COMMERCIAL AND INDUSTRIAL MATHEMATICS

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

Seminars 2009-10

Freddy Delbaen (ETH Zurich)
BSDE and time consistency
Abstract: the capital requirements for financial institutions should satisfy rules that encourage diversification, avoid risk concentration and take into account the time aspect of uncertainty. They also should allow to allocate, in a consistent way, the risk capital to the different branches of the financial institution. This is best done via convex, time consistent risk measures, a mathematical concept that is based on the theory of stochastic processes, backward stochastic differential equations and semi-linear PDE.

H. Mete Soner, Department of Mathematics, ETH Zurich
Financial markets with uncertain volatility.
Even in simple models in which the volatility is only known to stay in two bounds, it is quite hard to price and hedge derivatives which are not Markovian. The main reason for this difficulty emanates from the fact that the relevant set of probability measures is not dominated by one measure. In this talk we will prove a martingale representation theorem for this market. This result provides a complete answer to the questions of hedging and pricing. The main tools are the theory of nonlinear G-ex pectati ic analysis of Denis & Martini and the second order backward stochastic differential equations. This is joint work with Nizar Touzi from Ecole Polytechnique and Jianfeng Zhang from University of Southern California.

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.

Tom Hurd, McMaster University
Credit Risk via First Passage for Time Changed Brownian Motions
Audio and slides of Talk
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!

Traian Pirvu, McMaster University
Time Consistency in Portfolio Management
Audio and slides of Talk
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.

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)
Capital Requirements, Acceptable Risks and the Value of the Taxpayer Put

Limited liability for the firm in the presence of unbounded liabilities delivers a free put option to the firm that is rarely valued and accounted for. We christen this put option the taxpayer put. In addition the optimality of free markets is called into question by the introduction of adverse risk incentives exaggerated by compensation aligned to stock market values. In such a context we introduce the concept of socially acceptable risks, operationalized by a positive expectation after distortion of the distribution function for risky cash flows. This results in a definition of capital requirements making the risks undertaken acceptable to the wider community. Enforcing such capital requirements can mitigate the perverse risk incentives introduced by limited liability provided that the set of acceptable risks is suitably conservatively defined. Additionally the value of the free taxpayer put may be substantially reduced. We illustrate all computations for the six major US banks at the end of 2008.


Stan Uryasev (University of Florida)
Value-at-Risk vs. Conditional Value-at-Risk in Risk Management and Optimization
Joint talk with Konstantin Kalinchenko (Department of Industrial and Systems Engineering, Risk Management and Financial Engineering Lab, University of Florida) & Sergey Sarykalin and Gaia Serraino (American Optimal Decisions)

From mathematical perspective, risk management is a procedure for shaping a risk distribution. Popular functions for managing risk are Value at-Risk (VaR) and Conditional Value-at-Risk (CVaR). Reasons affecting the choice between VaR and CVaR are based on the differences in mathematical properties, stability of statistical estimation, simplicity of optimization procedures, acceptance by regulators, etc. We explain strong and weak features of these risk measures and illustrate them with examples. We demonstrate several risk management/optimization case studies conducted with Portfolio Safeguard decision support software. In particular, we will discuss how to calibrate risk preferences of investors doing risk management with options.

Bio:
Professor Stan Uryasev is director of the Risk Management and Financial Engineering Lab and director of the PhD Program with Concentration in Quantitative Finance at the University of Florida. His research is focused on efficient computer modeling and optimization techniques and their applications in finance and military projects. He published three books (monograph and two edited volumes) and about eighty research papers. He is a co-inventor of the Conditional Value-at-Risk and the Conditional Drawdown-at-Risk optimization methodologies. He is the founder of American Optimal Decisions (AORDA.com) developing optimization software in risk management area: VaR, CVaR, Default Probability, Drawdown, Credit Risk minimization.

Stan Uryasev is a frequent speaker at academic and professional conferences. He has delivered seminars on the topics of risk management and stochastic optimization. He is on the editorial board of a number of research journals and is Editor Emeritus and Chairman of the Editorial Board of the Journal of Risk.

- Benefits and limitations of rational models of asset pricing
- A alternative framework for asset price dynamics based on investor behavior
- Price-following and herding effects and their impact on the distribution of asset prices
- Fat tail effects. The impact of market feedback, credit cycles, crises and uncertainty
- Quantifying the risk of extreme events. From theory to practice.

Summary
A predominant view in the literature of market behavior is that investors are driven by rationality -that is, investors are presumed to make logical and sensible decisions all the time. Evidence accumulated over years, and in particular during turbulent times, has shown consistently that investors are far less rational in their decision making that the theory assumes. Understanding investors' behavior is important for developing more realistic models of asset price dynamics capable of capturing the extreme changes in prices observed during periods of market booms and crashes.

In the presentation I will elaborate on an asset pricing model based on market behavior and uncertainty. The phenomenological model of asset valuation dynamics includes investor behavior patterns based on price momentum and trends. The model produces fat-tail distributions of asset prices remarkably similar to the observed distributions, and provides a simple framework for understanding the potential divergence between fundamental and market valuations under uncertainty.

***Please note that the second talk by Igor Halperin has been canceled due to unforseen circumstances.