COMMERCIAL AND INDUSTRIAL MATHEMATICS

September 30, 2014

The Fields Institute 2005-2006
Seminar Series on Quantitative Finance

sponsored by

The Quantitative Finance Seminar has been a centerpiece of the Commercial/Industrial program at the Fields Institute since 1995. Its mandate is to arrange talks on current research in quantitative finance that will be of interest to those who work on the border of industry and academia. Wide participation has been the norm with representation from mathematics, statistics, computer science, economics, econometrics, finace and operations research. Topics have included derivatives valuation, credit risk, insurance and portfolio optimization. Talks occur on the last Wednesday of every month throughout the academic year and start at 5 pm. Each seminar is organized around a single theme with two 45-minute talks and a half hour reception. There is no cost to attend these seminars and everyone is welcome. To be informed of speakers and titles for upcoming seminars and financial mathematics activities, please subscribe to the Fields mail list.

Past seminars 2005-06

Upcoming Seminars

May 31, 2006

Tom Coleman, Dean, Faculty of Mathematics, University of Waterloo
Minimizing CVaR and VaR for a Portfolio of Derivatives
Value at Risk (VaR) and Conditional Value at Risk (CVaR) are frequently used as risk measures in risk management. We analyze the problem of computing the optimal VaR and CVaR portfolios - we illustrate that if the portfolios contain derivatives then the resultant optimization problems are typically ill-posed. We propose corrective measures for this problem and also look at some of the other computing challenges.

 


Past Seminars (Audio and Slides of talks)

April 26, 2006

Jean-Pierre Fouque, University of California Santa Barbara
Perturbation Methods in Default Modeling

We show that stochastic volatility incorporated in first passage models can create reasonable default probabilities over a wide range of
maturities. To achieve that, one has to carefully calibrate the time scales of volatility, and, to make this approach tractable, we show that regular and singular perturbations techniques associated to slow and fast time scales can be used. We then address the multi-name case and we show that default correlations created by stochastic volatility give interesting loss distributions. Perturbation techniques are gain used to compute these distributions and the related tranche prices.
Joint work with Ronnie Sircar (Princeton), Knut SOLNA (UC Irvine), and Stephen Zhou (PhD student, NC State University).

and

Sebastian Jaimungal, Department of Statistics, University of Toronto
Indifference Pricing for Equity-Linked Insurance and Reinsurance Options

Insurance companies are increasingly facing heavy exposure to capital market risks, due to the issuance of equity-linked insurance policies. There is now a growing need for coherent valuation and hedging methodologies that take into account the interwoven actuarial and financial risks. In this talk, I will demonstrate how the principle of equivalent utility provides equity-linked insurance premiums, and explain how to value double-trigger reinsurance options consistently; such contracts are crucial risk management vehicles since they provide the insurer with a means to offload unwanted risks. In addition, I will illustrate how utility indifference allows for the simultaneous treatment of counterparty risk. By solving the resulting HJB equations, I determine that both the premiums and prices satisfy Black-Scholes-like PDEs with non-linear and non-local risk-aversion correction terms. Numerical consequences will be explored throughout this talk to clarify the approach and to aid understanding.

March 29, 2006

Matheus Grasselli, McMaster University
Rational exercise of employee options.

At the core of the controversy surrounding the accounting status of employee options lies a lack of agreement on the correct valuation procedure for them. For this, we propose a discrete-time algorithm based on pricing techniques for derivatives in incomplete-markets. The two salient features of the method are that it takes into account the non-linearity inherent to risk preferences, as well as the possibility of partial hedge using a correlated instrument, such as a market index. The immediate effect of non-linearity is that the optimal exercise policy for the employee consists of partial exercise over en extended period of time, as opposed to immediate exercise as soon as the underlying reaches a threshold. The effect of trading on a correlated asset, on the other hand, counter-balances that of risk aversion and can be used to greatly increase the value of the option for the employee, and consequently its cost for the issuing firm.

and

Dan Rosen, Fields Institute
Economic Capital Allocation, Risk Contributions and Diversification in Credit Portfolios

Concentration risk is arguably the most important cause of major problems in banks, according to the Basel committee of Banking Supervision. The reverse side of the coin, diversification, is one of the key tools for managing the risk of credit portfolios. A thorough understanding of diversification/concentration risk is vital for allocating optimally economic credit capital. This is required for pricing, profitability assessment and limits, building optimal risk-return portfolios and strategies, performance measurement and risk based compensation.

