Seminar: Winter 2004

Eric Reiner (Managing Director, Group Market Risk, UBS AG)

Over-the-counter derivatives on time-averaged prices and/or portfolios of assets have become ubiquitous over the past decade. Nevertheless, a generalized approach to valuing such options in a Black-Scholes framework has yet to emerge. We report on research toward this goal.

We begin with an overview of Asian, basket, and related payoffs and their uses. Next, we perform a critical review of several of the available techniques for valuing such options, comparing accuracy and attempting to reveal the conceptual connections between methods. Third, we introduce a novel characteristic function expansion technique based on the Gram-Charlier approach but making greater use of the close relationship between target and reference distributions.

We examine numerical convergence properties of the new method; these are shown to be quite promising for Asian options, less so for basket options. Finally, we muse on some of the reasons why this might be so.

This is joint work with Dmitry Davydov and Rama Kumanduri.
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Tze Leung Lai (Stanford University)

Volatility modeling is a cornerstone of empirical finance, as portfolio theory, asset pricing and hedging all involve volatilities, and its fundamental importance has been recognized in this year's Nobel Prize award in Economics.

After a brief review of conventional approaches to modeling asset returns and their volatilities, we describe a new class of dynamic models that are stochastic regression models in which the regression parameters and error variances may undergo abrupt changes at unknown time points, while staying constant between adjacent change-points. Assuming conjugate priors, we derive closed-form recursive Bayes estimates of the regression parameters and error variances. Approximations to the Bayes estimates are developed that have much lower computational complexity and yet are comparable to the Bayes estimates in statistical efficiency.

We also address the problem of unknown hyperparameters and propose two practical methods for simultaneous estimation of the hyperparmeters, regression parameters and error variances. Applications of the methodology to simulated and real financial data show that it offers a promising alternative approach to modeling and forecasting asset returns and their volatilities.

This is joint work with Haipeng Xing and Haiyan Liu.
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Lisa Borland (Evnine-Vaughan Associates Inc)

We introduce a new model of stock returns that results in fat-tailed (power-law) Student distributions rather than Gaussians. These distributions are characterized by an index q, related to the Tsallis generalized entropy that we use to model the evolution of fluctuations. For q =1 the standard Black-Scholes case is recovered. Based on this model, which is a statistical feedback process for returns, one finds a martingale representation and simple closed form pricing equations for European calls. Most empirical distributions of returns are well-fitted with q around 1.5 (consistent with the so-called cubic law). Using that value of q in the option pricing formulas yields results which match empirically observed prices and volatility smiles very well, using just one value of sigma across all strikes. In a simple manner, we also show how this model can be extended to account for skew.
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William Ziemba (University of British Columbia)

Successful speculative investment requires strategies with positive expectation, plus optimization to provide desirable wealth paths over time. The Kelly or capital growth criterion, where the expected logarithm of wealth is maximized, has many desirable properties. As the horizon becomes increasingly long, the Kelly bettor has more and more wealth than any other essentially different bettor. However, in the short run, the essentially zero Arrow-Pratt risk aversion leads to very large bets and extreme volatility of wealth levels over time. Professional bettors in sports betting, racing syndicates, and financial markets, all of which are basically hedge funds, frequently use fractional Kelly strategies. This blend with cash lowers the bet size and leads to smoother wealth paths, but usually with lower final wealth. These strategies are essentially the negative power utility class that contains log as its limiting and most risky member. This is exact for lognormal assets and approximately correct otherwise.

I will show some examples, from simulated and actual betting, to illustrate the ideas from Lotto games with unpopular numbers, blackjack, futures options trading on the S&P 500, futures trading on the January turn-of-the-month effect, and some horseracing applications such as the Kentucky Derby and the Pick 6. These examples, plus the results of betting syndicates, the unofficial hedge funds of Lord Keynes, Warren Buffett and Ed Thorp, stress the behavioral and economic aspects of the construction of winning strategies.

There are a number of interesting unresolved mathematical problems which I would like to bring out in the talk. Calculation of optimal fractional Kelly strategies subject to constraints on wealth paths is one area of interest. One can choose the optimal fractions to stay above a wealth path with high probability. Other variants are possible as well.
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Angelos Dassios (London School of Economics)

In 1992 Miura suggested a new class of path dependent options based on the median of the asset price as opposed to the average (Asian option). Mathematically, this is as easy or as difficult to obtaining the distribution of any path quantile. Early results produced useful distribution identities for the Brownian motion that were then found to be true for the general Levy case.

In this seminar, we will survey results existing in the literture as well as new. We will also point out connections to other options based on occupation times and investigate some further issues of pricing and hedging.
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Peter Cotton (Morgan Stanley)

What mathematics is relevant to the practice of trading correlation-sensitive credit portfolio products? Is Sklar's theorem the key idea, or a red herring? Are Copulas the answer, or a cop-out?

I will argue that existing models are relevant insofar as they reflect the bare minimum for regulator and mark-to-market needs of banks -- maybe. I'll demonstrate a few views which market participants have gravitated to, and why they are all bad.

Then I will plea for help.
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Oren Cheyette (BARRA)

Using a bond's yield spread as a measure of credit risk, the "empirical credit risk" model explains its returns in terms of interest rate movements, the issuer's equity returns and residual bond market-specific factors.

High quality bond returns are largely explained by interest rate changes, while low quality bond returns are primarily explained by the issuers' equity returns. Intermediate credit quality bond returns are not significantly explained by either interest rate changes or equity returns, and appear to be attributable only to the bond market-specific factors.

I also describe evidence of an agency effect in the bond-equity return relationship.
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Russell Fuller and Mark Moon (Fuller & Thaler Asset Management)

This seminar will begin with a brief overview of how one investment management firm uses principles of behavioral finance to manage both long-only and hedged equity portfolios. We will then discuss some of the institutional problems we have encountered in forming short-portfolios -- most of these issues are ignored in the academic literature, but are quite important in implementing short strategies. A related issue concerning short positions is the timing of the short sale, which is much more important than the timing of the long purchase. We are currently exploring statistical techniques that might help in determining the appropriate time to initiate short sales. Interested seminar participants might review the book "Why Stock Markets Crash: Critical Events in Complex Systems," by Didier Sornette. While Sornette looks at the stock market as a whole, we are more interested in predicting "critical events" cross-sectionally for individual stocks.
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David Li (Citigroup)

An overview of the market development in the CDO market, especially the latest innovation of single tranche products will be given. Then I'll show how concretely to price these transactions by building credit curves, using copula function. Some practical computational issues on the model implementation will also be discussed. Lastly I'll discuss about the risk measurement and hedging issues. The shortcomings of the current model and possible future development on the CDO modeling front will be highlighted.
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