Archival
Journals

D. Ahn, N. Chen, and K.K. Kim. (2021). Robust Risk Quantification via Shock Propagation in Financial Networks. Minor revision under Operations Research.
[abstract]
We consider the EisenbergNoe model for financial networks, focusing on random shocks to financial insti tutions. Using duality, we characterize shock amplification caused by the network structure and find the condition when a specific group of banks (e.g., SIFI) fails. This finding enables us to improve our understanding of shock propagation in financial networks. To be specific, we obtain robust bounds of default probabilities when only partial network information is available, and we observe that the link structure of the network contains crucial information. This is also confirmed by looking at asymptotic default probabilities in a small shock regime. With such analytical tools, systemic risk capital which prevents the default of target banks is discussed using chanceconstrained optimization. All the claims are numerically illustrated by an actual European banking network.

N. Chen, X. Ma, Y. Liu, and W. Yu. (2022). Information Relaxation and A DualityDriven Algorithm for Stochastic Dynamic Programs. Minor revision under Operations Research.
[full pdf]
[abstract]
We use the technique of information relaxation to develop a dualitydriven iterative approach to obtaining and improving confidence interval estimates for the true value of finitehorizon 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 regressionbased 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.

A New Delta Expansion for Multivariate Diffusions via the ItoTaylor Expansion (with N. Yang
and X. Wan).
Journal of Econometrics Vol.209, pp. 256288, 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 ItˆoTaylor
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 timeinhomogeneous nonaffine
irreducible multivariate diffusions. We show that this new approach can be viewed as
an extension of A¨ıtSahalia (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. 427436, 2019.
[full pdf]
[abstract]
The stochasticalphabetarho (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 timechanged 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 ArbitrageFree Option Pricing under the
SABR Model (with N. Yang, Y. Liu and X. Wan).
Journal of Economic Dynamics and Control, Vol.83, pp.198214,
2017.
[full pdf]
[abstract]
The stochasticalphabetarho (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 closedform
approximations to the prices of vanilla options to incorporate the effect of such
a boundary condition. Different from the traditional normal distributionbased approximations,
our method stems from an expansion around a onedimensional 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
Debtinduced Collapse (with P. Glasserman, B. Nouri and M. Pelger). Review of Financial Studies
Vol.30, pp.39213969, 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 debtinduced 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,
Bailin, 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.931951, 2017.
[full pdf]
[abstract]
The stochastic alphabetarho (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 HartmanWatson and the noncentral chisquared 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
chisquared 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 semiexact 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 semiexact 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.10891108, 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. 297335，2015.
[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 twostage sequential dynamic twoperson 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 twoperson 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.
12121234，2015.
[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 reformulated 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 pricesensitivity estimators yielded from this approach demonstrate significant
advantages numerically in both highdimensional 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 byproduct 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. 591616, 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
acceptancerejection algorithm is devised based on the combination of
this decomposition and a localization technique. The numerical results
reveals that the meansquare 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, 40,
pp.554563, 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 Wienermeasuredecomposition 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
NonZeroSum Game Approach for Convertible Bonds: Tax Benefits,
Bankrupt Cost and Early/Late Call (with
M. Dai and X. Wan). Mathematical Finance,
Vol. 23, pp.5793, 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 nonzerosum
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 inthemoney
or outofthemoney. 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 JumpDiffusion Processes with Double
Exponential Jumps and the Pricing of Options (with
N. Cai and X. Wan). Mathematics of Operations
Research, Vol. 35, pp. 412437, 2010.
[full pdf]
[abstract]
In this paper, we provide Laplace transformbased analytical solutions
to pricing problems of various occupationtimerelated 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 closedform 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. 343378, 2009. [full pdf]
[abstract]
We propose a twosided 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 nonzero 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 twosided 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. 163167, 2008.
[full pdf]
[abstract]
In this paper we present a Laplace transformbased analytical solution for pricing doublebarrier options under
a flexible hyperexponential jump diffusion model (HEM).
The major theoretical contribution is that we prove nonsingularity
of a related highdimensional 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. 16891723, 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
discretetime 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. Timeaveraging of
multiple estimators emerges as a key feature in achieving convergence
to the continuoustime limit.
 Additive and Multiplicative Duals for American
Option Pricing (with
P. Glasserman). Finance and Stochastics, Vol. 11, pp. 153179, 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.
52:258270, 2004) and Rogers (Math. Finance 12:271286, 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.360367, 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(XZ)), 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 QuasiMonte Carlo Methods
2014, pp.229244, Springer Proceedings in Mathematics & Statistics.
[full pdf]
[abstract]
In this paper, we propose an importancesampling 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 timeconsuming 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.38713881, IEEE.
 Pathwise Derivative Method on SingleAsset American Option
Sensitivity Estimation (with
Y. Liu). Proceedings of the 2010 Winter Simulation
Conference, pp.27212731, IEEE.
 A
Wiener Measure Approach to Pricing Extreme Value Related
Derivatives (with
Z. Huang). Proceedings of the 2009 Winter Simulation Conference, pp.
12611271,IEEE.
 Monte Carlo Simulation in Financial Engineering
(with L.J. Hong).
Proceedings of the 2007 Winter Simulation Conference, pp. 919931,
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. 16361639.
[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
integrationbyparts 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.