


Mathematical Finance
Org: Len Bos and Anatoliy Swishchuk (Calgary) [PDF]
 ABEL CADENILLAS, University of Alberta, Dept. of Mathematical and Statistical
Sciences, Edmonton, Alberta T6G 2G1
Optimal RiskSharing with Effort and Project Choice
[PDF] 
We consider firstbest risksharing problems in which "the agent"
can control both the drift (effort choice) and the volatility of the
underlying process (project selection). In a model of delegated
portfolio management, it is optimal to compensate the manager with an
optiontype payoff, where the functional form of the option is
obtained as a solution to an ordinary differential equation. In the
general case, the optimal contract is a fixed point of a functional
that connects the agent's and the principal's maximization problems.
We apply martingale/duality methods.
Joint work with Jaksa Cvitanic and Fernando Zapatero.
 JOE CAMPOLIETI, Wilfrid Laurier University
New Families of Integrable Diffusion Models and their
Applications to Finance
[PDF] 
We present some recent developments in the construction and
classification of new families of analytically solvable
onedimensional diffusion models for which transition densities and
other quantities that are fundamental to financial modeling and
derivatives pricing are represented in closed form. Our approach
allows us to uncover new multiparameter processes that are mapped
into various simpler diffusions. From an asymptotic analysis of the
boundary behaviour of the processes, we arrive at a rigorous
characterization of the newly constructed diffusions with respect to
probability conservation and the martingale property. Specifically,
we analyze in detail three subfamilies of models arising from the
underlying squared Bessel process (Bessel family), the CIR process
(confluent hypergeometric family) and the Jacobi diffusion
(hypergeometric family). We show that the Bessel family is a superset
of the constant elasticity of variance (CEV) model. The former, in
turn, is generalized by the confluent hypergeometric family. For
these two families we find further subfamilies of conservative strict
supermartingales and absorbed martingales. For the new classes of
absorbed diffusions we also derive analytically exact firstpassage
time densities, as well as probability densities for the extrema of
the processes. Formulas are reduced to integral representations or
eigenfunction expansions involving special functions. New closedform
pricing formulas for standard Europeans, barrier options and lookback
options also follow. We conclude by discussing other mathematical
finance applications and possible extensions of our models to include
jumps and markov switching.
 MATT DAVISON, University of Western Ontario, Department of Applied Math
Some Optimal Control Problems in Natural Gas Storage
[PDF] 
Real options theory is used to derive nonlinear partial
integrodifferential equations for valuing and optimally operating
natural gas storage facilities. The equations incorporate a class of
spot price models which exhibit the same timedependent,
meanreverting dynamics and price spikes observed in energy markets.
The operational characteristics of real storage units, including
working gas capacities, variable delivery and injection rates, and
cycling limitations, are incorporated. The model is illustrated with
a solved numerical example of a salt cavern storage facility to
illustrate the similarity between storage facilities and financial
straddles. Depending on the amount of stored gas the relative
influence of the put and call components vary.
This is joint work with Matt Thompson and Henning Rasmussen.
 JEANMARIE DUFOUR, Université de Montréal
Testing portfolio efficiency with an unobservable zerobeta
rate and nonGaussian distributions: a finitesample
identificationrobust approach
[PDF] 
We consider the problem of testing portfolio efficiency when the
zerobeta rate is unknown [Black Capital Asset Pricing Model (BCAPM)].
It is well known that standard asymptotically justified tests and
confidence intervals are quite unreliable in this setup. We point out
that this feature is associated with the fact that the zerobeta rate
may be interpreted as a structural parameter that may be weakly
identified, leading to a breakdown of standard asymptotic procedures
based on estimated standard errors. The available exact procedures
for the BCAPM, however, rely heavily on the assumption that model
disturbances follow a Gaussian distribution, which does not appear to
be satisfied by many financial return series. We propose exact
simulationbased procedures for testing meanvariance efficiency of
the market portfolio and building confidence intervals for the unknown
zerobeta rate. The proposed methods are based on
likelihoodratiotype statistics, allow for a wide class of error
distributions (possibly heavytailed) and are robust to weak
identification of the zerobeta rate. Further, we suggest a general
method which yields tighter bounds in both Gaussian and nonGaussian
cases. In order to build confidence intervals for the zerobeta rate
in finite samples, a technique based on generalizations of the classic
Fieller method (for the ratio of two parameters) is proposed.
