Chen, Nan

Professor

Department of Systems Engineering and Engineering Management

The Chinese University of Hong Kong

Director

Bachelor of Engineering Programme in Financial Technology, CUHK

Centre for Financial Enginnering, CUHK

Master of Science Programme in Financial Engineering, CUHK(Shenzhen)

• Office: Room 709A, William M.W. Mong Engineering Building
• Phone: (852) 3943-8237       Fax: (852) 2603-5505
• Email: @se.cuhk.edu.hk

• Quantitative Methods in Finance and Risk Management
• Monte Carlo Simulation
• Applied Probability

My Google Scholar Citations  Citations: 1222, h-index: 17, i10-index: 17. (as of 4/10/2024)

My Social Science Research Network (SSRN)

Working Papers

• N. Chen and M. Liu. (2023). Adversarial Reinforcement Learning: A Duality-Based Approach to Solving Overlearning Issue in Deep Learning Approximation for Optimal Control.
• N. Chen, Y. Xu, R. Zhang, and M. Zhong. (2023). A Two Timescale Evolutionary Game Approach to Multi-Agent Reinforcement Learning.
• N. Chen, M. Dai, Q. Ding, and C. Yang. (2023). Patience is a Virtue: Optimal Investment in the Presence of Market Resilience. Major Revision with Management Science. [full pdf] [abstract]
  This paper investigates an optimal investment problem in an illiquid market, modeling explicitly the effects of three key features of market microstructure— market tightness, market depth, and finite market resilience — on the investor’s decision. By employing a Bachelier process to model the dynamic of the fundamental value of the asset and assuming CARA-type utility for the investor, we manage to obtain the investor’s optimal dynamic trading strategy in closed form by solving the resulting high-dimensional singular control problem. Furthermore, we extend the model to incorporate return-predicting signals and utilize an asymptotic expansion approach to derive approximate optimal trading strategies. The theoretical and numerical results emphasize the vital role of patience. Specifically, rather than dispersing small trades continuously over time as advocated by the existing literature, our findings suggest that investors should strategically time their trading activities to align with the aim portfolio in the presence of market resilience. To quantify this timing decision, we introduce a patience index that enables investors to strike a balance among various competing goals, including achieving currently optimal risk exposure, incorporating signals about future predictions, and minimizing trading costs, by leveraging market resilience.
• N. Chen, X. Wan, and N. Yang. (2024). Explicit Pathwise Expansion for Multivariate Diffusions and Its Application. Major Revision under Journal of Applied Probability. [full pdf] [abstract]
  In this paper, we provide expressions based on Hermite polynomials for the pathwise expansion method introduced by Watanabe (1987), Yoshida (1992b) and Li (2013). Our approach has two key innovations. First, we introduce a quasi-Lamperti transform that unitizes the process’ diffusion matrix at the initial time, as it corresponds to a multi-dimensional uncorrelated normal distribution, thus facilitating subsequent analysis. Second, by utilizing explicit expressions for the conditional expectation of the multiplication of iterated Itô integrals, we derive explicit formulas for the conditional expectation of the pathwise expansion of a general function on the transformed diffusion. Applying the newly derived method to the conditional expectation of the Dirac delta function and Hermite polynomial functions, respectively, we obtain alternative expressions for the pathwise based density expansion proposed by Li (2013) and the Hermite polynomials based density expansion introduced in Yang et al. (2019) and Wan and Yang (2021). We show that the formulas obtained from these two expansion methods are essentially the same by rearranging the terms according to the increasing order of Hermite polynomials.
• N. Chen and H. Xu. (2024). The Complementary Role of Digital Engagement in Social Activeness of Elderly Life: Evidence from Urban Region of China.
• N. Chen, X. Giroud, L. Qin, and N. Wang. (2024). Corporate Investment and Savings Demand. [abstract]
  Why do some firms save more than others? We revisit this important, widely studied question by developing a tractable continuous-time capital accumulation model for a financially constrained firm facing costly external equity. In addition to including the standard building blocks for a Q-theory of investment (Hayashi, 1982; Abel and Eberly, 1994): capital accumulation, capital adjustment costs, and a persistent stochastic productivity process, we assume that instantaneous profits (cashflows generated by productive capital) are stochastic. This assumption is a key difference between our model and widely used quantitative corporate finance models, e.g., Hennessy and Whited (2007) and Riddick and Whited (2009), in which instantaneous profits (conditional on productivity and capital stock) are deterministic. Firm value $V_t$ depends on capital stock $K_t$, cash balance $W_t$, and productivity $A_t$. Using the homogeneity property, we analytically characterize the solution with a variational inequality for $w_t = W_t / K_t$ and $A_t$. Importantly, we show that making instantaneous profits stochastic is key to generating quantitatively meaningful cash holdings. Consider the alternative that instantaneous profits are (locally) deterministic. Then the firm's cash balance at the end of the period would also be deterministic. This substantially reduces its precautionary savings demand, as locally the risk-neutral firm (as long as it is not too cash strapped) faces little liquidation risk and hence holding cash has limited value. Mathematically whether (conditional) instantaneous profits are stochastic implies whether the quadratic-variation $V_{WW}$ term appears in the Bellman equation, and economically this term has a first-order effect on precautionary savings demand.

