arXiv subjects

Tim Leung

Publications and source records attributed to Tim Leung.

At least 37 records · Page 2Linked to original sources

arXiv2018

Mean Reverting Portfolios via Penalized OU-Likelihood Estimation

We study an optimization-based approach to con- struct a mean-reverting portfolio of assets. Our objectives are threefold: (1) design a portfolio that is well-represented by an Ornstein-Uhlenbeck process with parameters estimated by maximum likelihood, (2) select portfolios with desirable characteristics of high mean reversion and low variance, and (3) select a parsimonious portfolio, i.e. find a small subset of a larger universe of assets that can be used for long and short positions. We present the full problem formulation, a specialized algorithm that exploits partial minimization, and numerical examples using both simulated and empirical price data.

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arXiv2018

Mean Reversion Trading with Sequential Deadlines and Transaction Costs

We study the optimal timing strategies for trading a mean-reverting price process with afinite deadline to enter and a separate finite deadline to exit the market. The price process is modeled by a diffusion with an affine drift that encapsulates a number of well-known models,including the Ornstein-Uhlenbeck (OU) model, Cox-Ingersoll-Ross (CIR) model, Jacobi model,and inhomogeneous geometric Brownian motion (IGBM) model.We analyze three types of trading strategies: (i) the long-short (long to open, short to close) strategy; (ii) the short-long(short to open, long to close) strategy, and (iii) the chooser strategy whereby the trader has the added flexibility to enter the market by taking either a long or short position, and subsequently close the position. For each strategy, we solve an optimal double stopping problem with sequential deadlines, and determine the optimal timing of trades. Our solution methodology utilizes the local time-space calculus of Peskir (2005) to derive nonlinear integral equations of Volterra-type that uniquely characterize the trading boundaries. Numerical implementation ofthe integral equations provides examples of the optimal trading boundaries.

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arXiv2017

Dynamic Index Tracking and Risk Exposure Control Using Derivatives

We develop a methodology for index tracking and risk exposure control using financial derivatives. Under a continuous-time diffusion framework for price evolution, we present a pathwise approach to construct dynamic portfolios of derivatives in order to gain exposure to an index and/or market factors that may be not directly tradable. Among our results, we establish a general tracking condition that relates the portfolio drift to the desired exposure coefficients under any given model. We also derive a slippage process that reveals how the portfolio return deviates from the targeted return. In our multi-factor setting, the portfolio's realized slippage depends not only on the realized variance of the index, but also the realized covariance among the index and factors. We implement our trading strategies under a number of models, and compare the tracking strategies and performances when using different derivatives, such as futures and options.

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arXiv2017

Optimal Execution of Limit and Market Orders with Trade Director, Speed Limiter, and Fill Uncertainty

We study the optimal execution of market and limit orders with permanent and temporary price impacts as well as uncertainty in the filling of limit orders. Our continuous-time model incorporates a trade speed limiter and a trader director to provide better control on the trading rates. We formulate a stochastic control problem to determine the optimal dynamic strategy for trade execution, with a quadratic terminal penalty to ensure complete liquidation. In addition, we identify conditions on the model parameters to ensure optimality of the controls and finiteness of the associated value functions. For comparison, we also solve the schedule-following optimal execution problem that penalizes deviations from an order schedule. Numerical results are provided to illustrate the optimal market and limit orders over time.

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arXiv2017

Understanding the Non-Convergence of Agricultural Futures via Stochastic Storage Costs and Timing Options

This paper studies the market phenomenon of non-convergence between futures and spot prices in the grains market. We postulate that the positive basis observed at maturity stems from the futures holder's timing options to exercise the shipping certificate delivery item and subsequently liquidate the physical grain. In our proposed approach, we incorporate stochastic spot price and storage cost, and solve an optimal double stopping problem to give the optimal strategies to exercise and liquidate the grain. Our new models for stochastic storage rates lead to explicit no-arbitrage prices for the shipping certificate and associated futures contract. We calibrate our models to empirical futures data during the periods of observed non-convergence, and illustrate the premium generated by the shipping certificate.

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arXiv2017

Timing Options for a Startup with Early Termination and Competition Risks

This paper analyzes the timing options embedded in a startup firm, and the associated market entry and exit timing decisions under the exogenous risks of early termination and competitor's entry. Our valuation approach leads to the analytical study of a non-standard perpetual American installment option nested with an optimal sequential stopping problem. Explicit formulas are derived for the firm's value functions. Analytically and numerically, we show that early termination risk leads to earlier voluntary entry or exit, and the threat of competition has a non-trivial effect on the firm's entry and abandonment strategies.

