COMMERCIAL AND INDUSTRIAL MATHEMATICS

2010-2011 Fields Quantitative Finance Seminar
Fields Institute, 222 College St., Toronto

Seminars 2010-2011

Xunyu Zhou (Oxford University)
Behavioural Portfolio Choice

I will first give a brief introduction on the motivation and background of behavioural finance theory, and then present an overview of the recent development on quantitative treatment of behavioural finance, primarily in the setting of portfolio choice under the cumulative prospect theory. Financial motivations and methodological challenges of the problem are highlighted. It is demonstrated that the solutions to the problem have in turn led to new financial and mathematical problems and machinery.

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Patrick Cheridito
Pricing and Hedging in Affine Models with Possibility of Default

We propose a general class of models for the simultaneous treatment of equity, corporate bonds, government bonds and derivatives. The noise is generated by a general affine Markov process. The framework allows for stochastic volatility, jumps, the possibility of default and correlations between different assets. We extend the notion of a discounted moment generation function of the log stock price to the case where the underlying can default and show how to calculate it in terms of a coupled system of generalized Riccati equations. This yields an efficient method to compute prices of power payoffs and Fourier transforms. European calls and puts as well as binaries and asset-or-nothing options can then be priced with the fast Fourier transform methods of Carr and Madan (1999) and Lee (2005). Other European payoffs can be approximated by a linear combination of power payoffs and vanilla options. We show the results to be superior to using only power payoffs or vanilla options. We also give conditions for our models to be complete if enough financial instruments are liquidly tradable and study dynamic hedging strategies. As an example we discuss a Heston-type stochastic volatility model with possibility of default and stochastic interest rates. Joint work with Alexander Wugalter.

Rafael Mendoza-Arriaga (The University of Texas at Austin)
Constructing Markov Processes with Dependent Jumps by Multivariate Subordination: Applications to Multi-Name Credit-Equity Modeling

We develop a new class of multi-name unified credit-equity models that jointly model the stock prices of multiple firms, as well as their default events, by a multi-dimensional Markov semimartingale constructed by multivariate subordination of jump-to-default extended constant elasticity of variance (JDCEV) diffusions. Each of the stock prices experiences state-dependent jumps with the leverage effect (arrival rates of large jumps increase as the stock price falls), including the possibility of a jump to zero (jump to default). Some of the jumps are idiosyncratic to each firm, while some are either common to all firms (systematic), or common to a subgroup of firms. For the two-firm case, we obtain analytical solutions for credit derivatives and equity derivatives, such as basket options, in terms of eigenfunction expansions associated with the relevant subordinated semigroups.

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Alfred Lehar (University of Calgary)
Macroprudential capital requirements and systemic risk
Full Paper Available Here

When regulating banks based on their contribution to the overall risk of the banking system we have to consider that the risk of the banking system as well as each bank’s risk contribution changes once bank equity capital gets reallocated. We define macroprudential capital requirements as the fixed point at which each bank’s capital requirement equals its contribution to the risk of the system under the proposed capital requirements. This study uses two alternative models, a network based framework and a Merton model, to measure systemic risk and how it changes with bank capital and allocates risk to individual banks based on fi;ve risk allocation mechanisms used in the literature. Using a sample of Canadian banks we find that macroprudential capital allocations can differ by as much as 70% from observed capital levels, are not trivially related to bank size or individual bank default prob- ability, increase in interbank assets, and differ substantially from a simple risk attribution analysis. We further find that across both models and all risk allocation mechanisms that macroprudential capital requirements reduce the default probabilities of individual banks as well as the probability of a systemic crisis by about 25%. Macroprudential capital requirements are robust to model risk and are positively correlated to future capital raised by banks as well as future losses in equity value. Our results suggest that financial stability can be substantially enhanced by implementing a systemic perspective on bank regulation.

Pierre Collin-Dufresne (Carson Family Professor of Finance, Columbia University)
On the Relative Pricing of long Maturity S&P 500 Index Options and CDX Tranches

We investigate a structural model of market and firm-level dynamics in order to jointly price long-dated S&P 500 options and tranche spreads on the five-year CDX index. We demonstrate the importance of calibrating the model to match the entire term structure of CDX index spreads because it contains pertinent information regarding the timing of expected defaults and the specification of idiosyncratic dynamics. Our model matches the time series of tranche spreads well, both before and during the financial crisis, thus offering a resolution to the puzzle reported by Coval, Jurek and Stafford (2009).

