that as price volatility increases, the value of actively managing inventory increases until some limit is reached.     

Optimal portfolios: new variations of an old theme

We survey some recent developments in the area of continuous-time portfolio optimization. These will include the use of options and of defaultable assets as investment classes and the presentation of a worst-case investment approach that takes the possibility of stock market crashes into account.      

Average shadow price and equilibrium price: A case study of tradable pollution permit markets

Shadow prices indicate implicit values of limited resources at the margin and provide important information in decision making for resource management. In continuous economic models, shadow prices associated with demand-supply balance constraints represent consumers’ willingness to pay and producers’ marginal cost, hence they correspond to market equilibrium prices. As well known, however, marginal analysis fails in the case of discrete optimization, such as mixed integer programming. An alternative concept has been introduced in the literature to measure the value of an extra unit of a limited resource in such cases. This concept is based on average rather than marginal values, thus called the average shadow price, and interpreted in the same way as conventional shadow prices. Whether average shadow prices in a discrete economic model can serve as market equilibrium prices has not been addressed in the related literature. The present paper addresses this issue in an empirical setting. Using a tradable pollution permit market as an example, where firms’ YES/NO type technology adoption decisions are represented by binary variables, we show that the average shadow price of tradable permits can be interpreted as the equilibrium price only when certain conditions related to the cost structure and emission levels hold. On the other hand, we show that an iterative procedure based on individual firms’ cost minimizing behavior presents a better approach for finding a price that can eliminate or reduce the gap between demand and supply of permits in the market.     

Estimating the efficient price from the order flow: A Brownian Cox process approach

At the ultra high frequency level, the notion of price of an asset is very ambiguous. Indeed, many different prices can be defined (last traded price, best bid price, mid price, etc.). Thus, in practice, market participants face the problem of choosing a price when implementing their strategies. In this work, we propose a notion of efficient price which seems relevant in practice. Furthermore, we provide a statistical methodology enabling to estimate this price from the order flow.     

Measuring brand equity in the car market: a hedonic price analysis

This paper designs and implements a hedonic model to investigate brand-name effects in the car market. The proposed model provides considerable empirical evidence in support of model-name premia, after controlling for observed product differentiation. Such premia incorporate not only manufacturer equity but also effects that are specific to individual models. The estimated price premia are remarkably intuitive and consistent across carmakers and models. The results yield important implications for brand and range management in the car market and suggest directions for future research.     

Convergence of estimated option price in a regime switching market

In an observed semi-Markov regime, estimation of transition rate of regime switching leads towards calculation of locally risk minimizing option price. Despite the uniform convergence of estimated step function of transition rate, to meet the existence of classical solution of the modified price equation, the estimator is approximated in the class of smooth functions and furthermore, the convergence is established. Later, the existence of the solution of the modified price equation is verified and the point-wise convergence of such approximation of option price is proved to answer the tractability of its application in Finance. To demonstrate the consistency in result a numerical experiment has been reported.     

A branch and price solution approach for order acceptance and capacity planning in make-to-order operations

Make-to-order (MTO) operations have to effectively manage their capacity to make long-term sustainable profits. This objective can be met by selectively accepting available customer orders and simultaneously planning for capacity. We model a MTO operation of a job-shop with multiple resources having regular and non-regular capacity. The MTO firm has a set of customer orders at time zero with fixed due-dates. The process route, processing times, and sales price for each order are given. Since orders compete for limited resources, the firm can only accept some orders. In this paper a Mixed-Integer Linear Program (MILP) is proposed to aid an operational manager to decide which orders to accept and how to allocate resources such that the overall profit is maximized. A branch-and-price (B&P) algorithm is devised to solve the MILP effectively. The MILP is first decomposed into a master problem and several sub-problems using Dantzig-Wolfe decomposition. Each sub-problem is represented as a network flow problem and an exact procedure is proposed to solve the sub-problems efficiently. We also propose an approximate B&P scheme, Lagrangian bounds, and approximations to fathom nodes in the branch-and-bound tree. Computational analysis shows that the proposed B&P algorithm can solve large problem instances with relatively short time.     

Optimal production in a competitive market under yield uncertainty

Motivated by the U.S. influenza vaccine market, we study the impact of random yield and production capacity on the number of firms, total supply, consumer surplus and social welfare in a market with identical suppliers. We formulate a two-stage game with endogenous entry, where each entering firm aims to maximize its profit under yield uncertainty subject to a production capacity constraint. We show that if firms produce full capacity in the equilibrium, then there are fewer firms in the equilibrium compared to the social optimum even for small levels of yield uncertainty. Furthermore, we prove that if firms do not produce full capacity in the equilibrium, they will not produce full capacity in the social optimum.     

