


Mathematical and Computational Finance / Mathématiques
financières (Org: Tahir Choulli and/et Jie Xiong)
 FELIPE AGUERREVERE, University of Alberta, School of Business
Equilibrium investment strategies and output price behavior: a
realoptions approach

Most of the real options applications that are concerned with strategic
exercise policies focus on the effect of competition on the value of
the option to invest and ignore the operating decisions that may arise
once the investment is completed. However, operating flexibility is a
valuable real option. This paper studies the effect of competitive
interactions on investment decisions and on the dynamics of the price
of a nonstorable commodity. We develop a model of incremental
investment with time to build and operating flexibility. We find that
an increase in uncertainty may encourage firms to increase their
capacity. Furthermore, we show that it may be optimal to invest in
additional capacity during periods in which part of the completed
capacity is being utilized. Our results on capacity choice and capacity
utilization are qualitatively the same under oligopoly and perfect
competition. However, the impact of competition on the properties of
the endogenous output price is dramatic. In particular, the dynamics of
capacity utilization induces heteroskadasticity in the output price
process, and the price volatility may be increasing in the number of
competitors in the industry. Our results on output price behavior
appear to conform to actual behavior in competitive wholesale
electricity markets.
 CRISTIN BUESCU, Department of Mathematical and Statistical Sciences,
University of Alberta, Edmonton, Alberta T6G 2G1
Optimal portfolio management when there are taxes and
transaction costs

(joint work with A. Cadenillas and S. Pliska)
We consider a financial market with a bank account, and a single risky
stock modelled as a geometric Brownian motion. The objective is to
maximize the longrun growth rate of the wealth resulting after paying
transaction costs and taxes. We apply the theory of optimal stopping
to determine the optimal investment strategy under the effect of taxes
and transaction costs.
 ABEL CADENILLAS, Department of Mathematical and Statistical Sciences, University
of Alberta, Edmonton, Alberta T6G 2G1
Optimal manager compensation with choice of effort and volatility

(Joint work with J. Cvitanic and F. Zapatero)
We consider the problem of optimal manager compensation in a
continuoustime framework. We assume that the manager can control both
the drift and the volatility of the stock. We solve this problem using
martingale/duality methods.
 JACQUES CARRIERE, Mathematical and Statistical Sciences, University of Alberta,
Edmonton, Alberta
Martingale valuation of cashflows for insurance and interest
models

Using a pricing axiom from financial economics, a martingale valuation
method is developed with the properties of numéraire invariance and
noarbitage. The pricing method is then applied to cashflows in
actuarial models like loans, bonds, insurances, annuities, reserves and
surplus processes. Special emphasis is given to the modelling of
portfolios of defaultable bonds where new results are given. Also, an
optimal repayment analysis of a common loan arrangement reveals that
the book and market rates have to be equal.
Please send requests to receive a copy of the entire manuscript
directly to the author at: jacques@gompertz.math.ualberta.ca
 SHUI FENG, McMaster University, Hamilton, Ontario L8S 4K1
The valuation of some American call options

In this talk, I will review results on the valuation of American call
options and American capped options. New results will also be presented
on the valuation of American call options on the minimum of two
dividendpaying assets. This is joint work with Jerome Detemple and
Weidong Tian.
 SEBASTIAN FERRANDO, Department of Mathematics, Ryerson University, Toronto,
Ontario M5B 2K3
Haar wavelets systems for pricing and hedging financial derivatives

We use Haar wavelets systems to construct pathwise approximations of
simple financial processes. This is in contrast to most techniques
which depend on weak approximations to the process. Our constructions
provide alternative computational approaches to price and hedge
financial derivatives. The approach allows the deployment, in an
stochastic process setting, of wavelets techniques such as:
compression, search for best Haar packets and denoising.
 BOB KIMMEL, Department of Economics, Princeton University, Princeton,
New Jersey 08544, USA
Market price of risk specifications for affine models: theory and
evidence

We extend the standard specification for the market price of risk for
affine yield models of the term structure, and estimate models using
the extended specification. For most models, likelihood ratio tests
indicate that the extended specification fits the data better than
standard specifications, often with huge statistical significance. The
squareroot process of Feller [1951] was first used in term structure
modeling by Cox, Ingersoll, and Ross [1985], who specified the market
price of risk as a constant times the square root of the interest rate.
With this specification, the interest rate follows an affine diffusion
under both the true and riskneutral probability measures. However, the
true and riskneutral dynamics of the interest rate are still affine
under both measures if the market price of risk also includes a term
that is inversely proportional to the square root of the interest rate.
This specification is never used in financial applications, probably
because it does not satisfy either the Novikov or Kazamaki criteria.
However, these criteria are sufficient but not necessary for the
existence of a Girsanov ratio. We show that the extended specification
does not permit arbitrage opportunities, provided that under either
measure the probability is zero that the interest rate process achieves
the boundary value of zero. We also extend our market price of risk
specification to the general multivariate affine yield model of Duffie
and Kan [1996], deriving the most general market price of risk
specification for which state variable dynamics are linear under both
measures. We estimate all affine yield models with three or fewer
factors, using the market price of risk specifications of Dai and
Singleton [2000], Duffee [2002], and our extended specification. We
apply likelihood ratio tests of our specification relative to both the
Dai/Singleton and Duffee specifications. There are six models for which
our extension is more general than Duffee's specification; we find our
extension is statistically significant for five of these models at the
conventional 95% confidence level, and at far higher levels for three
of the models. The results are particularly strong for affine
diffusions with multiple squareroot type variables, i.e., using
the notation of Dai and Singleton, Am(N) models with m > 1. Although
we focus on affine yield models, our extended market price of risk
specification also applies to any model in which Feller's process or a
multivariate extension is used to model asset prices.
 ALI LAZRAK, University of British Columbia
Information neutrality in the stochastic differential utility