This seminar presents a practical overview of the measurement of diversification and risk capital contributions in credit portfolios and their application to capital allocation. We stress several key points. First, marginal risk contributions provide a useful basis for allocating capital since they are additive and reflect the benefits of diversification within a portfolio. Second, the choice of the risk measure can have a substantial impact on capital allocation. In particular, the quantile level chosen for measuring VaR or expected shortfall (ES) can also have a significant impact on the relative amount of capital allocated to portfolio components. Third, diversification measures and risk contributions can be calculated analytically under certain models. These methods provide fast calculations and can be used to understand capital allocation strategies better, but they may present some practical limitations, as well. Finally, Monte Carlo methods may be required to compute risk contributions in more realistic credit models. Computing VaR and ES contributions is challenging, especially at the extreme quantiles typically used for credit capital definition.

February 22, 2006

Hyejin Ku, Department of Math and Statistics, York University
Liquidity Risk with Coherent Risk Measures

We consider questions related to the regulation of liquidity risk. Basically, the firm should be able to unwind its current position without too much loss of its wealth if it were required to do so. Liquidity risk is important in deciding whether a firm's position is "acceptable" or not. We develop a method to incorporate liquidity risk into risk measurement. We consider a portfolio to be acceptable if it can (by trading) be turned into an ?acceptable" cash-only position having positive future cash flows at some fixed date, and present an example of modeling liquidity.

and

Ajay Subramanian, Assistant Professor of Risk Management and Insurance
J. Mack Robinson College of Business, Georgia State University
Asymmetric Beliefs, Agency Conflicts, and Venture Capital Investment

We develop a dynamic principal-agent model to examine the interplay among risk, imperfect information, agency conflicts, and asymmetric beliefs on the characteristics of venture capital (VC) relationships---the economic value that they generate, the durations of relationships, the structures of long-term contracts between VCs and entrepreneurs (ENs), and the manner in which VC investment is staged over time. We show that the presence of asymmetric beliefs about project quality has a substantial beneficial impact on project value and the expected payoff to the VC implying that VCs have significant incentives to encourage entrepreneur optimism. We analytically characterize the effects of the project's characteristics---its systematic and technical risk, and the degree of asymmetry in beliefs about its quality---on the path of staged investments by the VC and the structure of the long-term contract between the VC and the EN. Consistent with empirical evidence, we predict that varying project characteristics lead to significant heterogeneity in contractual structures and investment schedules. The systematic and technical risks of projects have opposite effects on the durations and economic values of VC relationships. The duration, project value, and the expected payoff to the VC decrease with the project's systematic risk but increase with its technical risk, which leads to the striking implication that the value of the project and the expected payoff to the VC are actually enhanced when there is greater noise in the perception of project quality. Broadly, our study not only demonstrates that the interactions among agency conflicts, imperfect information, and asymmetric beliefs have a major impact on the VC-EN relationship, but also precisely describes the manner in which they affect this relationship.

Authors are Yahel Giat, Steven T. Hackman, and Ajay Subramanian .

January 25, 2006 -- 5:00 pm.

Roger Stein, Moodys
Better Predictions of Income Volatility Using a Structural Default Model
We propose a novel approach to predicting future volatility of company earnings, in this case EBITDA. Our approach combines predictions of a firm’s probability of default with insights from a structural model of default. The source of the probabilities of default can be econometric, structural, reduced-form or other models or agency ratings, provided the source has high predictive power. We use these probabilities to imply EBITDA volatility using a stylized, liquidity-based model of firm default similar in some ways to that originally proposed by Wilcox (1971). The method does not require market information and our out-of-sample testing suggests that our approach is more accurate in estimating future volatility than the historical volatility of EBIDTA. Importantly, the method also produced reasonable estimates of volatility when historical data is quite limited, for instance when no historical financial data are available for the firm. In addition in comparison with historical volatility estimates the implied volatility estimates appear provide incremental information useful in identifying those firms that are more likely to experience EBITDA. Beyond implied volatility, we explore extensions of the approach for estimating implied liquidity requirements and target growth rates for firms, given a starting capital structure and variable cash flow stream.

and

David Lando, Copenhagen Business School
Decomposing Swap Spreads
We analyze a six-factor model for Treasury bonds, corporate bonds, and swap rates and decompose swap spreads into three components: A convenience yield from holding Treasuries, a credit risk element from the underlying LIBOR rate, and a factor specific to the swap market. In the later part of our sample, the swap-specific factor is strongly correlated with hedging activity in the MBS market. The model further sheds light on the relationship between AA hazard rates and the spread between LIBOR rates and GC repo rates and on the level of the riskless rate compared to swap and Treasury rates.
(Joint work with Peter Feldhütter)

November 23, 2005 -- 5:00 p.m.

