Mathematical / Computational Finance
Org: Ruppa K. Thulasiram (Manitoba)
- RAJ APPADOO, University of Manitoba
Lattice Valuation Model Using O(m,n)-Trapezoidal Type Fuzzy
In this paper we discuss an option valuation model using O(m,n)
trapezoidal type fuzzy numbers. We demonstrate how fuzzy algebra
assisted by O(m,n) trapezoidal type fuzzy numbers can be
successfully applied to the discrete Cox-Ross-Rubinstein (1979)
binomial risk neutral option pricing model. Through option pricing
theory and fuzzy set theory we get results that allow us to
effectively price option in a fuzzy environment. The approach
developed in the paper is illustrated with the help of a numerical
Joint work with R. Thulasiram, C. R. Bector and S. K. Bhatt.
- JOE CAMPOLIETI, Wilfrid Laurier
New State Dependent Diffusions and their Application to
In recent years a new methodology (called "diffusion canonical
transformation") was developed for generating new families of
analytically solvable one-dimensional Markovian diffusions with
multiple adjustable model parameters. In this talk we discuss some of
the most recent, and ongoing, developments and applications of such
new diffusion models to option pricing. We show that certain
subfamilies of these processes properly describe forward (discounted)
asset prices as martingales under a risk-neutral measure.
Risk-neutral transition densities, first-passage time densities, and
option prices for standard Europeans, barrier options and lookback
options are derived in analytically closed-form. The newly derived
analytical formulas for these families are supersets of previously
derived formulas given for other popular models such as the CEV and
others. Implied volatility surfaces for these models, however,
exhibit a wide range of pronounced smiles and skews of the type
observed in the option markets. These diffusion models are also
applicable to pricing more generally path-dependent as well as
multi-asset options. We conclude by discussing some numerical
implementations for pricing Asian and Bermudan options via path
- TAHIR CHOULLI, University of Alberta
Two optimal pricing measures: A comparative study
In this talk, we will discuss two optimal martingale measures: The
minimal Hellinger martingale measure of order q (MHM(q) measure
hereafter) and the q-optimal martingale measure, for any q and any
semimartingale market model.
We will explain how these two optimal martingale measures are
obtained. Then we conduct our comparison for these two martingale
measures in two ways. The first comparison deals with comparing
`physically' the two optimal martingale measures. Precisely, in this
case, we show that the two optimal martingale measures coincide in the
case of Levy market models with known horizon, while they differ in
general. We also provide necessary and sufficient conditions for the
two optimal martingale measures to coincide in a general framework.
The second comparison addresses the question whether there exists a
model for which the MHM(q) measure of the underlying model is the
q-optimal martingale measure, and vice-versa. This last comparison
has a great impact in analysing uncertainty models. Finally, from the
very practical point of view, we analyze the MHM(q) measure for the
case of discrete-time market model.
- HAOHAN HUANG, York University, 4700 Keele Street, Toronto, ON
Estimating Value at Risk with Non-negative Matrix
In this talk I will introduce the Non-negative Matrix Factorization
(NMF) technique to estimate Value at Risk (VaR). VaR is a very
important methodology for measuring portfolio risk in finance.
Normally when calculating VaR, we need the correlations between each
product in our portfolio. But it's very time-consuming to get the
correlations when the number of products is large, besides, the
correlation cannot be very precise. The Non-negative Matrix
Factorization method has previously been shown to be a useful
decomposition for multivariate data. It is developed now to find
parts-based, linear representations of non-negative data. Then I use
this tool to deal with the data of the portfolio and gladly find I
can skip the step of calculating correlations when estimating VaR.
- ALI LAZRAK, University of British Columbia
Perfect competition among generations: growth theory under
This paper characterizes differentiable subgame perfect equilibria
in a continuous time intertemporal decision optimization problem
with non-constant discounting. It is assumed that successive
decision-makers are in a situation of perfect competition, so that
each of them takes the strategy of future generations as given. We
show that equilibrium strategies are characterized by a value
function, which must satisfy a certain equation. The equilibrium
equation takes two different forms, one of which is reminiscent of the
classical Hamilton-Jacobi-Bellman equation of optimal control, but
with a non-local term. We give a local existence result, and several
examples of equilibria, and we conclude that non constant 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.
