
MINISYMPOSIA 
Numerical
PDE Methods for Mathematical Finance
Organized by
Ken Jackson, Computer Science Department, University of Toronto

Many
problems in mathematical finance can be modeled by PDEs, most of which
cannot be solved in closed form. Therefore, numerical methods are needed
to solve them. As mathematical finance has developed over the past 30
or so years, the PDE models have become more sophisticated and the associated
numerical problems more challenging. This minisymposium consists of
a sample of four talks in this area.
SPEAKERS 

Peter
Forsyth
Cheriton School of Computer Science, University of Waterloo 
Methods
for Pricing American Options Under Regime Switching
We analyze a number of techniques for pricing American options under
a regime switching stochastic process. The techniques analyzed include
both explicit and implicit discretizations with the focus being
on methods which are unconditionally stable. In the case of implicit
methods we also compare a number of iterative procedures for solving
the associated nonlinear algebraic equations. Numerical tests indicate
that a fixed point policy iteration, coupled with a direct control
formulation, is a reliable general purpose method. Finally we remark
that we formulate the American problem as an abstract optimal control
problem, hence our results are applicable to more general problems
as well. 
Stephen
Tse
Cheriton School of Computer Science, University of Waterloo

Comparison
between the Mean Variance optimal and the Mean Quadratic Variation
optimal trading strategies
We compare optimal liquidation policies in continuous time
in the presence of trading impact by numerical solutions of Hamilton
Jacobi Bellman (HJB) partial differential equations (PDE). We show
quantitatively that the meanquadraticvariation strategy can be
significantly suboptimal in terms of meanvariance efficiency and
that the meanvariance strategy can be significantly suboptimal
in terms of meanquadraticvariation efficiency. Moreover, the meanquadraticvariation
strategy is on average more suboptimal than the meanvariance strategy,
in the above sense. In the semiLagrangian discretization used for
solving the HJB PDEs, we show that interpolating along the semiLagrangian
characteristics results in significant improvement in accuracy over
standard interpolation while still guaranteeing convergence to the
viscosity solution.

Christina
Christara
Computer Science Department, University of Toronto

PDE
modeling and pricing of Power Reverse Dual Currency swaps
Crosscurrency interest rate derivatives in general, and foreign
exchange (FX) interest rate hybrids in particular, are of great
practical importance. In particular, Power Reverse Dual Currency
(PRDC) swaps are one of the most widely traded and liquid FX interest
rate hybrids. These products are exposed to moves in both the spot
FX rate and the interest rates of the domestic and foreign currencies.
We discuss the modeling of PRDCs using two onefactor Gaussian models
for the two stochastic interest short rates, and a onefactor FX
skew model with a local volatility function for the spot FX rate.
The resulting timedependent PDE is threedimensional in space,
a fact that makes the pricing computationally challenging. We study
two numerical techniques for solving the PDE. The first uses Alternating
Direction Implicit (ADI) timestepping and the second a standard
CrankNicolson (CN) discretization in time, while both use centered
differences in space. The ADI technique requires solutions of tridiagonal
systems, while the CN discretization requires solution of blockbanded
systems. The latter are treated by GMRES with FFTbased preconditioning.
Numerical results indicate that the ADI method is faster in absolute
terms, while both methods are optimal or almost optimal in asymptotic
complexity.

Duy
Minh Dang
Computer Science Department, University of Toronto

Pricing
longdated foreign exchange interest rate hybrids via a PDE approach
with an efficient implementation on Graphics Processing Units
Power Reverse Dual Currency (PRDC) swaps are popular longdated
foreign exchange (FX) interest rate hybrids. A vanilla Power Reverse
Dual Currency (PRDC) swap, similar to a vanilla fixedforfloating
singlecurrency swap, involves an agreement between the issuer and
the investor, in which the issuer pays the investor a stream of
PRDC coupon amounts, and, in return, receives the investor's domestic
LIBOR payments. However, the PRDC coupons are linked to the spot
FX rate prevailing when the PRDC coupon rate is set. Variations
of PRDC swaps with exotic features, such as Bermudan cancelable,
or knockout, or FX Target Redemption (FXTARN), are much more popular
than ``vanilla'' PRDC swaps. The modeling of PRDC swaps using onefactor
Gaussian models for the domestic and foreign interest short rates,
and a onefactor skew model for the spot FX rate results in a timedependent
parabolic PDE in three space dimensions. While solution of the PDE
arising from vanilla PRDC swaps presents by itself a computational
challenge, the exotic features significantly increase the complexity
of the pricing. In this talk, we discuss the pricing via a PDE approach
of PRDC swaps with popular exotic features, namely Bermudan cancelable
and FXTARN. Our PDE pricing framework is based on partitioning
the pricing problem into several independent pricing subproblems
over each period of the swap's tenor structure, each of which requires
a solution of the modeldependent PDE. Finite differences and the
Alternating Direction Implicit (ADI) method are used for the spatial
and time discretizations, respectively, of this PDE. To handle the
increased computational requirements due to the exotic features,
we develop a Graphics Processing Unit (GPU) parallelization of the
pricing procedure on a multiGPU platform. Numerical results demonstrating
the efficiency and accuracy of the parallel numerical methods are
provided. 
