# THEMATIC PROGRAMS

October 22, 2014

### Visitors Seminar Series at the Fields Institute

All seminars will be held at the Fields Institute, 3rd Floor Stewart Library unless otherwise noted.

 Thematic Program Homepage Directions to Fields Affiliated Seminars

### Visitor Seminars in the Quantitative Finance Program Tuesday 1:30 pm - 3 pm

 PAST SEMINARS Tuesday February 2, 2010 1:30 pm - 2:15 pm Stewart Library     2:15 pm - 3:00 pm Stewart Library Hao Xing, PhD (Michigan) On the martingale property and the uniqueness of pricing equations in stochastic volatility models It is well known that the stock price process is only a strict local martingale in certain stochastic volatility models. The lack of the martingale property gives multiple solutions to the pricing equation of European call options. In this talk we will show that the pricing equation has a unique solution if and only if the stock price process is a martingale. On the other hand, the choice of the boundary condition for the pricing equation at vanishing volatility is a thorny issue. Different boundary conditions have been chosen for different models in numerical computation literature. We will explain the probabilistic interpretation of these boundary conditions. Joint work in progress with Erhan Bayraktar and Kostas Kardaras. Klaas Schulze, PhD (Bonn) Risk Measurement via the Indifference Risk Aversion Principle The phenomenon of risk plays a ubiquitous role in finance, insurance, and economics. We present a new principle for risk measurement, which refers to the level of risk aversion, which leaves an investor indifferent. Applying this principle in an elementary setting provides the indifference measure. This measure is shown to be a meaningful risk measure, as it characterizes all so-called dual risk measures, which respect comparative risk aversion. This characterization provides a decomposition and a construction method for dual risk measures. Tuesday February 9, 2010 1:30 pm - 2:15 pm Stewart Library         2:15 pm - 3:00 pm Stewart Library Dr. Xianhua Peng, Fields Ontario Postdoctoral Fellow (Fields/York) What Is a Good External Risk Measure: Bridging the Gaps between Robustness, Subadditivity, and Insurance Risk Measures The recently revised Basel II risk measure for market risk, which uses Value-at-Risk (VaR) with scenario analysis, does not belong to any existing theoretical framework of risk measures proposed in the academic literature. We propose new data-based risk measures, called natural risk statistics, by postulating a new set of axioms that only require subadditivity for commonotonic random losses. Natural risk statistics include (i) the tail conditional median that is more robust than the tail conditional expectation suggested by coherent risk measures, and (ii) VaR with scenario analysis, in particular the current and recently revised Basel II risk measures, as special cases. Hence, natural risk statistics provide axiomatic justification for Basel II risk measures. In addition, we emphasize that risk measures used for external regulation should be robust with respect to model misspecification and small changes in the data. We show that coherent risk measures are generally not robust with respect to data and insurance risk measures are generally not robust enough with respect to model misspecification. Andreas E. Kyprianou, Professor of Probability, The University of Bath Wiener-Hopf Monte-Carlo methods for Levy processes We introduce a completely new way of performing Monte-Carlo simulation using Wiener-Hopf factorization for functionals of a Levy process involving the joint law of the process and its maximum at a fixed time. The method has pertinence in numerically pricing barrier options. This is based on joint work with Juan Carlos Pardo and Kees van Schaik. Tuesday February 16, 2010 1:30 pm Stewart Library Dr. Vladimir Surkov, (Fields/University of Western Ontario) Efficient Fourier Transform-Based Pricing of Interest Rate Derivatives In this article, we introduce a new and efficient numerical method, called Fourier Space Time-stepping (irFST), for valuation of general interest rate contingent claims. While a wide array of models for the instantaneous short-rate exist in the literature, we focus on the Vasicek and Hull-White models, and their multi-factor and jump extensions. Even though closed-form solutions exist (for models without jumps) for zero-bond and swap options, the motivation for the irFST method is to efficiently price exotic, path-dependent, and early-exercise options. Recently, Fourier transform-based method have methods have began to gain traction in the area of pricing of interest rate derivatives. Duffie, Pan, Singleton (2000) and Chako, Das (2002) have used a Fourier inversion technique to obtain option prices for a wide range of problems. By adapting the approach of Lewis (2001) to short-rateinterest