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Seminar: Spring 2003

Stanford Financial Mathematics Seminar Schedule

Date Speaker Affiliation Talk Title
(click to see Abstract)
Comments
4/11 John Moody Professor of Computer Science and Electrical Engineering,Oregon Graduate Institute Minimizing Downside Risk via Direct Reinforcement Survey article (pdf)
4/18 Frank Riedel Lynen Research Fellow, Stanford University Dynamic Coherent Risk Measures [pdf]  
4/23 (Wed) 4:15 Andrew Mullhaupt Director of Research,
SAC Meridian Fund
Cantelli's Inequality and the Estimation of Transaction Costs [pdf] After the talk, meet company representatives and discuss career opportunities.
4/25 Peter Bossaerts Professor of Economics and Management and Professor of Finance, Caltech Prices And Portfolio Choices In Financial Markets: Theory And Experimental Evidence  
5/2 Sanjiv Das Professor of Finance, Santa Clara University Simple Comprehensive Models of Default Risk Paper: .pdf
5/9 Jaksa Cvitanic Professor of Mathematics and Economics, USC Revisiting Treynor and Black (1973): an Intertemporal Model of Active Portfolio Management  
5/16 Karim Khiar Director, Financial Modeling, Gifford Fong Associates Anatomy of Copula Default Models  
5/20 (Tue) 4:15 Eric Hillebrand Department of Mathematics, Stanford University Unknown Parameter Changes in GARCH and ARMA Models This is a joint Statistics / Financial Mathematics seminar.
5/23 Josh Gray Derivatives Specialist,
Financial Engineering Associates
Real Options: Theory and Practice  
5/30 Nassim Taleb Founder and Chairman of Empirica Capital On the Effect of Distributions Not Being Observable  
6/27 Nicole El Karoui Professor of Mathematics, Université de Paris VI.
Professor of Applied Mathematics, Ecole Polytechnique.
Optimal design of derivatives under dynamic risk measures This is a joint Probability / Financial Mathematics seminar.


Minimizing Downside Risk via Direct Reinforcement

John Moody ( Department of Computer Science, OGI School of Science & Engineering, Oregon Health & Science University)

Title: Presenter: Department of Computer Science OGI School of Science & Engineering Oregon Health & Science University Abstract:

I present reinforcement learning (RL) methods for optimizing portfolios, asset allocations and trading systems. In this approach, investment strategies are discovered in simulation via trial and error exploration.

RL has been developed in the machine learning, neural networks and control engineering communities since the 1950's. These methods can find approximate solutions of dynamic programming or stochastic control problems without a detailed model of the system. Direct Reinforcement (DR) algorithms are a promising alternative to standard RL methods. DR enables more natural problem representations, reduces Bellman's "curse of dimensionality" and offers compelling advantages in efficiency.

I will demonstrate how Direct Reinforcement can be used to directly optimize investment performance criteria such as profit, economic utility or risk adjusted returns. Traders with varying degrees of risk aversion can be modeled by the choice of performance measure. The approach is illustrated using the Sharpe ratio and a criterion that penalizes downside risk, the Downside Deviation ratio. Market frictions, such as transaction costs or market impact, can be included easily in the optimizations. The strategies discovered are shown to depend upon the level of transaction costs and choice of performance criterion.

Demonstrations of the proposed approach include a monthly asset allocation system and an intra-daily currency trader.

Speaker Biography:
John Moody is a Professor of Computer Science at the OGI School of Science & Engineering in Portland, Oregon. His research interests include machine learning, neural and statistical computing, time series analysis and computational finance. He recently served as Program Co-Chair of the IEEE Computational Intelligence in Financial Engineering 2003 in Hong Kong and Computational Finance 2000 in London. Moody founded OGI's very successful Computational Finance program in January 1996 and served as its Director until June 2002. He previously held positions in Computer Science and Neuroscience at Yale University and at the Institute for Theoretical Physics in Santa Barbara. Moody received his B.A. in Physics from the University of Chicago, and earned his Ph.D. in Theoretical Physics at Princeton.
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Prices And Portfolio Choices In Financial Markets: Theory And Experimental Evidence

Peter Bossaerts (Caltech)

This paper reports on experiments designed to test the basic principles of modern asset pricing theory. Prices in the experiments generally conform to the implications of standard asset-pricing theory (CAPM in particular), although risk premia are (perhaps surprisingly) large. However, portfolio choices in these experiments are strikingly at variance with the predictions of standard theory (portfolio separation). This is puzzling because the price predictions of standard theory depend critically on the portfolio predictions. The paper proposes a theoretical explanation of this price-portfolio paradox, extending the standard model to include perturbation terms in individual demands. When the number of traders is large --- as is the case in the experiments reported here --- these perturbation terms largely cancel out, and the model yields pricing predictions identical with those of standard theory but portfolio predictions that are quite different. The predictions of the extended model are consistent with the experimental data. The individual choices in the experiments shed light on the nature of the perturbation terms. Revealed preference analysis and analysis of expected-utility-equivalent experiments without uncertainty suggest that the errors do not arise from noise, subject confusion, lack of experience or inattention.
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Simple Comprehensive Models of Default Risk

Sanjiv Das (Santa Clara University)

In this talk, I will first introduce some important aspects about default risk modeling, and then present the paper titled "A simple unified model for pricing derivative securities with equity, interest rate, default and liquidity risk".

This paper develops a model for pricing securities that may be a function of several different sources of risk, namely, equity, interest-rate, default and liquidity risks. The model is also useful for extracting probabilities of default (PD) functions from market data.

