 |
|
|
|
|
||||
|
|
|
CIM PROGRAMS AND ACTIVITIES |
|
||||
| January 8, 2009 |
|
|||||||
|
The Actuarial Science and Mathematical Finance research group meets on a regular basis to discuss various problems and methods that arise in Finance and Actuarial Science. These informal meetings are held at the Fields Institute for Mathematical Sciences and are open to the public. Talks range from original research to reviews of classical papers and overviews of new and interesting mathematical and statistical techniques/frameworks that arise in the context of Finance and Actuarial Science. Meetings are normally held on Wednesdays from 2pm to 3:30pm, but check calendar for exceptions. If you are interested in presenting in this series please contact the seminar organizer: Prof. Sebastian Jaimungal (sebastian [dot] jaimungal [at] utoronto [dot] ca). Upcoming SeminarsWednesday, January 14, 2009 Adam Metzler, Department of Applied Mathematics, University
of Western Ontario In multiname extensions of the seminal Black-Cox model, dependence
between corporate defaults is typically introduced by correlating
the Brownian motions driving firm values. Despite its significant
intuitive appeal, such a framework is simply not capable of describing
market data. In this talk we investigate an alternative framework,
in which dependence is introduced via stochastic trend and volatility
in obligors’ credit qualities. We find that several specifications
of the framework are capable of describing market data for synthetic
CDO tranches, and compare calibrated parameters from both 2006 and
2008. Wednesday, January 21, 2009 Alexey Kuznetsov, Department of Mathematics and Statistics,
York University In this talk we will discuss some recent work on computing distributions
of various functionals of a Levy process. First, we will present
a method for computing the joint density of the first passage time
and the overshoot. This method is based on a numerical scheme for
solving Wiener-Hopf integral equations coupled with the local information,
provided by backward Kolmogorov equation. Second, we will discuss
some classical results on Wiener-Hopf factorization method and its
numerical implementation for a class of processes with phase-type
jumps. Finally, we will introduce a new class of Levy processes,
which is qualitatively similar to CGMY family, but for which the
Wiener-Hopf factors can be recovered almost explicitly (and very
efficiently from the computational point of view). We will also
present numerical results, possible applications in Mathematical
Finance, and discuss some future directions for research.
Past SeminarsWednesday, November 5, 2008 2:00 pm We consider the application of forward-backward stochastic differential equations (FBSDEs) to the problem of pricing and hedging various term structure derivatives. The underlying model assumed for the factors of the economy is a multi-factor affine diffusion. We consider affine term structure models (ATSMs) where the short-rate model is an affine function of the factors process and affine price models (APMs) where the price a risky asset is an exponential affine function of the factors process and the dividend yield is an affine function of the factors. Characterizing the underlying factor dynamics and derivative prices as FBSDEs allows for analytic solutions in certain cases and for the implementation of simulation-based numerical methods for solving FBSDEs. Wednesday, October 22, 2008 2:00 pm Angelo Valov, Department of Statistic, University of Toronto Some of the main tools in attacking the First Passage Time (FPT) problem for Brownian motion are integral equations of Voltera or Fredholm type. In this talk I will discuss a martingale method to construct such equations, generalize existing Voltera equations of the first kind and provide a simple alternative derivation of some known results. Furthermore I will discuss conditions for existence of a unique continuous solution for a subclass of Voltera equations. Finally I will present a partial solution to both the FPT problem and the corresponding inverse problem by introducing a random shift in the Brownian path. Thursday, September 25, 2008 - 2:00 P.M.Sidney Smith Hall SS2098 (enter through room SS2096). Matt Davison, Canada Research Chair in Quantitative Finance
Associate Professor of Applied Mathematics and of Statistical &
Actuarial Sciences, The University of Western Ontario. Energy Markets are a frontier areas of Mathematical Finance. They
differ from traditional financial mathematics in a number of ways.
