


Financial Mathematics / Mathématiques financières (Org: Joe Campolieti, David Vaughan, and/et Yongzeng Lai, Wilfrid Laurier University)
 JOE CAMPOLIETI, Wilfrid Laurier University
A Path Integral Formulation for Pricing Exotic Derivatives
under Alternative Price Processes

A path integral formulation for pricing generally exotic derivative
securities is presented. Some main features of the approach will be
discussed, such as its applicability for pricing virtually any type of
path dependent and early exercise option. In particular, we discuss
the use of new analytically tractable pricing kernels for valuing
exotics under multiparameter "hypergeometric" (i.e. alternative)
statedependent models as well as jump processes. Moreover, we show
how our approach presents a natural setting for exploiting various
Monte Carlo simulations as well as lattice methods using parallel
numerical algorithms for exotic option pricing.
 OLIVER CHEN, University of Toronto
Discrete Credit Barrier Models

A discrete lattice model appropriate for emulating credit rating
processesincluding the possibility of defaultis presented. This
includes a realworld process that matches historical migration
probabilities and historical default probabilities, and a riskneutral
process that matches market spread rates. The processes are
constructed such that the probability kernels can be expressed in
terms of the Hahn polynomials: a family of orthogonal polynomials of a
discrete variable, which enables numerically efficient calculation of
the kernels.
 ALEXEY KUZNETSOV, Department of Mathematics, University of Toronto
Unifying the Three Volatility Models

In my talk I will introduce a method for building analytically
tractable option pricing models which combine state dependent
volatility, stochastic volatility and jumps. The method of
eigenfunction expansion in orthogonal polynomials combined with
subordination is used to add stochastic volatility and jumps to
hypergeometric Brownian motions. I will also discuss how to construct
analytically tractable lattice approximations for these processes,
which is useful for pricing Americanstyle options. The numerical
results I will present show that such comprehensive unified models are
both easy to implement and are able to reflect the complexities of the
exotic option prices.
 GEORGE LAI, Wilfrid Laurier University
Simulating Financial Derivative Sensitivities by Malliavin
Calculus and QuasiMonte Carlo Methods

A lot of works have been done on Malliavin calculus since it was first
developed by Paul Malliavin in the 1970s. It is a tool for analysis
on Wiener space. In the past few years, Malliavin calculus was
applied in finance and showed its advantages over conventional
methods. But for many practitioners, and even for mathematicians,
Malliavin calculus is very theoretical and rather sophisticated. The
sensitivities (also called Greek letters) of financial derivatives
(such as options) are very important in financial risk management. I
will give an informal introduction to Malliavin calculus, and derive
formulas for financial derivative sensitivities. Monte Carlo (MC) and
quasiMonte Carlo (QMC) simulation methods are applied to obtain the
numerical values of sensitivities. The advantages of Malliavin
calculus combined with QMC method will also be demonstrated.
 STEPHAN LAWI, University of Toronto
Generating Functions for Stochastic Integrals

Generating functions for stochastic integrals have been known in
analytically closed form for just a handful of stochastic processes:
namely, the OrnsteinUhlenbeck, the CoxIngersollRoss (CIR) process
and the exponential of Brownian motion. In virtue of their analytical
tractability, these processes are extensively used in modeling
applications. In this paper, we construct broad extensions of these
process classes. We show how the known models fit into a
classification scheme for diffusion processes for which generating
functions for stochastic integrals and transition probability
densities can be evaluated as integrals of hypergeometric functions
against the spectral measure for certain selfadjoint operators. We
also extend this scheme to a class of finitestate Markov processes
related to hypergeometric polynomials in the discrete series of the
Askey classification tree.
 ROMAN MAKAROV, Wilfrid Laurier University, 75 University Avenue West,
Waterloo, Ontario, Canada
Efficient Monte Carlo Pricing of Path Dependent Options Using
High Performance Computing

We study path integral algorithms for pricing discretetime path
dependent options. For underlying asset price process a nonlinear
state dependent volatility model is considered. In particular, we use
as new family multiparameter volatility models, as the CEV model, for
which pricing kernels are related to an underlying squared Bessel
process.
Efficient Monte Carlo schemes for accurate evaluation of the integrals
are then presented. The algorithms combine variance reduction
techniques with distributed computing to significantly increase
efficiency. Moreover, we demonstrate how the Bessel bridge process can
be used for sampling discretetime paths. By rearranging variables in
path integral the increment of the process can be assigned to the
first sampling coordinates. Using subsequent splitting of the path
integral allows us to combine bridge sampling algorithm with either
adaptive Monte Carlo algorithms, or quasiMonte Carlo
techniques. Furthermore, the problem of using randomized quasiMonte
Carlo for sampling heavytailed distribution is investigated. The
technique which combines scrambling and random shift methods is
proposed.
 MARK REESOR, SHARCNET Research Chair in Financial Mathematics, Department
of Applied Mathematics, University of Western Ontario, London,
Ontario
Distortion, Relative Entropy Optimisation, and Copulas

Relative entropy optimisation and distortion are two methods commonly
used to reweight probability distributions. The connection between
these methods has recently been established and discussed in the
context of riskadjusted distributions and risk measurement (McLeish
and Reesor (2002)). Copulas provide a method of constructing
multivariate distributions from univariate marginals by imposing a
certain dependence structure. In many cases, this can be thought of
as a probability reweighting of the independent marginals to obtain a
desired joint distribution. Furthermore, for a given joint
distribution (copula) it can be desirable to reweight the
probabilities so that the distorted joint distribution satisfies
certain properties (Wang (1997), Genest and Rivest (2001), and Bennett
and Kennedy (2003)). Here, we examine the connection between copulas,
distortion, relative entropy optimisation, and risk.