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arXiv2017

Optimal Mean-Reverting Spread Trading: Nonlinear Integral Equation Approach

We study several optimal stopping problems that arise from trading a mean-reverting price spread over a finite horizon. Modeling the spread by the Ornstein-Uhlenbeck process, we analyze three different trading strategies: (i) the long-short strategy; (ii) the short-long strategy, and (iii) the chooser strategy, i.e. the trader can enter into the spread by taking either long or short position. In each of these cases, we solve an optimal double stopping problem to determine the optimal timing for starting and subsequently closing the position. We utilize the local time-space calculus of Peskir (2005a) and derive the nonlinear integral equations of Volterra-type that uniquely char- acterize the boundaries associated with the optimal timing decisions in all three problems. These integral equations are used to numerically compute the optimal boundaries.

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arXiv2016

Long-Term Growth Rate of Expected Utility for Leveraged ETFs: Martingale Extraction Approach

This paper studies the long-term growth rate of expected utility from holding a leveraged exchanged-traded fund (LETF), which is a constant proportion portfolio of the reference asset. Working with the power utility function, we develop an analytical approach that employs martingale extraction and involves finding the eigenpair associated with the infinitesimal generator of a Markovian time-homogeneous diffusion. We derive explicitly the long-term growth rates under a number of models for the reference asset, including the geometric Brownian motion model, GARCH model, inverse GARCH model, extended CIR model, 3/2 model, quadratic model, as well as the Heston and 3/2 stochastic volatility models. We also investigate the impact of stochastic interest rate such as the Vasicek model and the inverse GARCH short rate model. We determine the optimal leverage ratio for the long-term investor and examine the effects of model parameters.

q-fin.MF↗

arXiv2016

Asynchronous ADRs: Overnight vs Intraday Returns and Trading Strategies

American Depositary Receipts (ADRs) are exchange-traded certificates that rep- resent shares of non-U.S. company securities. They are major financial instruments for investing in foreign companies. Focusing on Asian ADRs in the context of asyn- chronous markets, we present methodologies and results of empirical analysis of their returns. In particular, we dissect their returns into intraday and overnight com- ponents with respect to the U.S. market hours. The return difference between the S&P500 index, traded through the SPDR S&P500 ETF (SPY), and each ADR is found to be a mean-reverting time series, and is fitted to an Ornstein-Uhlenbeck process via maximum-likelihood estimation (MLE). Our empirical observations also lead us to develop and backtest pairs trading strategies to exploit the mean-reverting ADR-SPY spreads. We find consistent positive payoffs when long position in ADR and short position in SPY are simultaneously executed at selected entry and exit levels.

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arXiv2016

Understanding the Tracking Errors of Commodity Leveraged ETFs

Commodity exchange-traded funds (ETFs) are a significant part of the rapidly growing ETF market. They have become popular in recent years as they provide investors access to a great variety of commodities, ranging from precious metals to building materials, and from oil and gas to agricultural products. In this article, we analyze the tracking performance of commodity leveraged ETFs and discuss the associated trading strategies. It is known that leveraged ETF returns typically deviate from their tracking target over longer holding horizons due to the so-called volatility decay. This motivates us to construct a benchmark process that accounts for the volatility decay, and use it to examine the tracking performance of commodity leveraged ETFs. From empirical data, we find that many commodity leveraged ETFs underperform significantly against the benchmark, and we quantify such a discrepancy via the novel idea of \emph{realized effective fee}. Finally, we consider a number of trading strategies and examine their performance by backtesting with historical price data.

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arXiv2016

Optimal Risk-Averse Timing of an Asset Sale: Trending vs Mean-Reverting Price Dynamics

This paper studies the optimal risk-averse timing to sell a risky asset. The investor's risk preference is described by the exponential, power, or log utility. Two stochastic models are considered for the asset price -- the geometric Brownian motion and exponential Ornstein-Uhlenbeck models -- to account for, respectively, the trending and mean-reverting price dynamics. In all cases, we derive the optimal thresholds and certainty equivalents to sell the asset, and compare them across models and utilities, with emphasis on their dependence on asset price, risk aversion, and quantity. We find that the timing option may render the investor's value function and certainty equivalent non-concave in price. Numerical results are provided to illustrate the investor's strategies and the premium associated with optimally timing to sell.

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arXiv2016

Speculative Futures Trading under Mean Reversion

This paper studies the problem of trading futures with transaction costs when the underlying spot price is mean-reverting. Specifically, we model the spot dynamics by the Ornstein-Uhlenbeck (OU), Cox-Ingersoll-Ross (CIR), or exponential Ornstein-Uhlenbeck (XOU) model. The futures term structure is derived and its connection to futures price dynamics is examined. For each futures contract, we describe the evolution of the roll yield, and compute explicitly the expected roll yield. For the futures trading problem, we incorporate the investor's timing option to enter or exit the market, as well as a chooser option to long or short a futures upon entry. This leads us to formulate and solve the corresponding optimal double stopping problems to determine the optimal trading strategies. Numerical results are presented to illustrate the optimal entry and exit boundaries under different models. We find that the option to choose between a long or short position induces the investor to delay market entry, as compared to the case where the investor pre-commits to go either long or short.