Published Papers

Archival Journals

• Information Relaxation and A Duality-Driven Algorithm for Stochastic Dynamic Programs (with X. Ma, Y. Liu, and W. Yu). Forthcoming in Operations Research. [full pdf] [abstract]
  We use the technique of information relaxation to develop a duality-driven iterative approach to obtaining and improving confidence interval estimates for the true value of finite-horizon stochastic dynamic programming problems. We show that the sequence of dual value estimates yielded from the proposed approach in principle monotonically converges to the true value function in a finite number of dual iterations. Aiming to overcome the curse of dimensionality in various applications, we also introduce a regression-based Monte Carlo algorithm for implementation. The new approach can be used not only to assess the quality of heuristic policies, but also to improve them if we find that their duality gap is large. We obtain the convergence rate of our Monte Carlo method in terms of the amounts of both basis functions and the sampled states. Finally, we demonstrate the effectiveness of our method in an optimal order execution problem with market friction and in an inventory management problem in the presence of lost sale and lead time. Both examples are well known in the literature to be difficult to solve for optimality. The experiments show that our method can significantly improve the heuristics suggested in the literature and obtain new policies with a satisfactory performance guarantee.
• Not All Market Participants Are Alike When Facing Crisis: Evidence from the 2015 Chinese Stock Market Turbulence (with C. Sui, and M. Yang). Pacific-Basin Finance Journal, Vol.82, 102164, 2023. [full pdf] [abstract]
  We propose a novel framework to analyze the potentially heterogeneous roles played by different market participants in the fire-sale process during market crash and illustrate the methodology with the 2015–16 Chinese stock market turbulence. Unlike conventional analysis focusing on one particular channel of fire sales, we establish a market-level measure of fire sales based on the decomposition of diffusion processes to quantitatively compare the contribution of various channels in driving stock prices to plummet. Empirical results identify mutual funds as the main fire-sale propagator, as well as the heterogeneities in response to the price crash among different market participants.
• Robust Risk Quantification via Shock Propagation in Financial Networks (with D. Ahn and K.-K. Kim). Operations Research, Vol. 72, pp. 1-18, 2023. [full pdf] [abstract]
  Given limited network information, we consider robust risk quantification under the Eisenberg–Noe model for financial networks. To be more specific, motivated by the fact that the structure of the interbank network is not completely known in practice, we propose a robust optimization approach to obtain worst-case default probabilities and associated capital requirements for a specific group of banks (e.g., systemically important financial institutions) under network information uncertainty. Using this tool, we analyze the effects of various incomplete network information structures on these worst-case quantities and provide regulatory insights into the collection of actionable network information. All claims are numerically illustrated using data from the European banking system.
• A New Delta Expansion for Multivariate Diffusions via the Ito-Taylor Expansion (with N. Yang and X. Wan). Journal of Econometrics, Vol. 209, pp. 256-288, 2019. [full pdf] [abstract]
  In this paper we develop a new delta expansion approach to deriving analytical approximation to the transition densities of multivariate diffusions using the Itˆo-Taylor expansion of the conditional expectation of the Dirac delta function. Our approach yields an explicit recursive formulas for the expansion coefficients and is universally applicable for a wide spectrum of models, particularly the time-inhomogeneous non-affine irreducible multivariate diffusions. We show that this new approach can be viewed as an extension of A¨ıt-Sahalia (2002) and Lee et al. (2014) to the case of multivariate models. The derived expansions are proved to converge to the true probability density as the observational time interval shrinks. The obtained approximations can thereby be used to carry out the maximum likelihood estimation for the diffusions with discretely observed data. Extensive numerical experiments demonstrate the accuracy and effectiveness of our approach.
  