First, both the spot price processes they engender and the financial
derivatives written on these prices tend toward complication. Second,
since energy assets are primarily consumption assets, the role of
supply demand balance and the physical realities of energy infrastructure
play a May 21, 2008 The expected overshoot in the case of normal variables with positive
mean is studied, and simpler self-contained derivations of the known
results are given. We also give new series expansions with better
convergence properties. Applications in finance are found in option
pricing, where overshoot corrections have been used in the pricing
of discrete barrier options. One way of analyzing insurance risk models is by making use of
the existing connections with stochastic fluid flows. Matrix-analytic
methods constitute a useful approach to the study of such fluid
flow models. In the present talk we illustrate the derivation of
several first passage probabilities whose numerical calculation
is very tractable, based on the structure and the probabilistic
meaning of certain matrices describing these fluid models. In the
end, we enumerate several classes of risk processes that can be
analyzed using these probabilistic tools. If realized volatility is a nonrandom constant, then of course the Black-Scholes implied volatility equals that constant realized volatility. If realized volatility is random, then how does it relate to implied volatility? We answer this question with respect to several notions of implied volatility -- the Black-Scholes definition, and two model-free definitions. We start by assuming only the positivity and continuity of the underlying price paths. Based on joint work with Peter Carr. In the first part of this talk I will present an asymptotic expansion
for the indifference price of equity-linked insurance contracts
in when the underlying financial asset follows a 2-factor stochastic
volatility model with fast mean reversion. For the second part of
the talk, I consider path-dependent contracts under stochastic interest
rates, obtain optimal investment strategies using stocks and bonds,
and present integral representations for the price of contracts
that depend exclusively on the paths of interest rates. Traditional structural models assume that firm value is a tradable security and proceed to value defaultable bonds as European or Barrier options on firm value. We introduce a model in which default is driven by a visible (but not tradable) credit worthiness index (CWI) that is correlated to the firm's equity value. Default occurs when the CWI falls below a critical level at which time equity drops to zero. Given the incomplete nature of this market setting, we adopt stochastic optimal control methods through utility indifference to extract the implied bond values and CDS spreads. [ joint work with Georg Sigloch ] In this talk, I will discuss the pricing problem for the European Asian options in jump diffusion models. Following the method I used to solve the problem for American options, a sequence of functions are also constructed to approximate the price of Asian options. However, because the pay-off functions are not necessarily bounded, new methods are introduced to prove the regularity of functions in this sequence. As a result, this sequence of functions converge unformly and exponentially fast to the price of Asian option on compact sets. This provides us a fast numerical algorithm. At the end of this talk, I will present the numerical performance of this algorithm for Merton's model and Kou's model. Joint work with Hao Xing. A dynamic asset-liability management model for defined-benefit
pension plans is developed. The plan sponsor exhibits features of
loss aversion and tolerance for limited shortfalls in assets under
management relative to the liability due. The optimal contribution
policy, the optimal dividend policy and the associated asset allocation
rule are derived and analyzed. Sound Asset-Liability Management
is shown to entail withdrawals as well as contributions from the
pension fund.
The talk begins with a qualitative description of CDO's and their usefulness in helping banks to shed the default risk of a loan portfolio. Then the iTraxx and CDO markets for CDO's are described. For these markets, there are a large number of market prices for CDO contracts of different maturities and different tranches established for a given underlying portfolio on a given day. The problem of calibrating a model to this large number of market prices has been one of the central problems of CDO research, and the loss surface approach to calibration is described. The impact of calibration across maturities on the determination
of the timing of defaults is discussed, as is the impact of the
timing of defaults on the marking to market of CDO contracts. In
so far as time permits, an introduction to the extension of the
loss surface model to a dynamic model, capable of being calibrated
to dynamics-sensitive contracts such as options on CDO's and leveraged
super-senior tranches, will be given. I will first review recent models of stochastic mortality and the associated problems in pricing mortality contingent claims under stochastic mortality age structures. The focus of my talk will then be on capturing the internal population-level cross-hedge between components of an insurer's portfolio, especially between life annuities and life insurance. I will derive and compare several linear mechanisms which value claims under various martingale measures, and then pass to exhaustive analysis of the exponential premium principle which is the representative nonlinear pricing rule in this framework. The results will be illustrated with a couple of numerical examples that show the relative importance of model parameters. Based on joint work with Erhan Bayraktar and Jenny Young (U of