Empirical results on NYSE returns show that exact confidence sets are
very different from the asymptotic ones, and allowing for nonGaussian
distributions substantially decreases the number of efficiency
rejections.
 ROBERT ELLIOTT, University of Calgary, Haskayne School of Business,
2500 University Drive NW, Calgary, AB T2N 1N4
New Results for Fractional Brownian Motion
[PDF] 
Mathematically Fractional Brownian Motion is a difficult process to
define. We shall present a new approach to Fractional Brownian
Motion extending recent work of Hu, Oksendal, Duncan and Pasik Duncan
to include processes with Hurst parameters 0 < H < 1. A central tool
for fractional Brownian motion is the analog of the Ito formula. We
give new proofs and results. We also answer some of the criticisms of
the use of fractional Brownian motion in financial modeling.
This is joint work with John van der Hoek of the University of
Adelaide, Australia.
 PETER FORSYTH, University of Waterloo, 200 University Avenue, Waterloo, Ontario
Dynamic Hedging Under Jump Diffusion with Transaction Costs
[PDF] 
It is well known that the standard model of asset price processes,
Geometric Brownian Motion, is not capable of reproducing the fat tails
in observed price distributions. From a risk management point of
view, the most troubling aspect of commonly used models is their
inability to provide a useful hedging strategy in the presence of
jumps.
If the price process follows a jump diffusion, it is known that a
perfect hedge is not possible with a finite number of hedging
instruments. It is also conventional wisdom that hedging with options
is too expensive, due to the large transaction costs typical of the
option market.
In this study, we suggest a dynamic hedging strategy based on hedging
with the underlying and liquid options. We solve an optimization
problem at each hedge rebalance date. We minimize both the "jump
risk" and the transaction cost.
Simulation studies of this strategy, using typical market bidask
spread data for the options, shows that using the underlying and
options in the hedge portfolio does an excellent job of minimizing
jump risk, as well as being not too costly in terms of total
cumulative transaction costs.
This is joint work with Shannon Kennedy and Ken Vetzal.
 ULRICH HORST, University of British Columbia, Department of Mathematics
Climate Risk, Securitization, and Equilibrium Bond Pricing
[PDF] 
We propose a method of pricing financial securities written on
nontradable underlyings such as temperature or precipitation levels.
To this end, we analyze a financial market where agents are exposed to
financial and nonfinancial risk factors. The agents hedge their
financial risk in the stock market and trade a risk bond issued by an
insurance company. From the issuer's point of view, the bond's
primary purpose is to shift insurance risks related to
noncatastrophic weather events to financial markets. As such its
terminal payoff and yield curve depend on an underlying climate or
temperature process whose dynamics is independent of the randomness
driving stock prices. We prove that if the bond's payoff function is
monotone in the external risk process, it can be priced by an
equilibrium approach. The equilibrium market price of climate risk
and the equilibrium price process are characterized as solution of
nonlinear backward stochastic differential equations. Transferring
the BSDEs into PDEs, we represent the bond prices as smooth functions
of the underlying risk factors.
The talk is based on joint work with Matthias Muller.
 ADAM KOLKIEWICZ, Department of Statistics and Actuarial Science, University of
Waterloo, Waterloo, Ontario
Inverse FirstPassage Problem with Applications to Credit
Risk Modeling
[PDF] 
In the paper we consider the inverse firstpassage problem for a
Brownian motion, which arises in the context of calibration of some
models for equity and credit markets. The objective of the problem is
to determine a deterministic barrier such that the corresponding exit
time of the process has a given distribution. In contrast to the
classical exit time problem, the inverse problem is less studied and
even for the standard Brownian motion the existing results are not
fully satisfactory. In the paper we identify shortcomings of existing
techniques and present a method that addresses these issues.