  ---江蘇省第十七屆哲學社會科學優秀成果獎二等獎 (This paper won Second place prize of the 17th Jiangsu Provincial Philosophy and Social Science Outstanding Research Achievement Award), 2023.
• The Principle of Not Feeling the Boundary for the SABR Model (with N. Yang). Quantative Finance, Vol. 19, pp. 427-436, 2019. [full pdf] [abstract]
  The stochastic-alpha-beta-rho (SABR) model is widely used in fixed income and foreign exchange markets as a benchmark. The underlying process may hit zero with a positive probability and therefore an absorbing boundary at zero should be specified to avoid arbitrage opportunities. However, a variety of numerical methods choose to ignore the boundary condition to maintain the tractability. This paper develops a new principle of not feeling the boundary to quantify the impact of this boundary condition to the distribution of underlying prices. It shows that the probability of the SABR hitting zero decays to 0 exponentially as the time horizon shrinks. Applying this principle, we further show that conditional on the volatility process, the distribution of the underlying process can be approximated by that of a time-changed Bessel process with an exponentially negligible error. This discovery provides a theoretical justification for many almost exact simulation algorithms for the SABR model in the literature. Numerical experiments are also presented to support our results.
• Approximate Arbitrage-Free Option Pricing under the SABR Model (with N. Yang, Y. Liu and X. Wan). Journal of Economic Dynamics and Control, Vol. 83, pp. 98-214, 201. [full pdf] [abstract]
  The stochastic-alpha-beta-rho (SABR) model introduced by Hagan et al. (2002) provides a popular vehicle to model the implied volatilities in the interest rate and foreign exchange markets. To exclude arbitrage opportunities, we need to specify an absorbing boundary at zero for this model, which the existing analytical approaches to pricing derivatives under the SABR model typically ignore. This paper develops closed-form approximations to the prices of vanilla options to incorporate the effect of such a boundary condition. Different from the traditional normal distribution-based approximations, our method stems from an expansion around a one-dimensional Bessel process. Extensive numerical experiments demonstrate its accuracy and efficiency. Furthermore, the explicit expression yielded from our method is appealing from the practical perspective because it can lead to fast calibration, pricing, and hedging.
• Contingent Capital, Tail Risk, and Debt-induced Collapse (with P. Glasserman, B. Nouri and M. Pelger). Review of Financial Studies, Vol. 30, pp. 3921-3969, 2017. [full pdf] [abstract]
  Contingent capital in the form of debt that converts to equity as a bank approaches financial distress offers a potential solution to the problem of banks that are too big to fail. This paper studies the design of contingent convertible bonds and their incentive effects in a structural model with endogenous default, debt rollover, and tail risk in the form of downward jumps in asset value. We show that once a firm issues contingent convertibles, the shareholders’ optimal bankruptcy boundary can be at one of two levels: a lower level with a lower default risk or a higher level at which default precedes conversion. An increase in the firm’s total debt load can move the firm from the first regime to the second, a phenomenon we call debt-induced collapse because it is accompanied by a sharp drop in equity value. We show that setting the contractual trigger for conversion sufficiently high avoids this hazard. With this condition in place, we investigate the effect of contingent capital and debt maturity on capital structure, debt overhang, and asset substitution. We also calibrate the model to past data on the largest U.S. bank holding companies to see what impact contingent convertible debt might have had under the conditions of the financial crisis.
  [ supplement appendix]
  ---A preliminary version of this paper was circulated under the title of CoCos, Bail-in, and Tail Risk. Click this link to the working paper series of Office of Financial Research, The Treasury Department, US.
  ---Accepted for presentation at the 2014 Western Finance Association Meeting (Acceptance rate: 144/1667).
  ---Accepted for presentation at the 2016 Annual Meeting of American Economic Association (Acceptance rate:<25%).
• Exact Simulation of the SABR Model (with N. Cai, Y. Song). Opeartions Research, Vol. 65, pp. 931-951, 2017. [full pdf] [abstract]
  The stochastic alpha-beta-rho (SABR) model becomes popular in the financial industry because it is capable of providing good fits to various types of implied volatility curves observed in the marketplace. However, no analytical solution to the SABR model exists that can be simulated directly. This paper explores the possibility of exact simulation for the SABR model. Our contribution is threefold. (i) We propose an exact simulation method for the forward price and its volatility in two special but practically interesting cases, i.e., when the elasticity β = 1, or when β< 1 and the price and volatility processes are instantaneously uncorrelated. Primary difficulties involved are how to simulate two random variables whose distributions can be expressed in terms of the Hartman-Watson and the noncentral chi-squared distribution functions, respectively. Two novel simulation schemes are proposed to achieve numerical accuracy, efficiency, and stability. One stems from numerical Laplace inversion and Asian option literature, and the other is based on recent developments in evaluating the noncentral chi-squared distribution functions in a robust way. Numerical examples demonstrate that our method is fast and accurate under various market environments. (ii) When β < 1 but the price and volatility processes are correlated, our simulation method becomes a semi-exact one. Numerical results suggest that it is still quite accurate when the time horizon is not long, e.g., no greater than one year. For long time horizons, a piecewise semi-exact simulation scheme is developed that reduces the biases substantially. (iii) For European option pricing under the SABR model, we propose a conditional simulation method, which reduces the variance of the plain simulation significantly, e.g., by more than 99%.
• An Optimization View of Financial Systemic Risk Modeling: The Network Effect and the Market Liquidity Effect (with X. Liu and D.D.Yao). Opeartions Research, Vol. 64, pp. 1089-1108, 2016. [full pdf] [abstract]
  Financial institutions are interconnected directly by holding debt claims against each other (the network channel), and they are also bound by the market when selling assets to raise cash in distressful circumstances (the liquidity channel). The goal of our study is to investigate how these two channels of risk interact to propagate individual defaults to a systemwide catastrophe. We formulate a constrained optimization problem that incorporates both channels of risk, and exploit the problem structure to generate the solution (to the clearing payment vector) via a partition algorithm. Through sensitivity analysis, we are able to identify two key contributors to financial systemic risk, the network multiplier and the liquidity amplifier, and to discern the qualitative difference between the two, confirming that the market liquidity effect has a great potential to cause systemwide contagion. We illustrate the network and market liquidity effects—in particular, the significance of the latter—in the formation of systemic risk with data from the European banking system. Our results contribute to a better understanding of the effectiveness of certain policy interventions. In addition, our algorithm can be used to pin down the changes of the net worth (marked to market) of each bank in the system as the spillover effect spreads, so as to estimate the extent of contagion, and to provide a metric of financial resilience as well. Our framework can also be easily extended to incorporate the effect of bankruptcy costs.
  