This paper develops in a Brownian information setting an approach for
analyzing the nonindifference for the timing of resolution of uncertainty, a
question that motivates the stochastic differential utility (SDU) due to
Duffie and Epstein (1992). For a class of Backward Stochastic Differential
Equations (BSDEs) including SDU, we formulate the information neutrality
property as an invariance principle when the filtration is coarser (or
finer) and characterize this property. Furthermore, we provide a concrete
example of heterogeneity in information that illustrates explicitly the
neutrality property for some particular BSDEs.
 ALEXANDER MELNIKOV, University of Alberta
On the pricing of lifeinsurance contracts based on risky assets

The talk is devoted to the pricing of socalled equitylinked
lifeinsurance contracts. These investment instruments reflect an
innovative trend in traditional insurance toward financial economics.
In the talk we consider discrete and continuous time variants of
BlackScholes models and present meanvariance and other
methodologies.
 PAT MULDOWNEY, University of Ulster
A generalized BlackScholes equation without Ito calculus

Using modern integration theory based on the generalized Riemann
integration of Henstock and Kurzweil, stochastic processes such as
those of financial theory can be modelled by elementary methods not
involving the Itô calculus. To demonstrate this, the following form
of the BlackScholes partial differential equation is established:

¶E^{m} (f)
¶t

+mx 
¶E^{m} (f)
¶x

+ 
1
2

s^{2} x^{2} 
¶^{2} E^{m} (f)
¶x^{2}

=r E^{m}(f). 

With the underlying asset price process x, with t as the initial
time and x the initial value (x(t) = x), and with r an
arbitrary discounting parameter, the expectation value E^{m}(f) is
the expected initial value of a discounted derivative asset price
process f relative to the probabilities which are induced in the
sample space by assuming an arbitrary growth rate m in the
underlying process x. This new approach also separates those
mathematical features which depend on the martingale property from
those which depend on the assumption of geometric Brownian motion.
 BRUNO REMILLARD, HEC, Montreal
Discrete time problems in hedging strategies and portfolio
optimization

In this talk I will discuss how martingales enters naturally in
dynamical selection problems like hedging strategies and investment
portfolio. I will also examine the impact of discrete time versus
continuous time in hedging strategies, option pricing and portfolio
optimization.
 LOUIS SECO, University of Toronto
Entropy methods for time series calibration

Determining the structure of distributions is at the heart of financial
risk management, specially when dealing with nongaussian markets that
exhibit heavy tail behavior. The problem lies at the ability to
extract information on extreme events from a limited sample data set.
This talk will present a methodology that uses entropy maximization
methods to provide with a framework that allows for the determination
of onedimensional probabilistic parametric structures.
 MICHAEL TAKSAR, Missouri
Ruin probability minimization and dividend distribution
optimization in diffusion models

We will consider a model of an insurance companies with different modes
of risk and financial control. Different types of reinsurance
correspond to the risk reduction techniques of the insurance, while
financial control corresponds to a more familiar portfolio rebalancing.
There are different objective which the company may pursue. One is the
classical minimization of the ruin probabilities. Another one is the
dividend payout maximization. The later merges with the classical
finance issue of utility optimization by a small investor, pioneered by
Merton. Diffusion approximation enables one to get a closed form
solution to many problems and see the structure of the optimal policy.
Mathematically, the problem becomes a mixed singular/regular control of
a diffusion process, whose analytical portion corresponds to a solution
of nonlinear ordinary or partial differential equations.
 RUPPA THULASIRAM, Department of Computer Science, University of Manitoba,
Winnipeg, Manitoba
Fast Fourier transform in option pricinga parallel algorithm