 ALI LAZRAK, University of British Columbia
Noncommitment in continuous time
[PDF] 
This paper characterizes differentiable subgame perfect equilibria in
a continuous time intertemporal decision optimization problem with
nonconstant discounting. The equilibrium equation takes two
different forms, one of which is reminiscent of the classical
HamiltonJacobiBellman equation of optimal control, but with a
nonlocal term. We give a local existence result, and several
examples in the consumption saving problem. The analysis is then
applied to suggest that nonconstant discount rates generate an
indeterminacy of the steady state in the Ramsey growth model. Despite
its indeterminacy, the steady state level is robust to small
deviations from constant discount rates.
 CRISTIANE LEMIEUX, University of Calgary, 2500 University Drive NW, Calgary, AB
T2N 1N4
QuasiMonte Carlo Methods and Finance
[PDF] 
QuasiMonte Carlo methods use highlyuniform point sets rather than
random sampling in an attempt to improve upon the Monte Carlo method.
These methods have found applications in different areas, most notably
in finance. In this talk, I will introduce the general principles
underlying quasiMonte Carlo methods, and discuss different aspects of
the interplay between these methods and financial problems. I will
conclude with a brief overview of quasiMonte Carlo constructions
tailored to financial applications.
 ALEX MELNIKOV, University of Alberta, Department of Math. and Stat. Sciences, Edmonton, AB
Partial Hedging for Valuation of FinanceInsurance Contracts
[PDF] 
The talk is devoted to hedging methods developed in the modern
financial mathematics and their applications to equitylinked life
insurance. We study mixed financeinsurance contracts with fixed and
flexible guarantees conditioned by survival status of the insured. In
our setting, these instruments are based on two risky assets of a
diffusion or jumpdiffusion market. The first asset is responsible
for the maximal size of a future profit while the second, more
reliable, asset provides a flexible guarantee for the insured. The
insurance company is considered as a hedger of a maximum of these
assets conditioned by remaining lifetime of a client. The main
attention is paid to quantile hedging, which, together with
BlackScholes (fixed guarantee) and Margrabe (flexible guarantee)
formulae, creates effective actuarial analysis of such contracts.
Some connections and further developments in mortality modeling and
risk measures are discussed. Finally, we give numerical examples
based on financial indices to demonstrate how our results can be
applied to actuarial riskmanagement.
 LUIS SECO, University of Toronto, Department of Mathematics,
40 St. George Street, Toronto, Ontario M5S 3G3
Collateralized Fund Obligations
[PDF] 
Collateralized fund obligations provide an inexpensive way to finance
leveraged fund investments. They have been quite popular in the low
rate environment of the last few years. As credit derivatives, their
pricing poses interesting challenges which can be studied with
traditional methods. This talk will survey the product and will
present a number of considerations involved in their pricing and risk
exposures.
 ANATOLIY SWISHCHUK, University of Calgary, 2500 University Drive NW, Calgary, AB
T2N 1N4
Change of Time Method in Mathematical Finance
[PDF] 
In this talk, we show how the changeoftime method works for
different kind of models and problems arising in financial
mathematics. We study the following three models in mathematical
finance: geometrical Brownian motion model for stock price,
meanreverting model for commodity asset price and stochastic
volatility model (that follows CoxIngersollRoss process) for
Heston model of stock price. We apply the changeoftime method to
derive (yet one more) the wellknown BlackScholes formula for
European call option and to derive the explicit option pricing formula
for European call option on meanreverting model of commodity asset.
We also derive the explicit formulas for variance and volatility swaps
for financial markets with stochastic volatility following
CoxIngersollRoss (1985) process (Heston (1993) model of stock
price). Two numerical examples on the S&P60 Canada Index (January
1997February 2002) to price variance and volatility swaps for Heston
model and on AECO Natural Gas Index (1 May 199830 April 1999) to
price European call option for meanreverting asset model will be
presented.
 TONY WARE, University of Calgary, 2500 University Drive NW, Calgary, AB
T2N 1N4
Swing options with continuous exercise
[PDF] 
Swing options with continuous exercise can be used to model
gasstorage contacts; options with discrete exercise opportunities and
American options, can be viewed as limiting cases. The value of such
an option may be found by solving a semilinear PDE, and I will discuss
the numerical solution of such equations.

