COMMERCIAL AND INDUSTRIAL MATHEMATICS

2009-2010
Fields Quantitative Finance Seminar
Fields Institute, 222 College St., Toronto

Seminars 2009-10

In this paper we introduce a novel approach to risk estimation based on nonlinear factor models - the "StressVaR" (SVaR). Developed to evaluate the risk of hedge funds, the SVaR appears to be applicable to a wide range of investments. The computation of the StressVaR is a 3 step procedure whose main components we describe in relative detail. Its principle is to use the fairly short and sparse history of the hedge fund returns to identify relevant risk factors amonga very broad set of possible risk sources. This risk profile is obtained by calibrating a collection of nonlinear single-factor models as opposed to a single multi-factor model. We then use the risk profile and the very long and rich history of the factors to asses the possible impact of known past crises on the funds, unveiling their hidden risks and so called "black swans".In backtests using data of 1060 hedge funds we demonstrate that the
SVaR has better or comparable properties than several common VaR measures - shows less VaR exceptions and, perhaps even more importantly, in case of an exception, by smaller amounts.The ultimate test of the StressVaR however, is in its usage as a fund allocating tool. By simulating a realistic investment in a portfolio of hedge funds, we show that the portfolio constructed using the StressVaR on average outperforms both the market and theportiolios constructed using common VaR measures.For the period from Feb. 2003 to June 2009, the StressVaR constructed portfolio outperforms the market by about 6% annually, and on average the competing VaR measures by around 3%.The performance numbers from Aug. 2007 to June 2009 are even more impressive. The SVaR portfolio outperforms the market by 20%, and the best competing measure by 4%.

Dilip Madan (University of Maryland)
TBA

Stan Uryasev (University of Florida)
TBA

We introduce a new semi-parametric multi-factor model for pricing and risk management of bespoke CDO tranches, with the main focus on bespokes that need to be mapped onto more than one reference portfolio. Our setting is intermediate between a top-down and a bottom-up approaches, with losses in certain sub-portfolios of index portfolios being the main modeling primitive.

Our calibration method amounts to a proper reweightening of an initial ("prior") measure obtained within a particular model (e.g. multi-factor Gaussian copula) using the Minimum Cross Entropy method. Because of reduction of dimensionality achieved in our approach, calibration reduces to convex optimization in a low dimensional space, making the model computationally efficient. Both the static (one-period) and dynamic versions of the model are presented.

Bio:

Igor Halperin is a Vice President in Quantitative Research at JP Morgan. He is responsible for model development and research in derivatives pricing across different asset classes. Igor holds a Ph.D. in Theoretical High Energy Physics, and worked as a quantitative researcher at Bloomberg L.P. before joining JP Morgan.

Jorge Sobehart (Citigroup)
TBA

Ulrich Horst, Humboldt University Berlin
Hidden Liquidity and the Optimal Placement of Iceberg Orders
Audio and Slides of the Talk
Almost all electronic trading systems are based on Limit Order Books (LOBs) in which all unexecuted limit orders are stored while awaiting execution. Not all the available liquidity is openly displayed, though. Most exchanges offer liquidity providers the option of shielding all or portions of their limit orders from public display. These modified order types, known as hidden orders or iceberg orders, meet the demands of traders who perceive benefits in obscuring their immediate trading needs from other market practitioners. We propose a simple mathematical model of a LOB within which to study the problem of the optimal display size of limit orders placed in the spread or at the top of the book. One of the most important determinants of the optimal display size is the amount of hidden liquidity with higher time priority of the hidden part of the submitted order. We estimated this quantity using recent NASDAQ data. Our empirical analysis shows that the spread together with the visible volume at the top of the book often well predicts the amount of hidden liquidity in the spread and on top of the book. We also report a couple of other empirical findings including the dependence of hidden liquidity on average daily trading volumes and average quote sizes, and the distribution of hidden liquidity in the spread.

The talk is based on joint work with Gökhan Cebiroglu (Humboldt University) and Mark DiBattista (Deutsche Bank AG)
&
Jeremy Graveline, University of Minnesota
G10 Swap and Exchange Rates
Audio and Slides of the Talk

In this talk we show how to extend single-currency dynamic term structure models to a multi-currency setting. When the risk-neutral pricing measures, or risk premia, are denominated in two different currencies they must differ by the covariance of the exchange with the other factors in the model. As an illustrative example, we provide estimates for a Gaussian model of the term structure of swap rates and exchange rates in the G10 countries. There are 9 exchange rates and each yield curve is described by 2 or 3 factors, for a total of 37 factors in the model. The parameters that govern the covariances and risk-neutral drifts are relatively easy to estimate. However, it is much harder to reliably estimate the risk premia parameters that relate the risk-neutral and statistical measures. We examine the performance of models for 7 years out-of-sample and show that models with a small number of priced risk factors provide a good in-sample fit and the best out-of-sample results. This talk discusses joint work with Scott Joslin at MIT.

We first present a general model for counterparty risk. We give are presentation formula for the Credit Value Adjustment (CVA) accounting for netting and collateralization in the context of bilateral counterparty risk. Then, we specify the results to the case of counterparty credit risk, where we consider a credit risky portfolio between two default prone counterparties. The underlying model for the dependence between defaults is based on the concept of Markov copula. Some numerical results illustrating computation of relevant quantities (such as CVA, EPE) will be presented.

Tom Hurd, McMaster University
Credit Risk via First Passage for Time Changed Brownian Motions

The first passage structural approach to credit risk, while very natural, is beset by technical difficulties that make it
inflexible in practice. Time changed Brownian motions (TCBMs) offer a simple but mathematically interesting way to circumvent these technicalities and open the door to a number of innovations. After a quick sketch of the basic properties of TCBM models, I show that they can give an excellent fit to the dynamics of credit default swaps
observed in the market. I then consider a more complex ``hybrid'' framework that can model the joint dynamics of equity and credit derivatives. Finally, I will touch briefly on how the TCBM framework extends to multiple firms, paving the way for a consistent ``bottom up'' approach to portfolio credit derivatives. This is a talk aimed at people who really work with credit default swaps and other credit risky securities, and their feedback will be welcomed!

The paper outlines some basic approaches to modeling liquidity, and its implications for asset pricing and portfolio strategy. These idea can be used to model a Risk Management system with liquidity problems. In addition they can be extended to explore Systemic Risks.

Traian Pirvu, McMaster University
Time Consistency in Portfolio Management

There are at least two examples in portfolio management that are time inconsistent. 1) Maximizing utility of intertemporal consumption and final wealth assuming a hyperbolic discount rate (the discount rate increases with time). 2) Mean-variance utility: This case is a continuous time version of the standard Markowitz investment problem, and the time inconsistencies are due to the wealth's variance (which is nonlinear and depends on the starting wealth). In this talk I will focus on the first example. There is strong evidence that individuals discount future utilities at nonconstant rates. The notion of optimality then disappears, because of time inconsistency and rational behaviour then centers around equilibrium strategies. I will investigate portfolio management with hyperbolic discounting, and I will show that this may explain some well known puzzles of portfolio management. This is joint work with Ivar Ekeland.