rate models, Bouziane (2008) develops a method that computes option prices for a range of strike prices. The irFST method is based on the mean-reverting Fourier Space Time-stepping algorithm of Jaimungal, Surkov (2008), but is tailored to interest rate models. It solves the partial integro-differential equation satisfied by the option price by first converting it into an ordinary differential equation (ODE) in Fourier space and then solves the resulting ODE analytically. The fast Fourier transform algorithm is used to efficiently switch between real and frequency spaces while the time step is performed in frequency space. The advantage of such approach is that European options can be priced in a single time step while Bermudan options do not require time-stepping between the monitoring dates. Since our approach is to apply the Fourier transform on the short-rate variable, we compute option prices for a range of short-rates, which enablesefficient pricing of path-dependent options as well. We also discuss an extension of the irFST method to efficiently compute option Greeks (sensitivities to changes in state variables or models parameters). Finally, we demonstrate the precision and flexibility of our numerical method through various examples. Tuesday February 23, 2010 1:30 pm Stewart Library Dr. Vladimir Vinogradov (Ohio University) Stochastic Models for Movements of Equities which Employ the Power-Variance Family The first group of models pertains to a class of Levy processes generated starting from the power-variance family of probability laws.For such models, we establish the exact asymptotics of the probabilities of large deviations, the distribution of the first passage time, and the rational price of the European call option. A subsequent application of the techniques of stochastic exponentials leads to an additional class of Levy processes. The ordinary exponentials of its members constitute the geometric Levy processes which we utilize for describing chaotic movements of equities. Namely, we consider a self-financing portfolio comprised of one bond and k equities assuming that the returns on all k equities belong to the latter class. We demonstrate that for a particular choice of constant portfolio weights, the combined movement of k equities is governed by a geometric Levy process, which belongs to the same class. The Merton-type allocation of constant weights, which we implement, coincides with those of fund managers. Although simpler, in the discontinuous case this approach is less profitable, than portfolio weight selection using an approach that maximizes the expected logarithmic utility. In a special case, we establish a converse of Merton's mutual fund theorem. We derive Pythagorean-type theorems for Sharpe portfolio performance measures emphasizing their relation to Merton-type weights and the additivity of shape parameter. Tuesday March 2, 2010 1:30 pm Stewart Library Kyoung-Kuk Kim, PhD (Columbia) Stability Analysis of Riccati Differential Equations for Affine Diffusions in Option Pricing Theory We study a class of generalized Riccati differential equations associated with affine diffusion processes. These diffusions arise in financial econometrics as candidate models for asset price dynamics. The generalized Riccati equations determine the Fourier transform of the diffusion’s transition law. We investigate stable regions of the dynamical systems and analyze their blow-up times. We discuss the implication of applying these results to affine diffusions and, in particular, to option pricing theory. Tuesday March 9, 2010 1:30 pm Stewart Library Rudra Jena (Centre de Mathématiques Appliquées, Ecole Polytechnique) Arbitrage Opportunities in Misspecified Stochastic Volatility Models Given a set of assumptions on the real-world dynamics of the underlying, the European options on this underlying are not efficiently priced in options markets, giving rise to arbitrage opportunities. In this paper, we study these scenarios for a generic stochastic volatility model and construct strategies to maximize the profit. When the misspecified model is Black Scholes, we make the folklore of butterflies and risk reversals being optimal strategies, precise and determine exactly which options must be bought and sold to maximize arbitrage gains, depending on model parameters. For the case of a specific misspecified stochastic volatility model, SABR $\beta=1$, we discuss perturbative results and provide the analytic expressions for arbitrage profits. We demonstrate the results of our strategies for a numerical example with and without including transaction costs for various options. Monday March 15, 2010 1:30-2:15 Room 230 Prof. Tom Salisbury (York University) Insurance and equity guarantees I'll survey some recent work on variable annuities, and the kind of equity guarantees insurance companies have