Steven Kou, Columbia University
Credit Spreads, Optimal Capital Structure, and Implied Volatility with Endogenous Default and Jump Risk

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) The jump 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; and although in general credit spreads and implied volatility tend to move in the same direction under exogenous default models, but this may not be true in presence of endogenous default and jumps. In terms of mathematical contribution, we give a proof of a version of the ``smooth fitting'' principle for the jump model, justifying a conjecture first suggested by Leland and Toft under the Brownian model.

and

Mary Hardy, University of Waterloo
Stock Return Models for Long Term Embedded Options'

Insurance companies in Canada and the USA have found that adding out-of-the-money guarantees to mutual fund type investments creates a product which is highly popular. The risk management of these contracts is challenging. A crucial part of the problem is finding a real world model for the mutual fund returns that adequately captures the tails.

In this talk I will briefly describe how actuaries approach the risk management of these contracts, and then will present some of the many models proposed for equity returns. It emerges that very small tweaks in the equity model can make a significant difference to the resulting regulatory capital. Using (and abusing) a bootstrap approach we show how to determine which of these models can be justified using the historic data.

October 26, 2005 -- 5:00 p.m.

Nizar Touzi, University Paris I-Pantheon-Sorbonne
Modelling continuous-time financial markets with capital gains taxes
We formulate a model of continuous time financial market consisting of a bank account with constant interest rate and one risky asset subject to transaction costs and capital gains taxes. The taxation rule is linear so that it allows for tax credits when capital losses are experienced. We consider the problem of maximizing expected utility from future consumption in infinite horizon. We first derive lower and upper bounds on the value function in erms of the corresponding value function in the tax free and frictionless model. In particular, these bounds allow to obtain an explicit first order expansion of our value function for small interest rate and tax rate coefficients. We next provide a characterization of the value function in terms of the associated dynamic programming equation, and we suggest a numerical approximation scheme based on finite differences and the Howard
algorithm. The numerical results show that the first order Taylor expansion is very accurate for reasonable market data.

and

Marco Frittelli
, Università degli Studi di Firenze
A Unifying Framework for Utility Maximization Problems with Unbounded SemimartingalesDuring the past twenty years, the theory of expected utility maximization in continuous-time stochastic incomplete markets has
constantly improved, but a case has been left apart: exactly the situation examined in this talk where the semi-martingale X, describing the price evolution of a finite number of assets, can be possibly unbounded. This is a non-trivial extension, from a mathematical but also from a financial point of view.

In fact, in highly risky markets (i.e. with unbounded losses in the trading: think of X as a Compound Poisson process on a finite horizon, with unbounded jumps) the traditional approach to the problem leads to trivial maximization: the optimal choice for the agent would be investing the initial endowment entirely in the risk free asset. However, it could happen that some of the investors are willing to take a greater risk: mathematically speaking, they accept trading strategies that may lead to unbounded losses. This risk-taking attitude gives them the concrete possibility of increasing their expected utility from terminal wealth.

In a unified framework, we consider the utility maximization problem for utility functions that can be finite valued on the whole real line or only on the positive semi axes and we select a generalized class of trading strategies which allows for unbounded stochastic integrals. By duality methods we prove existence of the optimal solution to both the dual and the primal problems.
As it is widely known, the utility maximization problem is linked to derivative pricing through the so-called indifference pricing technique. Such a technique is far from being a theoretical speculation, since it is currently used by financial institutions to price new and/or illiquid derivatives. The results here presented allow tackling this problem in the general case of a non-necessarily locally bounded semi-martingale price process.

September 28, 2005 -- 5:00 p.m.

Dmitry Kramkov, Carnegie Mellon
Sensitivity analysis of utility based prices and risk-tolerance wealth processes
In the general framework of a semimartingale financial model and a utility function U defined on the positive real line we compute the first order expansion of marginal utility based prices with respect to a ``small'' number of random endowments. We show that this linear approximation has some important qualitative properties if and only if there is a risk-tolerance wealth process. In particular, they hold true in the following polar cases:
(i) for any utility function U if and only if the set of state price densities has a greatest element from the point of view of second order stochastic dominance
(ii) for any financial model if and only if U is a power utility function (U is an exponential utility function if it is defined on the whole real line).
The presentation is based on a joint paper with Mihai Sirbu.

and

Mark Reesor, University of Western Ontario
A Debt Strategy Simulation Framework and Interest-rate Model Risk
Debt strategy is the manner in which governments (or agencies and corporations) issue bonds to cover their funding requirements. The issuer has some control over the relative amounts of bond issuance across the maturity spectrum, making this problem analogous to the typical portfolio selection problem. Furthermore, there are additional constraints that are unique to the problem of managing a large portfolio of public debt. We discuss how this bond issuance problem can be formulated as a (constrained) stochastic optimal control problem, along with a simulation framework that allows for its analysis. Clearly a model for interest rates is one of the main components of such a framework. Using a simple example, we investigate the issue of model risk in the debt strategy analysis.
This is joint work with Shudan Liu, a PhD student in the Dept of Applied Math, UWO.

back to top