

 Seminar: Winter 2004
Stanford Financial Mathematics Seminar Schedule
A Rapidly Convergent Expansion Method for Asian and Basket Options


Eric Reiner (Managing Director, Group Market Risk, UBS AG)
Overthecounter derivatives on timeaveraged prices and/or portfolios
of assets have become ubiquitous over the past decade. Nevertheless, a
generalized approach to valuing such options in a BlackScholes 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
GramCharlier 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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Dynamic Models with TimeVarying Volatilities and Regression Parameters
and Their Applications to Financial Time Series


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 changepoints. Assuming conjugate priors, we
derive closedform 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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A Theory of NonGaussian Option Pricing: Capturing the Smile and the Skew


Lisa Borland (EvnineVaughan Associates Inc)
We introduce a new model of stock returns that results in fattailed
(powerlaw) 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 BlackScholes 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 wellfitted
with q around 1.5 (consistent with the socalled 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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Security Market Imperfections and Optimal Betting Strategies


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 ArrowPratt 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
turnofthemonth 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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Quantiles of Levy processes and related path dependent options.


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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Trading Correlation


Peter Cotton (Morgan Stanley)
What mathematics is relevant to the practice of trading
correlationsensitive credit portfolio products?
Is Sklar's theorem the key idea, or a red herring?
Are Copulas the answer, or a copout?
I will argue that existing models are relevant insofar as
they reflect the bare minimum for regulator and marktomarket 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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Empirical Credit Risk


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 marketspecific 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 marketspecific
factors.
I also describe evidence of an agency effect in the
bondequity return relationship.
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Using Principles of Behavioral Finance to Manage LongOnly and Hedged
Portfolios


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
longonly and hedged equity portfolios. We will then discuss some of
the institutional problems we have encountered in forming
shortportfolios  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" crosssectionally for individual stocks.
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Pricing and Hedging of Synthetic CDO Transactions


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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