been building into retirement savings products. We'll look at hedging issues, from the point of view of the issuer. The challenge here is that the guarantees involve long-dated embedded options, and blend mortality risk with equity risk. We'll also consider some of the issues clients have in optimally managing portfolios that include these products. My work in this area is joint with Huaxiong Huang and Moshe Milevsky, both of York University. Monday March 15, 2010 2:15-3:00 Room 230 Dr. Fouad Marri, (Fields Institute/York University) Pricing compound Poisson processes with the Farlie-Gumbel-Morgenstern dependence structure In this paper we consider an extension to the classical compound Poisson risk model. Historically, it has been assumed that the loss amounts and claim inter-arrival times are independent. In this contribution, a dependence structure between the claim amount and the interclaim time is introduced through a Farlie-Gumbel-Morgenstern copula. In this framework, the moment generating function of the aforementioned dependent processes is derived and studied. The moments of the compound Poisson risk model are also derived. Various implications of the dependence are discussed and exemplified numerically. This is joint work with Edward Furman (York University). Tuesday April 6, 2010 1:30 - 2:15 Stewart Library Alexey Kuznetsov (York University) 12 functionals of Levy processes you always wanted to know how to compute (but were afraid to ask) We discuss the problem of computing marginal and joint distribution of such functionals of a Levy process as the first passage time, overshoot, undershoot, extrema, last extrema before the first passage, etc. First we discuss a rather simple case of a process with Gaussian component and two-sided hyper-exponential jumps, and then present our recent results on processes with meromorphic characteristic exponent, including beta, theta and hypergeometric families of processes. These families are qualitatively similar to the well-known CGMY (KoBoL) family and they allow for a flexible modelling of small jumps. We will also discuss some applications of these results to pricing various exotic options in Mathematical Finance. This is based on joint work with Andreas Kyprianou and Juan Carlos Pardo. Tuesday April 13, 2010 1:30 pm Stewart Library Arash Fahim A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs We introduce a probabilistic numerical scheme for fully nonlinear PDEs, and show that it can be introduced naturally as a combination of Monte Carlo and finite differences scheme without appealing to the theory of backward stochastic differential equations. We mention the convergence of the discrete-time approximation and derives a bound on the discretization error in terms of the time step. An explicit implementable scheme requires to approximate the conditional expectation operators involved in the discretization. This induces a further Monte Carlo error. We also mention the result which proves the convergence of the latter approximation scheme, and derives an upper bound on the approximation error. Numerical experiments are performed for the approximation of the solution of the mean curvature flow equation in dimensions two and three, and for two and five-dimensional (plus time) fully-nonlinear Hamilton-Jacobi-Bellman equations arising in the theory of portfolio optimization in financial mathematics. We also provide a generalization to nonlocal fully nonlinear PDEs (Integro-differential PDEs) Tuesday April 20, 2010 1:30 - 2:15 pm Stewart Library Xianhua Peng (Fields/York University) Default Clustering and Valuation of Collateralized Debt Obligations The recent financial turmoil has witnessed the powerful impact of the default clustering effect (i.e., one default event tends to trigger more default events in the future and cross-sectionally), especially on the market of collateralized debt obligations (CDOs). We propose a model based on cumulative default intensities that can incorporate the default clustering effect. Furthermore, the model is tractable enough to provide a direct link between single-name credit securities, such as credit default swaps (CDS), and multi-name credit securities, such as CDOs. The result of calibration to the recent market data, when Bear Sterns, Lehman Brothers, etc. collapsed and default correlation among firms was substantially high, shows that the model is promising. Tuesday April 27, 2010 1:30 pm - 2:15 pm Stewart Library Dr. Anke Wiese (Heriot-Watt University) Stochastic expansions and Hopf algebras We study solutions to nonlinear stochastic differential systems driven by a multi-dimensional Wiener process. A useful algorithm for strongly simulating such stochastic systems is the Castell-Gaines method, which is based on the exponential Lie series. When the diffusion vector fields commute, it has been proved that at low orders