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, 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.