q-fin.MF↗

arXiv2015

Optimal Static Quadratic Hedging

We propose a flexible framework for hedging a contingent claim by holding static positions in vanilla European calls, puts, bonds, and forwards. A model-free expression is derived for the optimal static hedging strategy that minimizes the expected squared hedging error subject to a cost constraint. The optimal hedge involves computing a number of expectations that reflect the dependence among the contingent claim and the hedging assets. We provide a general method for approximating these expectations analytically in a general Markov diffusion market. To illustrate the versatility of our approach, we present several numerical examples, including hedging path-dependent options and options written on a correlated asset.

q-fin.MF↗

arXiv2015

An analytic recursive method for optimal multiple stopping: Canadization and phase-type fitting

We study an optimal multiple stopping problem for call-type payoff driven by a spectrally negative Levy process. The stopping times are separated by constant refraction times, and the discount rate can be positive or negative. The computation involves a distribution of the Levy process at a constant horizon and hence the solutions in general cannot be attained analytically. Motivated by the maturity randomization (Canadization) technique by Carr (1998), we approximate the refraction times by independent, identically distributed Erlang random variables. In addition, fitting random jumps to phase-type distributions, our method involves repeated integrations with respect to the resolvent measure written in terms of the scale function of the underlying Levy process. We derive a recursive algorithm to compute the value function in closed form, and sequentially determine the optimal exercise thresholds. A series of numerical examples are provided to compare our analytic formula to results from Monte Carlo simulation.

q-fin.MF↗

arXiv2015

Optimal Multiple Stopping with Negative Discount Rate and Random Refraction Times under Levy Models

This paper studies a class of optimal multiple stopping problems driven by Lévy processes. Our model allows for a negative effective discount rate, which arises in a number of financial applications, including stock loans and real options, where the strike price can potentially grow at a higher rate than the original discount factor. Moreover, successive exercise opportunities are separated by i.i.d. random refraction times. Under a wide class of two-sided Lévy models with a general random refraction time, we rigorously show that the optimal strategy to exercise successive call options is uniquely characterized by a sequence of up-crossing times. The corresponding optimal thresholds are determined explicitly in the single stopping case and recursively in the multiple stopping case.

q-fin.MF↗

arXiv2015

Optimal Mean Reversion Trading with Transaction Costs and Stop-Loss Exit

Motivated by the industry practice of pairs trading, we study the optimal timing strategies for trading a mean-reverting price spread. An optimal double stopping problem is formulated to analyze the timing to start and subsequently liquidate the position subject to transaction costs. Modeling the price spread by an Ornstein-Uhlenbeck process, we apply a probabilistic methodology and rigorously derive the optimal price intervals for market entry and exit. As an extension, we incorporate a stop-loss constraint to limit the maximum loss. We show that the entry region is characterized by a bounded price interval that lies strictly above the stop-loss level. As for the exit timing, a higher stop-loss level always implies a lower optimal take-profit level. Both analytical and numerical results are provided to illustrate the dependence of timing strategies on model parameters such as transaction cost and stop-loss level.

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arXiv2015

ESO Valuation with Job Termination Risk and Jumps in Stock Price

Employee stock options (ESOs) are American-style call options that can be terminated early due to employment shock. This paper studies an ESO valuation framework that accounts for job termination risk and jumps in the company stock price. Under general Lévy stock price dynamics, we show that a higher job termination risk induces the ESO holder to voluntarily accelerate exercise, which in turn reduces the cost to the company. The holder's optimal exercise boundary and ESO cost are determined by solving an inhomogeneous partial integro-differential variational inequality (PIDVI). We apply Fourier transform to simplify the variational inequality and develop accurate numerical methods. Furthermore, when the stock price follows a geometric Brownian motion, we provide closed-form formulas for both the vested and unvested perpetual ESOs. Our model is also applied to evaluate the probabilities of understating ESO expenses and contract termination.

q-fin.PR↗

arXiv2015

Optimal Multiple Trading Times Under the Exponential OU Model with Transaction Costs

This paper studies the timing of trades under mean-reverting price dynamics subject to fixed transaction costs. We solve an optimal double stopping problem to determine the optimal times to enter and subsequently exit the market, when prices are driven by an exponential Ornstein-Uhlenbeck process. In addition, we analyze a related optimal switching problem that involves an infinite sequence of trades, and identify the conditions under which the double stopping and switching problems admit the same optimal entry and/or exit timing strategies. Among our results, we find that the investor generally enters when the price is low, but may find it optimal to wait if the current price is sufficiently close to zero. In other words, the continuation (waiting) region for entry is disconnected. Numerical results are provided to illustrate the dependence of timing strategies on model parameters and transaction costs.

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