  ---Xin Liu was selected to finalists (top 5) of the Best Student Research Paper Competition, Session of Financial Service, INFORMS,2015.
  ---Xin Liu won second prize of the Best Student Research Paper Competition, the 3rd Asian Quantitative Finance Conference, 2015.
• Game Options Analysis of the Information Role of Call Policies in Convertible Bonds (with C. M. Leung and Y. K. Kwok). Applied Mathematical Finance, Vol. 22, pp. 297-335，2015. [full pdf] [abstract]
  In debt financing, existence of information asymmetry on the firm quality between the firm management and bond investors may lead to significant adverse selection costs. We develop the two-stage sequential dynamic two-person game option models to analyze the market signaling role of the callable feature in convertible bonds. We show that firms with positive private information on earning potential may signal their type to investors via the callable feature in a convertible bond. We present the variational inequalities formulation with respect to various equilibrium strategies in the two-person game option models via characterization of the optimal stopping rules adopted by the bond issuer and bondholders. The bondholders’ belief system on the firm quality may be revealed with the passage of time when the issuer follows his optimal strategy of declaring call or bankruptcy. Under separating equilibrium, the quality status of the firm is revealed so the information asymmetry game becomes a new game under complete information. To analyze pooling equilibrium, the corresponding incentive compatibility constraint is derived. We manage to deduce the sufficient conditions for the existence of signaling equilibrium of our game option model under information asymmetry. We analyze how the callable feature may lower the adverse selection costs in convertible bond financing. We show how low quality firm may benefit from information asymmetry and vice versa, underpricing of the value of debt issued by a high quality firm.
• Optimal Double Stopping of a Brownian Bridge (with E. J. Baurdoux, B. A. Surya and K. Yamazaki). Advances in Applied Probability, Vol. 47, pp. 1212-1234，2015. [full pdf] [abstract]
  We study optimal double stopping problems driven by a Brownian bridge. The objective is to maximize the expected spread between the payoffs achieved at the two stopping times. We study several cases where the solutions can be solved explicitly by strategies of threshold type.
• American Option Sensitivity Estimation via a Generalized IPA Approach (with Y. Liu). Operations Research, Vol.62, pp.616–632, 2014. [full pdf] [abstract]
  In this paper, we develop ecient Monte Carlo methods for estimating American option sensitivities. The problem can be re-formulated as how to perform sensitivity analysis for a stochastic optimization problem with model uncertainty. We introduce a generalized infinitesimal perturbation analysis (IPA) approach to resolve the difficulty caused by discontinuity of the optimal decision with respect to the underlying parameter. The IPA estimators are unbiased if the optimal decisions are explicitly known. To quantify the estimation bias caused by untractable exercising policies in the case of pricing American options, we also provide an approximation guarantee which relates the sensitivity under the optimal exercise policy to that computed under a suboptimal policy. The price-sensitivity estimators yielded from this approach demonstrate significant advantages numerically in both high-dimensional environments and various process settings. We can easily embed them into many of the most popular pricing algorithms without extra simulation effort to obtain sensitivities as a by-product of the option price. Our generalized approach also casts new insights on how to perform sensitivity analysis using IPA: we do not need pathwise continuity to apply it.
• Localization and Exact Simulation of Brownian Motion Driven Stochastic Differential Equations (with Z. Huang). Mathematics of Operations Research, Vol. 38, pp. 591-616, 2013. [full pdf] [abstract]
  A typical application of Monte Carlo simulation in financial engineering usually starts by simulating sample paths of stochastic dfiferential equations (SDE). Discretization is a popular approximate approach to generating those paths: it is easy to implement but prone to simulation bias. This article presents a new simulation scheme to exactly generate samples for SDEs. The key observation is that the law of a general SDE can be decomposed into a product of standard Brownian motion and a doubly stochastic Poisson process. An acceptance-rejection algorithm is devised based on the combination of this decomposition and a localization technique. The numerical results reveals that the mean-square error of the proposed method is in the order of O(t^{-1/2}), which is superior to conventional discretization schemes. Furthermore, the proposed method also can generate exact samples for SDE with boundaries which the discretization schemes usually find difficulty in dealing with.
• Brownian Meanders, Importance Sampling and Unbiased Simulation of Diffusion Extremes (with Z. Huang). Operations Research Letters, Vol. 40, pp.554-563, 2012. [full pdf] [abstract]
  Computing expected values of functions involving extreme values of diffusion processes can find wide applications in financial engineering. Conventional discretization simulation schemes often converge slowly. We propose a Wiener-measure-decomposition based approach to construct unbiased Monte Carlo estimators. Combined with the importance sampling technique and the Williams path decomposition of Brownian motion, this approach transforms simulating extreme values of a general diffusion process to simulating two Brownian meanders. Numerical experiments show this estimator performs effciently for diffusions with and without boundaries.
• A Non-Zero-Sum Game Approach for Convertible Bonds: Tax Benefits, Bankrupt Cost and Early/Late Call (with M. Dai and X. Wan). Mathematical Finance, Vol. 23, pp.57-93, 2010. [full pdf] [abstract]
  Convertible bonds are hybrid securities that embody the characteristics of both straight bonds and equities. The conflicts of interest between bondholders and shareholders affect the security prices significantly. In this paper, we investigate how to use a non-zero-sum game framework to model the interaction between bondholders and shareholders and to evaluate the bond accordingly. Mathematically, this problem can be reduced to a system of variational inequalities and we explicitly derive the Nash equilibrium to the game. Our model shows that credit risk and tax benefit have considerable impacts on the optimal strategies of both parties. The shareholder may issue a call when the debt is in-the-money or out-of-the-money. This is consistent with the empirical findings of "late and early calls" (Ingersoll (1977), Mikkelson (1981), Cowan et al. (1993), Asquith (1995)). In addition, the optimal call policy under our model offers an explanation for certain stylized patterns.
  
