|
Probability
and Statistics in Complex Systems: Genomics, Networks, and Financial
Engineering, September 1, 2003 - June 30, 2004
Abstracts:
IMA
Workshop:
May
3-7, 2004
Photo
Gallery Material
from Talks
Pierre
Collin-Dufresne (University of California, Berkeley)
dufresne@andrew.cmu.edu
Identification
and Estimation of 'Maximal' Affine Term Structure Models: An
Application to Stochastic Volatility
Slides: pdf
Paper:
pdf
We
propose a canonical representation for affine term structure
models where the state vector is comprised of the first few
Taylor-series components of the yield curve and their quadratic
[4] (co-)variations. With this representation: (i) the state
variables have simple physical interpretations such as level,
slope and curvature, (ii) their dynamics remain affine and tractable,
(iii) the model is by construction `maximal' (i.e., it is the
most general model that is econometrically identifiable), and
(iv) model-insensitive estimates of the state vector process
implied from the term structure are readily available. (Furthermore,
this representation may be useful for identifying the state
variables in a squared-Gaussian framework where typically there
is no one-to-one mapping between observable yields and latent
state variables). We find that the `unrestricted' A1(3)
model of Dai and Singleton (2000) estimated by `inverting' the
yield curve for the state variables generates volatility estimates
that are negatively correlated with the time series
of volatility estimated using a standard GARCH approach. This
occurs because the `unrestricted' A1(3) model imposes
the restriction that the volatility state variable is simultaneously
a linear combination of yields (i.e., it impacts the cross-section
of yields), and the quadratic variation of the spot rate process
(i.e., it impacts the time-series of yields). We then investigate
the A1(3) model which exhibits `unspanned stochastic
volatility' (USV). This model predicts that the cross section
of bond prices is independent of the volatility state variable,
and hence breaks the tension between the time-series and cross-sectional
features of the term structure inherent in the unrestricted
model. We find that explicitly imposing the USV constraint on
affine models significantly improves the volatility estimates,
while maintaining a good fit cross-sectionally.
Ron
S. Dembo
(Founding Chairman, Algorithmics Incorporated) dembo@algorithmics.com
Risk
Measurement; Risk Architecture and the Bank of the Future
Slides: html
pdf
ps
ppt
Measuring
the risk of a large financial institution is a gargantuan task.
There have been major improvements in doing so over the past
few years. These have resulted in the ability of institutions
to take on more and more complexity, thereby keeping the risk
management treadmill alive and well. We discuss how risk is
actually measured, some major new accomplishments, such as real-time
risk based on simulation and highlight some of the interesting
research problems that are being addressed, such as real-time
bank-wide optimization.
Gregory
R. Duffee
(Haas School of Business, University of California-Berkeley)
duffee@haas.berkeley.edu
http://faculty.haas.berkeley.edu/duffee/
Estimation
of Dynamic Term Structure Models
Slides:
duffee_ima_present.pdf
Paper: duffee_stanton.pdf
This
talk discusses the finite sample properties of some of the standard
techniques used to estimate modern term structure models. For
sample sizes similar to those used in most empirical work, I
note three surprising conclusions. First, maximum likelihood
produces strongly biased parameter estimates. Second, despite
having the same asymptotic efficiency as maximum likelihood,
the small sample performance of Efficient Method of Moments
(a commonly used method for estimating complicated models) is
unacceptable even in the simplest term structure settings. Third,
the linearized Kalman filter is a tractable and reasonably accurate
estimation technique that I recommend in settings where maximum
likelihood is impractical.
Philip
H. Dybvig
(Washington University in Saint Louis) pdybvig@dybfin.wustl.edu
Mandatory
or Voluntary Retirement
Slides:
pdf
Saving
for retirement is a primary end purpose of many parts of the
financial sector, including pension plans, life insurance, and
indeed much of retail banking and brokerage. As a step towards
understanding these markets, we solve the lifetime consumption
and investment problem of a competitive agent who faces voluntary
or mandatory retirement. The model includes such realistic features
as stochastic age-dependent wage, age-dependent life-table mortality
and age-dependent preferences for working as well as a constraint
that prevents borrowing against future labor income. The tightly
approximated model is solved parametrically in the dual, in
closed form up to determination of some constants. The solution
uses the technique of Carr (for American Options) and Liu and
Loewenstein (for transaction costs) of making the nonstationary
problem into a sequence of stationary problems by approximating
a fixed horizon by a sequence of stationary random horizons.