Computational Finance (CF) is relatively young and rapidly growing
field. CF knowledge and skills are in increasing demand in the finance
industry. This area is a coalescence of many disciplines, such as
Mathematics, Computer Science & Engineering, Numerical Techniques and
the theory of Financial Economics. Pricing of derivatives is one of
the central problems in CF. Since the theory of derivative pricing is
highly mathematical, numerical techniques such as lattice approach,
finitedifference and finiteelement techniques among others have been
employed. Recently, Fast Fourier Transform (FFT) has been used for
derivative pricing. In this talk we describe the FFT relation to
option pricing problem and describe the design and development of a
parallel algorithm for option pricing with FFT. For a data size of N
and P processors, a blocked data distribution for the FFT in general
produces log(N)log(P) iterations of local communications and
log(P) iterations of remote communications. We discuss the
performance of our algorithm first and identify some of the
shortcomings. We then improve our algorithm by exploiting data
locality to reduce communication overheads inherent in the parallel FFT
algorithm. Compared to the original FFT algorithm, the modified
algorithm with data swap network improves the performance by about 15%
for large data sizes.
 HAO WANG, University of Oregon, Eugene, Oregon 974031222, USA
Calibrated models for heavytailed risk factors

We have seen the deficiencies of normal and other thintailed
distribution models in risk management with extreme observations. In
this talk, we will introduce some models to handle heavytailed
phenomena. First, a calibrated scenario generation model for
multivariate risk factors with heavy tailed distribution will be
discussed. This model is a generalization of the standard model of
scenario generation developed by J. P. Morgan and later on updated by
RiskMetrics to simulate risk factors with heavytailed distribution.
In this model, a rotation method is introduced to calibrate the
covariance matrix and a mixture of normal distributions is used to
calibrate and fit to each marginal distribution. Then, based on the
scenario generation, a nonparametric method is applied to estimate
portfolio extreme ValueatRisk and ValueatRisk confidence interval.
The generalization of BlackScholes formula is also considered for the
stock or asset prices with heavytailed distributions.
 LIXIN WU, Claremont Graduate University
LIBOR market model: from deterministic to stochastic volatility

LIBOR market model is the benchmark model for interestrate
derivatives. It has been a challenge to extend the standard LIBOR
market model so as to cope with the volatility smiles and/or skews that
are pronounced in the swaption markets. In this talk we extend the
standard LIBOR market model, which takes forward rate or swap rate as
state variables, by adopting stochastic volatility. Specifically, we
adopt a multiplicative stochastic factor for the volatility functions
of all relevant forward rates. The stochastic factor follows a
squaredroot diffusion process, and it can be correlated with the
forwardrate processes. We derive approximate processes for swap rates
after the change to forward swap measures, and develop a closedform
formula for swaption prices in terms of Fourier integrals. We then
develop a fast Fourier transform algorithm for the implementation of
the formula. The approximations are well supported by pricing
accuracy. By adjusting the correlation between the forward rates and
the volatility in a way consistent with intuition, we can generate
volatility smiles or skews of the swaption prices similar to those
observed in the markets. Calibration of the model will also be
discussed.
 YONG ZENG, Department of Mathematics and Statistics, University of Missouri,
Kansas City, Missouri 64110, USA
A cass of partiallyobserved models for micromovement of
asset orices with Bayesian inference via filtering

A general class of micromovement models that describes the
transactional price behavior is proposed. The model ties the sample
characteristics of micromovement and macromovement in a consistent
manner. An important feature of the model is that it can be transformed
to a filtering problem with counting process observations.
Consequently, the complete information of price and trading time is
captured and then utilized in Bayesian inference via filtering for the
parameter estimation and model selection. The filtering equations are
derived. Recursive algorithms are constructed via the Markov chain
approximation method to compute (1) the approximate posterior for
tradebytrade Bayes estimation and (2) the Bayes factor for Bayesian
model selection. The consistency (or Robustness) of the recursive
algorithms is proven. Two models are studied in detail. One is the
model built on geometric Brownian motion (GBM) and the other is on the
GBM plus jumping stochastic volatility. Simulation results show that
the Bayes estimates for timeinvariant parameters are consistent, the
Bayes estimates for stochastic volatility capture the movement of
volatility, and the Bayes factor can effecitvely and efficiently
selects the right model. Realworld applications to Microsoft
transaction data are also provided.
 XINGQIU ZHAO, University of Alberta
Option pricing with modified fractional Brownian motion

The BlackScholes model introduced by Black and Scholes (1973) and
Merton (1973) has become synonymous with modern finance theory. It
assumes that the dynamics of stock prices is well described by
exponential Brownian motions; that is, stock price returns behave like
a sequence of independently idential distributed Gaussian random
variables. The hypothesis is not consistent with empirical stock price
returns. Many authers (e.g., Mandelbrot (1967), Greene and
Fielitz(1977), Lo and Mackinlay (1998), Willinger, Taqqe and Teverovsky
(1999), etc.) found empirical evidence of longrange dependence
in stock price returns. This paper develops a numerical model for
option pricing under the hypothesis that the underlying asset price
satisfies a stochastic differential equation driven by a modified
fractional Brownian motion. The empirical results indicate that the
model is better than the BlackScholes model.