  ---Second place prize of the Best Student Research Paper Award Competition of the Financial Services Section, INFORMS, 2010.
• Occupation Times of Jump-Diffusion Processes with Double Exponential Jumps and the Pricing of Options (with N. Cai and X. Wan). Mathematics of Operations Research, Vol. 35, pp. 412-437, 2010. [full pdf] [abstract]
  In this paper, we provide Laplace transform-based analytical solutions to pricing problems of various occupation-time-related derivatives such as step options, corridor options, and quantile options under Kou's double exponential jump diffusion model. These transforms can be inverted numerically via the Euler Laplace inversion algorithm, and the numerical results illustrate that our pricing methods are accurate and efficient. The analytical solutions can be obtained primarily because we derive the closed-form Laplace transform of the joint distribution of the occupation time and the terminal value of the double exponential jump diffusion process. Beyond financial applications, the mathematical results about occupation times of a jump diffusion process are of more general interest in applied probability.
• Credit Spread, Implied Volatility, and Optimal Capital Structures with Jump Risk and Endogenous Defaults (with S. Kou). Mathematical Finance, Vol. 19, pp. 343-378, 2009. [full pdf] [abstract]
  We propose a two-sided jump model for credit risk by extending the Leland–Toft endogenous default model based on the geometric Brownian motion. The model shows that jump risk and endogenous default can have significant impacts on credit spreads, optimal capital structure, and implied volatility of equity options: (1) Jumps and endogenous default can produce a variety of non-zero credit spreads, including upward, humped, and downward shapes; interesting enough, the model can even produce, consistent with empirical findings, upward credit spreads for speculative grade bonds. (2) The jump risk leads to much lower optimal debt/equity ratio; in fact, with jump risk, highly risky firms tend to have very little debt. (3) The two-sided jumps lead to a variety of shapes for the implied volatility of equity options, even for long maturity options; although in general credit spreads and implied volatility tend to move in the same direction under exogenous default models, this may not be true in presence of endogenous default and jumps. Pricing formulae of credit default swaps and equity default swaps are also given. In terms of mathematical contribution, we give a proof of a version of the “smooth fitting” principle under the jump model, justifying a conjecture first suggested by Leland and Toft under the Brownian model.
  