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Kostas Kardaras (Boston University)
Pricing and hedging barrier options in diffusion models via 3-dimensional Bessel bridges

Due to the discontinuous payoff of barrier options, finite difference methods typically lead to large error for the price function and spatial derivatives near expiry date and the barrier. Furthermore, usual Monte-Carlo estimators for their price and sensitivities typically have significant variance. In this work, we consider alternative representations for barrier option prices in terms of the 3-dimensional Bessel bridge, and show how this leads to better estimators, especially for short maturities where we are able to increase the estimator efficiency dramatically.

We also discuss the related problem of efficient estimation of the density of first-passage times for diffusions. Even though the density estimation problem is essentially non-parametric, our method achieves (the typical Monte-Carlo) square-root order of convergence.

Fernando Zapatero (Marshall School of Business, University of Southern California)
Executive Stock Options as a Screening Mechanism

Coauthors: Abel Cadenillas (Department of Mathematical and Statistical Sciences, University of Alberta) & Jaksa Cvitanic (Division of Humanities and Social Sciences, Caltech)

We study how and when option grants can be the optimal compensation to screen low-ability executives. In a dynamic setting, we consider the problem of a risk-neutral firm that tries to hire a risk-averse executive whose actions can affect the expected return and volatility of the stock price. Even if the optimal compensation for all types of executives is stock under complete information, it might be optimal to offer options under incomplete information. We show that the likelihood of using options increases with the dispersion of types and the size of the firm, and decreases with the availability of growth opportunities for the firm.

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Emanuel Derman (Columbia University)
Metaphors, Models & Theories in Science and Finance

There has been a great deal of confusion about the role of models in the financial crisis. In this talk I want to discuss the possible ways of describing and explaining the world.
Scientific theories deal with the natural world on its own terms, and can achieve great truth and accuracy. They are very rare. Models in finance are not theories; they are closer to metaphors that try to describe the object of their attention by comparing it to something else they already understand via theories. Models are idealizations that always sweep dirt under the rug, and good models tell you what kind of dirt it is, and where it lies.

Sept. 29, 2010
Room 230, 5pm
Audio &
Slides of the Talks

Liuren Wu (Professor of Economics and Finance, Zicklin School of Business, Baruch College)
A New Approach to Constructing Implied Volatility Surfaces
Coauthors: Peter Carr

Standard option pricing often specifies the dynamics of the security price and the instantaneous variance rate, and derives its no-arbitrage implication for the option implied volatility surface. Market models have also been proposed to start with an initial implied volatility surface and a diffusion specification for the implied volatility dynamics, and derive the no-arbitrage constraints on the risk-neutral drift of the dynamics. This paper proposes a new approach, which specifies the security price dynamics, but leaves the instantaneous variance rate dynamics unspecified while specifying implied volatility dynamics instead. The allowable shape for the initial implied volatility surface is then derived based on dynamic no-arbitrage arguments. Two parametric specifications for the implied volatility dynamics lead to particularly tractable solutions for the whole implied volatility surface, as the surface can be represented as solutions to simple quadratic equations. The paper also proposes a dynamic calibration methodology and calibrates the two models to over-the-counter currency option and equity index option implied volatility surfaces over an 11-year period. The pricing performance is similar to standard option pricing models of similar complexities, but calibrating them is 100 times faster.

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Alexey Kuznetsov (York University)
Meromorphic Levy processes and their applications in Finance and Insurance

What is the distribution of the first passage time of the Variance Gamma process? What is the price of the double barrier option in the CGMY model? How can I compute the Gerber-Shiu function for
something more interesting than a compound Poisson process with exponential jumps? We all know that these are very hard questions, and despite a multitude of research papers published in this area there is still no consensus on what are the right answers. So if we can't find the answer, let's modify the question: can we find an interesting and large enough class of Levy processes for which all these problems can be solved? In this talk we will answer this last question in the affirmative by introducing meromorphic Levy processes. This is joint work with A.E.Kyprianou (University of Bath, UK), M.Morales (University of Montreal, Canada) and J.C.Pardo (CIMAT, Mexico).