Author
Tim Leung
Other affiliations: Johns Hopkins University, Columbia University
Bio: Tim Leung is an academic researcher from University of Washington. The author has contributed to research in topics: Optimal stopping & Trading strategy. The author has an hindex of 21, co-authored 163 publications receiving 1532 citations. Previous affiliations of Tim Leung include Johns Hopkins University & Columbia University.
Papers
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TL;DR: In this article, the authors present a valuation framework that captures the main characteristics of employee stock options (ESOs), which financial regulations now require to be expensed in firms' accounting statements.
Abstract: We present a valuation framework that captures the main characteristics of employee stock options (ESOs), which financial regulations now require to be expensed in firms' accounting statements. The value of these options is much less than Black–Scholes prices for corresponding market-traded options due to the suboptimal exercising strategies of the holders, which arise from risk aversion, trading and hedging constraints, and job termination risk. We analyze the combined effect of all of these factors along with the standard ESO features of multiple exercising rights, and vesting periods. This leads to the study of a chain of nonlinear free-boundary problems of reaction-diffusion type. We find that job termination risk, vesting, finite maturity and non-zero interest rates are significant contributors to the ESO cost. However, we find that in the presence of vesting, the impact of allowing multiple exercise rights on ESO cost is negligible.
116 citations
TL;DR: In this article, an optimal double stopping problem is formulated to analyze the timing to start and subsequently liquidate the position subject to transaction costs, and a stop-loss constraint is incorporated to limit the maximum loss.
Abstract: 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.
80 citations
TL;DR: In this paper, an optimal double stopping problem is formulated to analyze the timing to start and subsequently liquidate the position subject to transaction costs, and a stop-loss constraint is incorporated to limit the maximum loss.
Abstract: 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 costs and stop-loss level.
79 citations
TL;DR: In this paper, the authors studied the fair valuation of drawdown insurance contracts and derived the investor's optimal cancellation timing in terms of a two-sided first passage time of the underlying drawdown process.
Abstract: This paper studies the stochastic modeling of market drawdown events and the fair valuation of insurance contracts based on drawdowns. We model the asset drawdown process as the current relative distance from the historical maximum of the asset value. We first consider a vanilla insurance contract whereby the protection buyer pays a constant premium over time to insure against a drawdown of a pre-specified level. This leads to the analysis of the conditional Laplace transform of the drawdown time, which will serve as the building block for drawdown insurance with early cancellation or drawup contingency. For the cancellable drawdown insurance, we derive the investor’s optimal cancellation timing in terms of a two-sided first passage time of the underlying drawdown process. Our model can also be applied to insure against a drawdown by a defaultable stock. We provide analytic formulas for the fair premium and illustrate the impact of default risk.
48 citations
Posted Content•
TL;DR: In this paper, the authors provide a systematic study on the optimal timing of trades in markets with mean-reverting price dynamics, and present a financial engineering approach that distills the core mathematical questions from different trading problems and also incorporates the practical aspects of trading, such as model estimation, risk premia, risk constraints, and transaction costs, into their analysis.
Abstract: This book provides a systematic study on the optimal timing of trades in markets with mean-reverting price dynamics. We present a financial engineering approach that distills the core mathematical questions from different trading problems, and also incorporates the practical aspects of trading, such as model estimation, risk premia, risk constraints, and transaction costs, into our analysis. Self-contained and organized, the book not only discusses the mathematical framework and analytical results for the financial problems, but also gives formulas and numerical tools for practical implementation. A wide array of real-world applications are discussed, such as pairs trading of exchange-traded funds, dynamic portfolio of futures on commodities or volatility indices, and liquidation of options or credit risk derivatives.A core element of our mathematical approach is the theory of optimal stopping. For a number of the trading problems discussed herein, the optimal strategies are represented by the solutions to the corresponding optimal single/multiple stopping problems. This also leads to the analytical and numerical studies of the associated variational inequalities or free boundary problems. We provide an overview of our methodology and chapter outlines in the Introduction.Our objective is to design the book so that it can be useful for doctoral and masters students, advanced undergraduates, and researchers in financial engineering/mathematics, especially those who specialize in algorithmic trading, or have interest in trading exchange-traded funds, commodities, volatility, and credit risk, and related derivatives. For practitioners, we provide formulas for instant strategy implementation, propose new trading strategies with mathematical justification, as well as quantitative enhancement for some existing heuristic trading strategies.
43 citations
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01 Jan 1998
TL;DR: In this paper, the authors explore questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties, using diffusion processes as a model of a Markov process with continuous sample paths.
Abstract: We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.
2,446 citations
TL;DR: The theory of Fourier integrals arises out of the elegant pair of reciprocal formulae The Laplace Transform By David Vernon Widder as mentioned in this paper, which is the basis of our theory of integrals.
Abstract: THE theory of Fourier integrals arises out of the elegant pair of reciprocal formulae The Laplace Transform By David Vernon Widder. (Princeton Mathematical Series.) Pp. x + 406. (Princeton: Princeton University Press; London: Oxford University Press, 1941.) 36s. net.
743 citations
01 Mar 1979
TL;DR: In this article, the authors developed the theory behind Krishnaiah and Schuurmann's theoretical work reported in their report Approximations to the Distributions of the Traces of Complex Multivariate Beta and F Matrices.
Abstract: : One use of spectral analysis of time series is to determine if two different time series are realizations from the same process This thesis develops the theory behind Krishnaiah and Schuurmann's theoretical work reported in their report Approximations to the Distributions of the Traces of Complex Multivariate Beta and F Matrices We take the trace of a test statistic calculated from the spectral density matrices of the time series and test it The thesis applies the theory to two small sample simulations (Author)
683 citations
554 citations
Posted Content•
TL;DR: In this paper, the authors show that hedge fund performance fees are valuable to money managers, and conversely represent a claim on a significant proportion of investor wealth, and provide a closed-form solution to the high-water mark.
Abstract: Incentive or performance fees for money managers are frequently accompanied by high-water mark provisions which condition the payment of the performance fee upon exceeding the maximum achieved share value. In this paper, we show that hedge fund performance fees are valuable to money managers, and conversely represent a claim on a significant proportion of investor wealth. The high-water mark provisions in these contracts limit the value of the performance fees. We provide a closed-form solution to the high-water mark
447 citations