  ---Second place prize of the Best Student Research Paper Award Competition of the Financial Services Section, INFORMS, 2006.
• Pricing Double Barrier Options under a Flexible Jump Diffusion Model (with N. Cai and X. Wan). Operations Research Letters, Vol. 37, pp. 163-167, 2008. [full pdf] [abstract]
  In this paper we present a Laplace transform-based analytical solution for pricing double-barrier options under a flexible hyper-exponential jump diffusion model (HEM). The major theoretical contribution is that we prove non-singularity of a related high-dimensional matrix, which guarantees the existence and uniqueness of the solution.
• Malliavin Greeks without Malliavin Calculus (with P. Glasserman). Stochastic Processes and their Applications, Vol. 117, pp. 1689-1723, 2007. [full pdf] [abstract]
  We derive and analyze Monte Carlo estimators of price sensitivities ("Greeks") for contingent claims priced in a diffusion model. There have traditionally been two categories of methods for estimating sensitivities: methods that differentiate paths and methods that differentiate densities. A more recent line of work derives estimators through Malliavin calculus. The purpose of this article is to investigate connections between Malliavin estimators and the more traditional and elementary pathwise method and likelihood ratio method. Malliavin estimators have been derived directly for diffusion processes, but implementation typically requires simulation of a discrete-time approximation. This raises the question of whether one should discretize first and then differentiate, or differentiate first and then discretize.We show that in several important cases the first route leads to the same estimators as are found through Malliavin calculus, but using only elementary techniques. Time-averaging of multiple estimators emerges as a key feature in achieving convergence to the continuous-time limit.
• Additive and Multiplicative Duals for American Option Pricing (with P. Glasserman). Finance and Stochastics, Vol. 11, pp. 153-179, 2007. [full pdf] [abstract]
  We investigate and compare two dual formulations of the American option pricing problem based on two decompositions of supermartingales: the additive dual of Haugh and Kogan (Oper. Res. 52:258-270, 2004) and Rogers (Math. Finance 12:271-286, 2002) and the multiplicative dual of Jamshidian (Minimax optimality of Bermudan and American claims and their Monte- Carlo upper bound approximation. NIB Capital, The Hague, 2003). Both provide upper bounds on American option prices; we show how to improve these bounds iteratively and use this to show that any multiplicative dual can be improved by an additive dual and vice versa. This iterative improvement converges to the optimal value function.We also compare bias and variance under the two dual formulations as the time horizon grows; either method may have smaller bias, but the variance of the multiplicative method typically grows much faster than that of the additive method. We show that in the case of a discrete state space, the additive dual coincides with the dual of the optimal stopping problem in the sense of linear programming duality and the multiplicative method arises through a nonlinear duality.