Optimal repair-cost limit for a consumer following expiry of a warranty

This paper develops a methodology for obtaining the optimalrepair-cost limit for the repair-replace decision that a consumerfaces on the expiry of a product warranty, where the producthas been minimally repaired on failure during the warranty.Three cases are considered; (i) an exponentially distributedtime to failure with an arbitrary repair-cost distribution,(ii) an exponentially distributed time to failure with a uniformrepair-cost distribution, and (iii) a Weibull-distributed timeto failure with a beta-distributed repair cost. Numerical optimizationsare performed for the third case, and conclusions drawn as tothe sensitivity of the optimal repair-cost limit on the parametervalues of the distributions involved.     

Optimal partial hedging in a discrete-time market as a knapsack problem

We present a new approach for studying the problem of optimal hedging of a European option in a finite and complete discrete-time market model. We consider partial hedging strategies that maximize the success probability or minimize the expected shortfall under a cost constraint and show that these problems can be treated as so called knapsack problems, which are a widely researched subject in linear programming. This observation gives us better understanding of the problem of optimal hedging in discrete time.     

Optimal initiation of a GLWB in a variable annuity: No Arbitrage approach

This paper offers a financial economic perspective on the optimal time (and age) at which the owner of a Variable Annuity (VA) policy with a Guaranteed Lifetime Withdrawal Benefit (GLWB) rider should initiate guaranteed lifetime income payments. We bypass issues related to utility, bequest and consumption preference by treating the VA as liquid and tradable. This allows us to use an American option pricing framework to derive a so-called optimal initiation region. Our main practical finding is that given current design parameters in which volatility (asset allocation) is restricted to less than 20%, while guaranteed payout rates (GPR) as well as bonus (roll-up) rates are less than 5%, GLWBs that are in-the-money should be turned on by the late 50s and certainly the early 60s. The exception to the rule is when a non-constant GPR is about to increase to a higher age band, in which case the optimal policy is to wait until the new GPR is hit and then initiate immediately. Also, to offer a different perspective, we invert the model and solve for the bonus (roll-up) rate that is required to justify delaying initiation at any age. We find that the required bonus is quite high and more than what is currently promised by existing products. Our methodology and results should be of interest to researchers as well as to the individuals that collectively have over $1 USD trillion in aggregate invested in these products. We conclude by suggesting that much of the non-initiation at older ages is irrational (which obviously benefits the insurance industry).     

Selection of contract suppliers under price and demand uncertainty in a dynamic market

In this paper, we consider a supply contracting problem in which the buyer firm faces non-stationary stochastic price and demand. First, we derive analytical results to compare two pure strategies: (i) periodically purchasing from the spot market; and (ii) signing a long-term contract with a single supplier. The results from the pure strategies show that the selection of suppliers can be complicated by many parameters, and is particularly affected by price uncertainty. We then develop a stochastic dynamic programming model to incorporate mixed strategies, purchasing commitments and contract cancellations. Computational results show that increases in price (demand) uncertainty favor long-term (short-term) suppliers. By examining the two-way interactions of contract factors (price, demand, purchasing bounds, learning and technology effect, salvage values and contract cancellation), both intuitive and non-intuitive managerial insights in outsourcing strategies are derived.     

Zero-dispersion limit to the Korteweg-de Vries equation: a dressing chain approach

An asymptotic solution of the KdV equation with small dispersion is studied for the case of smooth hump-like initial condition with monotonically decreasing slopes. Despite the well-known approaches by Lax-Levermore and Gurevich-Pitaevskii, a new way of constructing the asymptotics is proposed using the inverse scattering transform together with the dressing chain technique developed by A. Shabat [1]. It provides the Whitham-type approximaton of the leading term by solving the dressing chain through a finite-gap asymptotic ansatz. This yields the Whitham equations on the Riemann invariants together with hodograph transform which solves these equations explicitly. Thus we reproduce an uniform in x asymptotics consisting of smooth solution of the Hopf equation outside the oscillating domain and a slowly modulated cnoidal wave within the domain. Finally, the dressing chain technique provides the proof of an asymptotic estimate for the leading term.      

Valuation of large variable annuity portfolios under nested simulation: A functional data approach

A variable annuity (VA) is equity-linked annuity product that has rapidly grown in popularity around the world in recent years. Research up to date on VA largely focuses on the valuation of guarantees embedded in a single VA contract. However, methods developed for individual VA contracts based on option pricing theory cannot be extended to large VA portfolios. Insurance companies currently use nested simulation to valuate guarantees for VA portfolios but efficient valuation under nested simulation for a large VA portfolio has been a real challenge. The computation in nested simulation is highly intensive and often prohibitive. In this paper, we propose a novel approach that combines a clustering technique with a functional data analysis technique to address the issue. We create a highly non-homogeneous synthetic VA portfolio of 100,000 contracts and use it to estimate the dollar Delta of the portfolio at each time step of outer loop scenarios under the nested simulation framework over a period of 25 years. Our test results show that the proposed approach performs well in terms of accuracy and efficiency.