COMMERCIAL AND INDUSTRIAL MATHEMATICS

July 31, 2014

The Fields Institute 2007-2008
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, finance 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 (Audio of talks)

Apr. 30, 2008

Leif Andersen, Bank of America Securities
Markov modeling of seasonal commodities
In this talk, we examine low-dimensional Markov models for practical trading of gas futures and options. We aim for a realistic (seasonal) correlation structure and for perfect replication of the strongly seasonal form of gas option volatilities. We also investigate techniques to incorporate seasonal volatility smiles into the model, without losing analytical tractability. For motivation, the talk is sprinkled with empirical data from the US gas market.

and

Michael Gordy, Federal Reserve Board
Nested Simulation in Portfolio Risk Measurement (with Sandeep Juneja)
Risk measurement for derivative portfolios almost invariably calls for nested simulation. In the outer step one draws realizations of all risk factors up to the horizon, and in the inner step one re-prices each instrument in the portfolio at the horizon conditional on the drawn risk factors. Practitioners may perceive the computational burden of such nested schemes to be unacceptable, and adopt a variety of second-best pricing techniques to avoid the inner simulation. In this paper, we question whether such short cuts are necessary. We show that a relatively small number of trials in the inner step can yield accurate estimates, and analyze how a fixed computational budget may be allocated to the inner and the outer step to minimize the mean square error of the resultant estimator. Finally, we develop jackknife and dynamic allocation methods for bias reduction.

March 26, 2008 Peter Cotton, Julius Finance Corporation (Audio of talks)
Bowling Alone. Do Copula Models Washout?
Mathematical models for structured credit products enjoy a privileged and protected existence. The dependence structure amongst corporate defaults may have many causes and descriptions but delineation between these is largely defeated by an absence of empirical data. There are very few realizations of the system over any reasonable time frame and no guarantee of stationarity. One might argue that there have been no realizations in fact, whereas presumably many thousands are required in any rigorous testing of causality. Consequently, one mans' contagion is another man's macro-economic co-dependence, and so forth.
We inject some data into the debate - albeit entirely irrelevant data. We examine a small collection of dependent objects whose joint distribution can be interrogated by tedious but strangely satisfying experiments. The objects are pins. There are ten of them. They display a tendency to fall over in much the same manner as companies. Their defaults, as we might call them, have knock-on effects which can sometimes bring down one or more of their colleagues.

and

Roger Lee, Department of Mathematics, University of Chicago (Audio of talks)
Hedging Options on Realized Variance
Contracts on realized variance have become a leading tool for managing exposure to volatility risk. In particular, variance swaps pay realized variance, typically defined as the sum of squared daily returns of the underlying. Moreover, a second generation of variance contracts has emerged, including variance _options_ (calls and puts on realized variance) which offer portfolio managers even greater control over their volatility risk profile. However, they present greater pricing/hedging problems to the dealer. We take a robust model-free approach to these problems. Assuming essentially only the positivity and continuity of the underlying share price, we hedge continuously-monitored variance options by dynamically trading the underlying shares, and statically holding European options. These hedges lead to upper and lower bounds on variance option values. (Joint with Peter Carr)

Feb. 27, 2008

Rama Cont, Columbia University (Audio of talks)
Calibration of portfolio credit risk models: solution of an inverse problem via intensity control.
Pricing models for portfolio credit derivatives such as CDOs involves the construction of a stochastic process for the losses due to defaults which is compatible with a set of observations of market spreads for CDO tranches. We propose an efficient and stable algorithm to solve this inverse problem by transforming it into a stochastic control probem. We formalize the problem in terms of minimization of relative entropy with respect to a prior jump process under calibration constraints and use convex duality techniques to solve the problem. The dual problem is shown to be an intensity control problem. We show that the corresponding nonlinear Hamilton Jacobi system of differential equations can be represented in terms of a nonlinear transform of a linear system of ODEs and thus easily solved. Our method allows to construct a Markovian jump process for defaults which leads to CDO tranche spreads consistent with the observations. We illustrate our method ITRAXX index data: our results reveal strong evidence for the dependence of loss transitions rates on the past number of defaults, thus offering quantitative evidence for contagion effects in the risk-neutral loss process.