Michigan). We propose a model for stock price dynamics that explicitly incorporates
(random) waiting times, also known as duration, and show how option
prices are calculated. We use ultra-high frequency data for blue-chip
companies to justify a particular choice of waiting time or duration
distribution and then calibrate risk-neutral parameters from options
data. We also show that implied volatilities may be explained by
the presence of duration between trades. We consider the model of European stock with jumps. A partial integro differential equation, which related the price of a calendar spread to the prices of butterfly spreads, is derived. The functions describing the evolution of the process are also given. The evolution functions are the forward local variance rate and forward local default arrival rate. We specialize the case where the only jump which can occur reduces the underlying stock price by a fixed fraction of its pre-default value. In particular using a few calendar dates, we derive closed form expressions for both the local variance and the local default arrival rate. [ This is a review of the article by Peter Carr and Alireza Javaheri
] Option pricing based on GARCH models is typically obtained under the assumption that the random innovations are standard normal (normal GARCH models). However, these models fail to capture the skewness and the leptokurtosis observed in financial data, so a number of various other distributions have been proposed. Since under GARCH models the markets are incomplete, there are an infinite number of risk neutral measures for pricing contingent claims. The impact of the choice of an appropriate martingale measure on option pricing has yet to be addressed in these setups. The present work investigates the applicability of some well-known risk neutral measures for various GARCH models. Since only a few papers have studied the pricing performance of
non-normal driving noise, we propose a new semiparametric GARCH
option pricing model. Our approach is to compute option prices based
on a non-parametric density estimator for the unknown distribution
of the innovations based on standardized residuals. An empirical
study regarding European Call option valuation on S&P500 Index
shows our semiparametric model outperforms the normal GARCH option
pricing models We consider the jump-diffusion that is obtained if an independent
Wiener process is added to the surplus process of classical ruin
theory. In this model, we examine the expected discounted value
of a penalty at ruin. It can be shown that the solution satisfies
a defective renewal equation which has probabilistic interpretation.
As an application, we determine the optimal exercise boundary for
a perpetual put option. This talk approaches optimal control problems for discrete-time
controlled Markov processes by representing the value of the problem
in a dual Lagrangian form. This approach is a completely novel way
to look any stochastic optimal control problem, independent of (but
complementing) the classical dynamic-programming/value-function
approach. The representation obtained opens up the possibility of
numerical methods based on Monte Carlo simulation which may be advantageous
in high-dimensional problems, or in problems with complicated constraints. The data from financial markets show that the correlation, which
is typically assumed to be constant, is a stationary stochastic
process. Very little has been published on stochastic correlation
models so far. In this talk, I will discuss the obstacles for considering
correlation as a stochastic process and illustrate how to price
options with stochastic correlations. This talk will provide an overview for the GARCH and Heston Model, including their mathematical formulation, stylized facts, and methods for model calibration. Many time series are affected by a hidden process. An interesting
example can be found in the financial markets which experience in
alternance periods of stress and calm; and accordingly period of
high and low volatility. When modelling the volatility of stock
returns it is sensible to take into consideration the above mentioned
hidden process. The goal of this presentation is to explain how
we can identify the hidden process which is responsible for the
fluctuation of volatility between two states (high and low) by adopting
a Bayesian approach. We then use simulation to asses the efficiency
of our method. In this talk, I will discuss some analytical methods developed
in the past few years for insurance risk models. One of the advantages
for using such analytical methods is that they require little probabilistic
argument and hence can easily be understood by non-probabilists.
These methods also allow us to utilize results in analysis and differential
equations. Another is that it can some time handle more complex
risk models, especially the risk models with dividend policies,
for which probabilistic reasoning might be difficult. I will also
briefly discuss some potential applications in option pricing. The discussion will present and contrast affine and quadratic risk-free
rate term structure models. It will highlight the key differences
in the models both in terms of financial interpretation and mathematical
representation. Specific attention will be paid to the representative
Riccati equations. Issues related to parameter estimation and numerical
modelling will be discussed. Comments regarding extensions to corporate
bond modelling will also be provided. This presentation will draw
from two primary references, (Dai and Singleton, 2000) and (Ahn
et al, 2002), and results related to research requiring the use
of the key results of these papers. |
|
||||||
| ||||||||