Jean-Pierre
Fouque
(Department of Mathematics, North Carolina State University)
fouque@math.ncsu.edu
http://www.math.ncsu.edu/~fouque
Variance
Reduction for MC Methods to Evaluate Option Prices Under Multi-Factor
Stochastic Volatility Models
Slides:
pdf
We
present variance reduction methods for Monte Carlo simulations
to evaluate European and Asian options in the context of multiscale
stochastic volatility models. European option price approximations,
obtained from singular and regular perturbation analysis [J.P.
Fouque, G. Papanicolaou, R. Sircar and K. Solna: Multiscale
Stochastic Volatility Asymptotics, SIAM Journal on Multiscale
Modeling and Simulation {\bf 2(1)}, 2003], are used in important
sampling techniques, and their efficiencies are compared. Then
we investigate the problem of pricing arithmetic average Asian
options (AAOs) by Monte Carlo simulations. A two-step strategy
is proposed to reduce the variance where geometric average Asian
options (GAOs) are used as control variates. Due to the lack
of analytical formulas for GAOs, it is then necessary to consider
efficient Monte Carlo methods to estimate the unbiased means
of GAOs. The second step consists in deriving formulas for approximated
prices based on perturbation techniques, and in computing GAOs
by using important sampling. Numerical results illustrate the
efficiency of our method.
Joint
work with Chuan-Hsiang (Sean) Han.
Paul
Glasserman (Graduate School of Business, Columbia
University) pg20@columbia.edu
Monte Carlo Pricing of American Options: Overview
and New Results
Slides: pdf
Paper: pdf
An American option allows the holder to choose the time of exercise,
so valuing such an option entails solving an optimal stopping
problem. This "free boundary" problem presents a challenge for
Monte Carlo methods. The first part of this talk will be an
overview of methods developed in recent years to address this
problem. These methods apply weighted backward induction to
simulated paths, with weights defined through likelihood ratios,
through calibration, or implicitly through regression. The second
part of this talk analyzes conditions for convergence as both
the number of paths and number of basis functions for regression
grow. Using polynomials in the regressions, the number of paths
must grow exponentially with the number of basis functions to
assure convergence when applied to Brownian motion, faster when
applied to geometric Brownian motion. This analysis is based
on joint work with Bin Yu.
David
C. Heath (Department
of Mathematical Sciences, Center for Computational Finance,
Carnegie Mellon University) heath@red.math.cmu.edu
Efficient
Option Valuation Using (Non-Recombining) Trees
Paper: pdf
Joint
work with Stefano Herzel.
We
propose an algorithm for the discrete approximation of continuous
market price processes which uses trees instead of lattices.
We show that it is convergent when used to price both European
and American options and that it is more efficient, for some
models, than the usual recombining schemes.
Dmitry
Kramkov
(Center for Computational Finance, Carnegie Mellon University)
kramkov@andrew.cmu.edu
Risk-Tolerance
Wealth Processes and Sensitivity Analysis of Utility Based
Prices
Slides:
pdf
We
present the asymptotic analysis of the marginal utility based
prices of contingent claims in incomplete financial models
with respect to the number of these claims held in the portfolio.
Our main result states that such an approximation preserves
a number of important qualitative properties of the original
utility based prices if and only if there is a risk-tolerance
wealth process. The talk is based on a joint paper with Mihai
Sîrbu.
Joseph
Langsam (Morgan Stanley) Joseph.Langsam@morganstanley.com
Changing
Dynamics in the Securities Market
Slides: html pdf ps ppt
Mathematical
finance has evolved since the early days of Black-Scholes
with the assumptions of lognormal dynamics, constant interest
rates, and constant volatility. The growth of the derivatives
market and product innovation in new markets has forced "Wall
Street" to confront the complexities of far more generalized
dynamics. In this talk, I will review the dynamics, complexities,
and apparent conundrums in the modeling of a variety of financial
products including interest rate, foreign exchange, equity,
electricity, and credit products. Many questions will be asked
and many problems posed, but few answers will be given and
fewer solutions will be offered.
Michael
Ludkovski
(Department of Operations Research and Financial Engineering,
Princeton University) mludkovs@Princeton.EDU
Convenience
Yield Model with Partial Observations and Exponential Utility
(poster session)
Joint
work with Rene Carmona.
We
consider the problem of pricing claims for delivery of crude
oil or natural gas to a given location. We work with a three
factor model for the asset spot, the convenience yield and
the locational basis. The convenience yield is taken to be
unobserved and must be filtered. Our methodology is indifference
pricing with exponential utility. Assuming the basis is independent
from the spot, the partially observed stochastic control problem
can be expressed as a Feynman-Kac expectation. If the basis
is also independent from the convenience yield, the resulting
indifference price is trivial. Otherwise, we show how to numerically
compute the expectation using a Kalman or particle filter.
The basic model may be generalized to include nonlinear dynamics.
We finish by performing comparative statics and relating the
results to the full information setting.