this method is more accurate in the mean-square error than corresponding stochastic Taylor methods. However it has also been shown that when the diffusion vector fields do not commute, this is not true for strong order one methods. Here we prove that when the diffusion vector fields do not commute, the exponential Lie series is usurped by the sinh-log series. In other words, the mean-square error associated with a numerical method based on the sinh-log series, is always smaller than the corresponding stochastic Taylor error, in fact to all orders. Our proof utilizes the underlying Hopf algebra structure of these series, and a two-alphabet associative algebra of shuffle and concatenation operations. We illustrate the benefits of the proposed series in numerical studies. Joint work with S.J. Malham, Heriot-Watt University. Tuesday May 4, 2010 1:30 - 2:15 pm Stewart Library Eddie Ng Kernel-based Copula Processes The field of time-series analysis has made important contributions to a wide spectrum of applications such as tide-level studies in hydrology, natural resource prospecting in geo-statistics, speech recognition, weather forecasting, financial trading, and economic forecasts and analysis. Nevertheless, the analysis of the non-Gaussian and non-stationary features of time-series remains challenging for the current state-of-art models. This work proposes an innovative framework which leverages the theory of copula, combined with a probabilistic framework from the machine learning community, to produce a versatile tool for multiple time-series analysis. I coined this new model Kernel-based Copula Processes (KCPs). Under the new proposed framework, various idiosyncracies can be modeled parsimoniously via a kernel function for individual time-series, and long-range dependency can be captured by a copula function. The copula function separates the marginal behavior and serial dependency structures, thus allowing them to be modeled separately and with much greater flexibility. Moreover, the codependent structure of a large number of time-series with potentially vastly different characteristics can be captured in a compact and elegant fashion through the notion of a binding copula. This feature allows a highly heterogeneous model to be built, breaking free from the homogeneous limitation of most conventional models. The KCPs have demonstrated superior predictive power when used to forecast a multitude of data sets from meteorological and financial areas. Finally, the versatility of the KCP model is exemplified when it was successfully applied to non-trivial classification problems unaltered. Tuesday May 11, 2010 1:30 pm - 2:15 pm Stewart Library Prof. Luis Seco (University of Toronto) Correlations: the importance of being stochastic The events of 2007 and 2008 show that correlation regimes have transcendental impact in finance; I will present some mathematical considerations around this phenomenon and discuss some of its applications. Tuesday June 1, 2010 1:30 pm - 2:15 pm Stewart Library     2:15 pm - 3:00 pm Stewart Library Pavel Gapeev (London School of Economics) Constructing of L\’evy driven analogues of diffusions Stochastic differential equations driven by a class of L\'evy processes are considered and the question of finding closed form solutions is studied. Being based on smooth invertible transformations of the current states of the underlying processes, a reducibility criterion is presented for such stochastic differential equations to ones which are solvable using ordinary differential equations. A method is proposed for constructing L\'evy driven analogues of the initial continuous diffusions, which is based on this reducibility criterion. The action of this method is illustrated on the construction of some well known diffusions, and related applications to finance are discussed. Jean-Francois Renaud (University of Waterloo) A simple discretization scheme for nonnegative diffusion processes, with applications to option pricing A discretization scheme for nonnegative diffusion processes is proposed and the convergence of the corresponding sequence of approximating processes is proved using the martingale problem framework. Motivations for this scheme come typically from finance, especially for path-dependent option pricing. The scheme is simple: one only needs to find a nonnegative distribution whose mean and variance satisfy a simple condition to apply it. Then, for virtually any (path-dependent) payoff, Monte Carlo option prices obtained from this scheme will converge to the theoretical price. Examples of models and diffusion processes for which the scheme applies will be presented. This is joint work with Chantal Labbé and Bruno Rémillard (HEC Montréal) Wednesday June 9, 2010   1:30 pm - 2:15 pm Stewart Library       2:15 - 3:00 pm Stewart Library Lung Kwan Tsui (University of Pittsburgh) Multi-Factor