The model is not based on the stochastic process for the value of the firm, but on the stochastic process for interest rates and the equity price, which are observable. The model comprises a risk-neutral setting in which the joint process of interest rates and equity are modeled together with the default conditions for security payoffs. The model is embedded on a recombining lattice which makes implementation of the pricing scheme feasible with polynomial complexity. We present a simple approach to calibration of the model to market observable data. The framework is shown to nest many familiar models as special cases.

The model is extensible to handling correlated default risk and may be used to value distressed convertible bonds, debt-equity swaps, and credit portfolio products such as CDOs. We present several numerical and calibration examples to demonstrate the applicability and implementation of our approach.
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Revisiting Treynor and Black (1973): an Intertemporal Model of Active Portfolio Management

Jaksa Cvitanic (University of Southern California)

Authors:
Jaksa Cvitanic, Ali Lazrak, Lionel Martellini and Fernando Zapatero

In the line of Treynor and Black (1973), we provide a closed-form solution to the problem of an investor with non-myopic CRRA utility who selects an optimal portfolio when assets offer an abnormal expected return.

In the model of portfolio optimization with partial information in continuous time, we assume that the investor has a normal prior on the abnormal returns of the securities and upgrades those priors in a Bayesian way. We allow the priors to be correlated and show that the correlation between priors is an important parameter in the determination of optimal holdings. The time horizon of the investor is another important parameter. We also account for the presence of portfolio constraints. These results provide a formal model for long/short investment strategies and have implications for the active management industry.

Our findings suggest that low beta hedge funds may serve as natural substitutes for a significant portion of an investor risk-free asset holdings. We calibrate the model to a database of hedge funds data, and find that the optimal portfolios roughly agree with the market practice.
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Anatomy of Copula Default Models

Karim Khiar (Gifford Fong Associates)

Default models are critical for pricing bonds, credit derivatives and structured finance products driven by defaults. In this framework, practitioners extensively use default models.

In this discussion, I will focus on the copula default models. These models (and more precisely the normal copula model) are very common in credit default modeling. These models are dominant due to their tractability. We will examine their mathematical properties.

First, we will analyze the different components necessary to implement a default model. We will examine the correlation parameter, the asset model, and term structure of default.

Next, we will describe the single step default model based on multivariate normal distribution and we will illustrate its limitations.

Then, we will describe the multi-step approach. The approach will explicitly introduce a time dimension in the default model. We will characterize the advantages and its limitations. To illustrate important differences between these models, the single step outputs are compared to a multi step model.

Finally, through several examples, we will emphasize the impact of these different parts (such as maturity, rating, correlation, etc.) on the multi-step model. These examples are based on current practices in the field of structured finance to design and manage the risk of credit portfolios. Based on this simple example, I will illustrate the misuse and misconception of current practices.
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Unknown Parameter Changes in GARCH and ARMA Models

Eric Hillebrand (Stanford University)

A common finding in the empirical financial literature is that the volatility of financial data exhibits high persistence, or slow mean reversion of the order of months. In GARCH models, this is represented by the fact that the sum of the estimates of the autoregressive parameters is close to one when sample periods are considered that cover several years.

It has been documented in simulations, however, that parameter changes in the data-generating process lead to exactly this phenomenon. It has also been reported that segmentations of financial time series and local GARCH estimations on the segments considerably reduce the estimated persistence, that is, the sum of the estimated autoregressive parameters is below one. One possible explanation is that the data-generating persistence within segments of constant parameters is relatively low and that the globally measured high persistence is caused by the parameter changes.

I will show that in GARCH models of orders up to GARCH(2,2), unknown parameter changes in the data-generating process in fact cause the sum of the estimated autoregressive parameters to be close to one. This is a consequence of the geometry of the estimation problem, not of the statistical properties of the estimators, about which little is known. This particular estimation problem is not confined to GARCH models, the arguments are readily generalized to ARMA models of orders up to ARMA(2,2). I will illustrate this effect using synthetic and market data.
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Real Options: Theory and Practice

Josh Gray (Financial Engineering Associates)

Real Options Theory is an attempt to map an asset, a contract or a project into the parlance of the Black-Scholes treatment of options. As such, the theory is widely used in energy markets where the valuation and hedging of certain contracts, with embedded or explicit optionality, are managed through this simplifying and tractable translation of the real world into the financial.

In this talk, we will discuss several examples of real options theory, including swing contracts (american options), natural gas storage (calendar spread options), and tolling agreements (spark spread options), while pointing out the strengths, weaknesses and uncharted areas for exploration.
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On the Effect of Distributions Not Being Observable

Nassim Taleb (Empirica Capital)

The generator of a random process is seldom observable in the social sciences, only its realizations, which may or may not reveal its nature. Yet knowledge about the generator is what is needed to infer the true moment properties of the process (expectation, variance). Distinguishing between Knightian risk and Knightian uncertainty becomes impossible. We present through the use of Gedanken experiments mathematical and epistemological problems that arise in the process of attaining knowledge about such needed properties.
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Optimal design of derivatives under dynamic risk measures

Nicole El Karoui (Université Paris VI & Ecole Polytechnique)

We develop a methodology to optimally design a financial issue F to reduce the exposure of an agent A towards a non tradable risk X. The constraint is that an agent B only enters the transaction if her risk level remains below a given threshold. Both agents have also the opportunity to invest on financial markets.

In an entropic framework, the optimal solution is to transfer a constant proportion of the initial exposure. This constant is a function of the risk aversion coefficient, and not of the distribution of the risks. The key assumption is that both agents may invest in the same market.

In the general case, the same results hold if both agents assess their risk through convex risk measures (in the sense of Foellmer and Schied) with proportional penalty functions. If not, the solution is more complex. For dynamic risk measures characterized by their local decomposition as semimartingales, we characterize the solution in terms of Backward Stochastic Differential equations.

Joint work with Pauline Barrieu.
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