Refereed Conference Proceedings

• Unbiased Monte Carlo Computation of Smooth Functions of Expectations via Taylor Expansions (with J. Blanchet and P.W. Glynn). Proceedings of the 2015 Winter Simulation Conference, pp.360-367, IEEE. [full pdf] [abstract]
  Many Monte Carlo computations involve computing quantities that can be expressed as g(EX), where g is nonlinear and smooth, and X is an easily simulatable random variable. The nonlinearity of g makes the conventional Monte Carlo estimator for such quantities biased. In this paper, we show how such quantities can be estimated without bias. However, our approach typically increases the variance. Thus, our approach is primarily of theoretical interest in the above setting. However, our method can also be applied to the computation of the inner expectation associated with Eg(E(X|Z)), and in this setting, the application of this method can have a significant positive effect on improving the rate of convergence relative to conventional "nested schemes" for carrying out such calculations.
• Unbiased Simulation of Distributions with Explicitly Known Integral Transforms (with D. Belomestny and Y. Wang). Monte Carlo and Quasi-Monte Carlo Methods 2014, pp.229-244, Springer Proceedings in Mathematics & Statistics. [full pdf] [abstract]
  In this paper, we propose an importance-sampling based method to obtain an unbiased simulator to evaluate expectations involving random variables whose probability density functions are unknown while their Fourier transforms have an explicit form.We give a general principle about how to choose appropriate importance samplers under different models. Compared with the existing methods, our method avoids time-consuming numerical Fourier inversion and can be applied effectively to high dimensional financial applications such as option pricing and sensitivity estimation under Heston stochastic volatility model, high dimensional affine jumpdiffusion model, and various Levy processes.
• Sensitivity Estimation of SABR Model via Derivative of Random Variable (with Y. Liu). Proceedings of the 2011 Winter Simulation Conference, pp.3871-3881, IEEE.
• Pathwise Derivative Method on Single-Asset American Option Sensitivity Estimation (with Y. Liu). Proceedings of the 2010 Winter Simulation Conference, pp.2721-2731, IEEE.
• A Wiener Measure Approach to Pricing Extreme Value Related Derivatives (with Z. Huang). Proceedings of the 2009 Winter Simulation Conference, pp. 1261-1271,IEEE.
• Monte Carlo Simulation in Financial Engineering (with L.J. Hong). Proceedings of the 2007 Winter Simulation Conference, pp. 919-931, IEEE.

Refereed Book Chapters

• Sensitivity Computations: Integration by Parts. Encyclopedia of Quantitative Finance, edited by Rama Cont, Johns Wiley & Sons, Ltd., West Sussex, UK, pp. 1636-1639. [full pdf] [abstract]
  Derivative price sensitivities, or greeks, play an important role in the practice of risk management to quantify the potential effects of the changes of underlying market parameters on the values of derivatives. However, how to calculate them efficiently is a challenging problem for computational finance. An obvious approach is to simulate replications of the model at perturbed parameters and then to use finite difference to form estimators. While this method has its own merits depending on the circumstances, it usually yields estimators with often unacceptably high variances, unless major computational efforts are made in terms of long calculation times. To obtain estimators with lower variance, traditional methods either differentiate the payoff functions of derivatives or differentiate the probability density of the underlying price. The former approach fails when the payoff functions are discontinuous while the latter meets difficulty if the explicit form of the density is not available. The integration-by-parts method overcomes both shortcomings of the traditional methods. It shifts the differential operator from the payoffs to the underlying diffusions in order to remove the smoothness requirement on the payoff functions. This method can be traced back to the Malliavin calculus in the field of stochastic analysis.