Jim Gatheral, Merrill Lynch and Courant Institute (Audio of talks)
Developments in Volatility Derivatives Pricing.
Over the last two years, we have seen increased trading in both options on VIX and options on variance swaps. Also, there has been increased interest in modeling volatility with more than just one factor (Lorenzo Bergomi's two-factor model for example); rather than model instantaneous or implied volatility, the current tendency is to model variance swaps (or curves). Hans Buehler has derived consistency conditions on variance curve models in general, concentrating on the so-called Double Heston model in numerical examples. Bergomi's model on the other hand has (double) lognormal dynamics.

In this talk, we place the Double Heston and Double Lognormal models in their historical context, explain Buehler's consistency condition and explore implications for the pricing of VIX options. We see that market pricing of VIX options excludes Heston dynamics but is roughly consistent with lognormal dynamics.

Finally, we extract time series of the two volatility factors from historical option prices and compare statistical and risk-neutral parameters, finding that many of the unrealistic features of one-factor stochastic volatility models are eliminated

February 6, 2008 - 5:00 p.m.
5:00 p.m. Reception
5:30 p.m

**The Seminar Series On Quantitative Finance for tonight, February 6, 2008 has been cancelled due to weather conditions.**
. Rama Cont, Columbia University
Calibration of portfolio credit risk models: solution of an inverse problem via intensity control.
Pricing models for portfolio credit derivatives such as CDOs involves the construction of a stochastic process for the losses due to defaults which is compatible with a set of observations of market spreads for CDO tranches. We propose an efficient and stable algorithm to solve this inverse problem by transforming it into a stochastic control probem. We formalize the problem in terms of minimization of relative entropy with respect to a prior jump process under calibration constraints and use convex duality techniques to solve the problem. The dual problem is shown to be an intensity control problem. We show that the corresponding nonlinear Hamilton Jacobi system of differential equations can be represented in terms of a nonlinear transform of a linear system of ODEs and thus easily solved. Our method allows to construct a Markovian jump process for defaults which leads to CDO tranche spreads consistent with the observations. We illustrate our method ITRAXX index data: our results reveal strong evidence for the dependence of loss transitions rates on the past number of defaults, thus offering quantitative evidence for contagion effects in the risk-neutral loss process.

November 28, 2007

Michael Walker, University of Toronto (Audio of talks)
A Calibratable Dynamic Model for CDO's: Application to Leveraged-Super-Senior Tranche Valuation
The talk begins with an introduction describing the rapid rise in the use of credit derivatives in the banking and related industries. CDO's, i.e. collateralized debt obligations (one of the most important credit derivatives), are also introduced. Two ideas that allow the construction of easily calibratable models for the risk-neutral valuation of CDO's, and for the valuation of related dynamics-sensitive derivatives, are then described. Finally, illustrative results for the valuation of an exotic, path-dependent derivative, the leveraged super-senior tranche, are presented.

Thaleia Zariphopoulou, University of Texas Austin (Audio of talks)
Stochastic pdes in portfolio choice
A new approach to investment performance measurement is introduced via the so-called forward performance processes. A related stochastic pde is derived and analyzed for various cases of utility coefficients. Applications of the new approach to risk measures and its connection to the traditional expected utility maximization problems are discussed.

October 31, 2007 Martin Schweizer, ETH Zurichand (Audio of talks)
Arbitrage-free joint models for assets and derivatives
In today's financial markets, many instruments are traded so liquidly that they can be viewed like primary assets. A typical example are plain vanilla options which are often used as hedging instruments when determining prices for other, more exotic derivatives. Hence one might want to have a model where one can specify dynamics (SDEs) not only for the underlying stocks, but also for a family of options written on that stock. However, this is not entirely straightforward because there are no-arbitrage constraints between these price processes. A closer look even shows that this problem is surprisingly tricky.

We present some recent advances on the construction of arbitrage-free market models for option prices. This includes a presentation of the problem and of our goals, issues of parametrization and existence of models, and comparisons to related work in the literature. While we have some answers, many questions remain, and hopefully the audience can contribute to better understanding as well.

This is joint work with Johannes Wissel (ETH Zurich).

David X. Li, Barclays Capital (Audio of talks)
Dynamical Competing Risk Model for Home Equity Loan Securities:
We present a new approach to model Home Equity Loan (HEL) securities. We take prepayment and default as two fundamental variables and model them in a dynamic competing risk model framework. The valuation of single name ABS CDS, ABX, and ABS CDO will be discussed.

back to top