Bottom-Up Model for Pricing Credit Derivatives In this note we continue the study of the stress event model, a simple and intuitive dynamic model for credit risky portfolios, proposed by Duffie and Singleton (1999). The model is a bottom-up version of the multi-factor portfolio credit model proposed by Longstaff and Rajan (2008). By a novel identifcation of independence conditions, we are able to decompose the loss distribution into a series expansion which not only provides a clear picture of the characteristics of the loss distribution but also suggests a fast and accurate approximation for it. Our approach has three important features: (i) it is able to match the standard CDS index tranche prices and the underlying CDS spreads, (ii) the computational speed of the loss distribution is very fast, comparable to that of the Gaussian copula, (iii) the computational cost for additional factors is mild, allowing for more flexibility for calibrations and opening the possibility of studying multi-factor default dependence of a portfolio via a bottom-up approach. We demonstrate the tractability and e?ciency of our approach by calibrating it to investment grade CDS index tranches. Alex Kreinin (Algorithmics) Combinatorics of Mills' Ratio The Mill's ratio, R(t), is defined as the ratio of the probability P(X>t) to the density of the standard normal random variable, X, computed at t. This function plays an important role in Statistics and Stochastic Modeling. In this talk we discuss analytical and combinatorial properties of this function. Our approach is based on the complete monotonicity of the function R(t). Tuesday June 15, 2010 1:30 - 2:15 pm Stewart Library               2:15 - 3:00 pm Stewart Library Joe Campolieti (Wilfrid Laurier University) Analytically Solvable Families of Local Volatility Diffusion Models:Spectral Expansions and Applications to Asset Pricing We present some recent developments in the construction, classification and application of new families of solvable diffusion models with affine drift and nonlinear volatility functions. The solvable diffusions admit closed-form analytical expressions for various transition densities, first hitting time distributions, distributions of various extrema of the processes, and other quantities that are fundamental to financial derivatives pricing. Our approach is based on so-called diffusion canonical transformations which exploit the use of measure changes (i.e. Doob transforms) in combination with elementary (Ito) transformations. The mathematical framework produces a large class of analytically tractable multi-parameter nonlinear local volatility diffusion models that are mapped onto various simpler underlying diffusions. In particular, in this talk we present a spectral theory for transformed diffusions and derive closed-form spectral expansions for first-hitting time densities and transition probability densities for three new main families of one-dimensional diffusions with (and without) imposed killing. The rapidly convergent spectral expansions lead to various applications in asset pricing. As an example of an application, in this talk we specifically examine the so-called Confluent-U asset pricing model for both credit risk modeling and option pricing. An equity-based structural first-passage time default model is constructed based on the Confluent-U model with efficient closed-form formulas for computing default probabilities. The model robustness is tested by its calibration to market credit default swap (CDS) spreads for four companies with various credit ratings. It is shown that the model can be quite accurately calibrated to single-firm market credit spreads as well as to the corresponding market put and call option prices across various strikes and maturities. Finally, we investigate the linkage between \textsf{CDS} spreads and out-of-the-money put options. Hao Xing (Boston University) Portfolio turnpike in incomplete markets Portfolio turnpike is an intuitive property of the portfolio choice problem. It states that if preferences of two agents are similar at large wealth levels, then their investment strategies are similar as horizon increases. This problem dates back to 1968 and it has been proven in different market settings. But all results assume the completeness of the market. In this talk, we will discuss whether this property holds in an incomplete market. We show that when the investment strategy of one agent is myopic, the turnpike property holds in a general incomplete market with semimartingale dynamics. When the optimal strategy is not myopic, we study a specific market model whose asset prices are driven by a common factor. In this market model, we will discuss the relationship between turnpike property, h-transform, and ergodic theory. This is a joint work with Paolo Guasoni, Kostas Kardaras, and Scott Robertson.