Curt
Randall
( SciComp Inc.) randall@scicomp.com
Software
Synthesis - Pricing without Programming
Software
synthesis methods applied to the development of derivative
pricing and hedging models allow quantitative analysts, researchers,
and risk managers to rapidly generate models without programming.
The high level language developed for SciFinance will be used
to illustrate how a financial compiler generates source code.
This presentation will show how a 10 line specification for
a complex financial derivative can generate a ready to use
model to price the instrument often comprising thousands of
lines of source code. Examples will be shown over several
asset classes using both PDE and Monte Carlo methods. Attendees
will be given a web link to access papers and a free sample
pricing code that illustrates the use of software synthesis.
Mathias
Rousset (
Lab. Statistique et Probabilités, Université Paul Sabatier)
rousset@cict.fr
Sampling
Prescribed Distributions with Interacting Particle Systems
We
present a new class of interacting Metropolis models having
a prescribed limiting distribution. In contrast to traditional
Monte-Carlo methods, and when the population size is large,
the decay to equilibrium does not depend on the target distribution.
Some conclusive simulations are presented, focusing on diffusive
models and their implementation.
Louis
Scott (Morgan Stanley & Co.) Louis.Scott@morganstanley.com
Stochastic
Volatility and Jumps: Risk Management and Hedging Strategies
Slides:
LOS_Slides_SVJ_2004.pdf
The
talk covers stochastic volatility and jumps from a risk management
perspective. Various topics include gap risk in the underlying
prices, gap risk in the option implied volatilites, and some
analysis of the effects on the greeks (risk exposures). Examples
for stock index options are covered. Question: what should
you do with the skew in a stress test for stock prices? A
typical stress test is to decrease stock pirces by 20% and
increase the at-the-money implied volatilities by something
more than 20%.
1)
Overview
Volatility
Risk and Jump Risk (Gap Risk)
Discipline
and Risk Management
| The
Role of Models |
| |
Greeks:
delta, gamma, kappa/vega, theta, PV01 |
| |
Stress Tests and Scenario Analyses: revalue portfolios
for extreme, but plausible, market changes |
| |
Valuation
and Relative Value Trading |
| |
Additional
Tools for Understanding Market Dynamics and Risks |
| New
Markets: CDO's and Basket Default Swaps |
| |
Elements of gap risk |
| |
Stress
Tests: what happens to tranches when one or several names
go into financial distress? |
| |
Model
Correlation for Relative Value Trading |
2)
Overview of Stochastic Volatility/Jump Models
3)
Hedging Stochastic Volatility Risk
| Measuring
and Hedging Kappa/Vega Risk |
| |
Compute
kappa for all options and manage overall kappa exposure
|
| |
Compute
kappa from Black-Scholes model |
| |
Bucket
kappa exposure by type and term |
| Hedging
Stochastic Volatility |
| |
Treat
stochastic volatility as another random state variable
and compute its partial |
| |
Compute
vol exposure from stochastic volatility model |
4)
Managing Jump Risk
Need
to balance long/short positions in options
| Brute
Force: simulate jumps and revalue all positions, both
options and hedges |
| |
Compute
a 99% VaR Loss |
| |
May
need to use additional measures of tail risk |
| Run
scenarios which incorporate plausible jump risks, or the
ones that could be most damaging |
| |
Equity
markets down 30% and increase equity implied volatilities
|
| |
Recompute kappa or stochastic volatility risk under each
scenario |
5)
Several Examples for Stressing Equity Option Skew Curves
6)
Evaluating the Risks of Option Writing
| Net
short equity options, delta hedge using spot or futures |
| |
This
trading strategy receives the equity volatility risk premium. |
| |
Exposure is volatility risk and jump risk. |
| |
The risk premium reflects the fact that these risks occur
at bad times. |
| These
risks are highly correlated with negative returns on market
portfolios. |
| Sell
puts and delta hedge with long spot/futures positions
|
| |
Stocks drop suddenly and implied volatilities increase.
|
| |
Strategy
is a double loser under this scenario. |
| |
|
| Sell
calls and delta hedge with short spot/futures positions |
| |
Stock
prices drop: gain on short call positions, but lose on
hedges. |
| |
Delta on the calls decreases and the gain on the calls
is smaller because of option gamma. |
| |
The increase in implied volatility reduces the gain on
the call position even further. |
Long
residential mortgages, short an interest rate option
No
risk premium for FX volatility and interest rate volatility.