Co-author: Ivar Ekeland.
- ALEX PASEKA, University of Manitoba
Information in Option Prices and the Underlying Asset
We jointly estimate stochastic volatility (Heston) model parameters in
both the physical and equivalent martingale measures, by exploiting
all the information in a broad cross-section of option prices along
with the dynamics of the underlying asset, using the exact
probabilistic framework of the model. To this end, we derive the
necessary joint transition density to draw from the latent volatility
process conditioned on the observed returns. We answer questions
along two dimensions. First, no study to date has used all the
information in the exact transition densities from the stochastic
volatility (Heston) model. Second, in the context of efficient
estimation, how much of the mispricing may be attributable to
parameter and state variable uncertainty? Finally, our metric for
assessing the fit of the model is the predictive posterior on the
implied volatility smile. Using this metric we assess directly the
model's ability to capture the empirical smile, which is an important
criterion in evaluating an option pricing model.
For the deep in-the-money, short-term call, parameter and state
variable uncertainty give rise to a 90%ile band in the predictive
posterior density of the implied volatility of 0.56%. But the data
are on average 3.6% away from the predictive posterior density mean.
By contrast a slightly in-the-money, intermediate term option has
average pricing errors relative to the posterior mean of 0.26%, and
parameter and state variable uncertainty imply that the predictive
posterior density's 90%ile band width is 0.58% (suggesting an
excellent model fit).
- LUIS SECO, University of Toronto, Department of Mathematics, 40 St. George Street, Toronto M5S 3G3
Modeling stochastic correlation in financial markets
This talk will review some old and new models of stochastic
correlation, with applications to the pricing and risk sensitivities
of some correlation-sensitivie financial instruments, such as spread
options and CDO's.
- LIQUN WANG, University of Manitoba, Department of Statistics, Winnipeg,
Manitoba, R3T 2N2
First Hitting Time Distribution for Diffusion Processes and
Time-Dependent Double Barriers
The first hitting time distributions (or boundary crossing
probabilities) play an important role in pricing barrier options and
other financial derivatives. In particular, evaluation of
time-dependent barrier options leads to the first hitting time
distribution for nonlinear boundaries. Because no explicit formula
exists in such situations, the entailed numerical evaluation is a
difficult task. The computation of boundary crossing probabilities
arises also in many other scientific fields, e.g., in biology,
epidemiology, econometrics and statistics.
In this talk, I will present explicit formulae for the probabilities
that a Brownian motion crosses piecewise linear boundaries. Then I
will use this formula to approximate the crossing probabilities for
general nonlinear boundaries. This technique is further extended to a
class of more general diffusion processes, including
Ornstein-Uhlenbeck processes and geometric Brownian motion with
time-dependent drift. The numerical computation is done using Monte
Carlo integration which is straightforward and easy to implement.
Some numerical examples will be presented to illustrate this technique.
- XIAONAN WU, Memorial University of Newfoundland,St John's, NL
Anomaly Detection in Financial Fraud Data
Building effective fraud detectors can significantly reduce billions
of losses of financial institutions due to fraudulent activities. One
popular anomaly detection method is to check changes in users'
behavior that might hint at suspicious activity. Boundary that
distinguishes normal and abnormal space determines detection accuracy.
In this talk, we will discuss how to accurately define this boundary
inspired by an interesting natural phenomenon: species in nature
undergo intensive competitions and interactions with environment, and
finally come into balance. In our approach, we define two sets of
rules: positive rules for normal space and negative rules for abnormal
space, and refer each of them as a species, which evolves
independently. Meanwhile they control each other's evolutionary
environment, such as selection pressure and crossover/mutation rates.
During the evolutionary process, positive rules and negative rules
will move towards the boundary. Any rule which is over the boundary
will be punished. In the end, both of the rule sets will converge
around the boundary. Hence the boundary between normal and abnormal
space can be accurately defined, thus helps to avoid the
over-generalization problem which exists in methods that consider
normal space only and "hole" (insufficient detectors) problem
which is caused by considering abnormal space only. We have applied
this method to real financial data to detect fraud. The preliminary
results indicate that this approach provides good accuracy and is able
to scan financial databases quickly.