End
Steven
E. Shreve (Department of Mathematical Sciences,
Carnegie Mellon University) shreve@matt.math.cmu.edu
A Two-Person Game for Pricing Convertible Bonds
Slides:
pdf
ps
A
firm issues a convertible bond. At each subsequent time, the
bondholder must decide whether to keep the bond, thereby collecting
coupons. or to convert the bond for stock. The bondholder
wishes to choose a conversion strategy to maximize the bond
value. Subject to some restrictions, the bond can be called
by the issuing firm, which acts to maximize equity value and
thus minimize bond value. This creates a two-erosn game, and
we model the bond price as the value of this game. We show,
however, that under the assumption that dividends are paid
at a lower rate than the short-term interest rate, this game
reduces to one of two optimal stopping problems, and which
is the relevant problem can be determined apriori.
Because
the dividends paid depend on the value of equity, which in
turn depends on the value of the bond, the dynamics of the
firm value cannot be specified until the bond pricing problem
is solved. As a result, the optimal stopping problems which
must be solved lead to nonlinear partial differential equations.
These can be solved by a fixed-point method.
This
work is Mihai Sîrbu's Ph.D.
dissertation.
Mihai
Sîrbu (Department of Mathematical Sciences,
Carnegie Mellon University) msirbu@andrew.cmu.edu
http://www.math.cmu.edu/users/msirbu/
Perpetual
Convertible Bonds (poster
session)
Slides: pdf
In a similar model to the presentation of Steven E. Shreve
("A Two-Person Game for Pricing Convertible Bonds"), we consider
the problem of pricing a convertible bond that has no maturity
date. The problem reduces to solving a nonlinear ODE and to
a min-max argument. The Perpetual Convertible Bond represents
the asymptotic behavior for the finite maturity case. The
presentation is based on joint work with Igor
Pikovsky and Steven E. Shreve.
Srdjan
D. Stojanovic
(Department of Mathematical Sciences, University of Cincinnati)
srdjan@math.uc.edu
http://math.uc.edu/~srdjan/
Pricing
Options Under Stochastic Volatility: Complete Solution
Paper: pdf
We
have found, at least from the practical point of view, the
complete solution of the option pricing problem for underlying
securities obeying stochastic volatility price dynamics. In
particular, we have found the exact expression for the "market
price of volatility risk." The pricing problem is reduced
to solving an uncoupled system of a Monge-Ampère type PDE
and a Black-Scholes type PDE. The general problem of hedging
in such an environment is solved too. Results of computational
experiments will be presented as well.
Yong
Zeng
(Department of Mathematics and Statistics, University of Missouri
at Kansas City) zeng@mendota.umkc.edu
http://mendota.umkc.edu
A
General Equilibrium Model of the Term Structure of Interest
Rates Under Regime-Switching Risk
Slides:
pdf
ps
Paper: pdf
ps
This
work incorporates the systematic risk of regime shifts into
a general equilibrium model of the term structure of interest
rates. The model shows that there is a new source of time-variation
in bond term premiums in the presence of regime shifts. This
new component is a regime-switching risk premium that depends
on the covariations between discrete changes in marginal utility
and bond prices across different regimes. A closed-form solution
for the term structure of interest rates is obtained under
an affine model using log-linear approximation. The model
is estimated by Efficient Method of Moments. The regime-switching
risk is found to be statistically significant and mostly affect
the long-end of the yield curve. This is a joint work with
Shu Wu at the University of Kansas.
Yong
Zeng
(Department of Mathematics and Statistics, University of Missouri
at Kansas City) zeng@mendota.umkc.edu
http://mendota.umkc.edu
A
Class of Micro-Movement Models of Asset Price with Continuous-Time
Bayesian Inference via Filtering (poster
session)
A
rich class of micro-movement models that describe the transactional
price behavior is proposed. The model ties the sample characteristics
of micro-movement and macro-movement in a consistent manner.
An important feature of the model is that it can be transformed
to a filtering problem with counting process observations.
Consequently, the complete information of price and trading
time is captured and then is utilized in Bayesian inference
via filtering for the parameter estimation and model selection.
The evolution equations characterizing likelihoods, posteriors,
and Bayes factors are derived. Recursive algorithms are constructed
via the Markov chain approximation method to compute likelihoods,
posteriors, and Bayes factors. The consistencies (or robustness)
of the recursive algorithms are proven. Two micromovement
models are studied in detail. One is the model built on geometric
Brownian motion (GBM) and the other is on the GBM plus jumping
stochastic volatility. Simulation results show that the Bayes
estimates for time-invariant parameters are consistent, the
Bayes estimates for stochastic volatility capture the movement
of volatility, and the Bayes factor can effectively selects
the right model. Real-world applications to Microsoft transaction
data are also provided.
Photo
Gallery Material
from Talks
Probability
and Statistics in Complex Systems: Genomics, Networks, and Financial
Engineering, September 1, 2003 - June 30, 2004
|
