Hélyette Geman: Stochastic Time Changes, Lévy Processes and Option Pricing
The classical Black-Scholes model assumes constant volatility and continuity of
trajectories. From classical observations of the financial markets however, it is clear that
these assumptions are not satisfied. Fat tails, volatility smile, jumps in skewness, in option
prices are examples of deviations of market prices from the Black-Scholes assumptions.
A natural and parsimonious way of capturing the non constancy of volatility is to
introduce a stochastic clock where time accelerates during periods of high volatility. From
a financial perspective, the number of trades is exhibited to be the quantity driving the
transaction time. From a mathematical perspective, it is shown that under no arbitrage,
asset prices are time-changed Brownian motions. This representation allows to make a
further case for jump processes to show the drawbacks of the Merton (1976) jump
diffusion model, and to generate pure jump Lévy processes which include the CGMY
(Carr - Geman - Madan - Yor) model. Stochastic volatility in the CGMY will finally be
introduced (under the form of a time change) to allow for an excellent fit of the option
volatility surface across strikes and maturities, using the characteristic function of the
process and the fast Fourier transform in strike or the option prices.
Lastly, the topic of incomplete markets will be discussed and the paradigm of acceptable
risk introduced; electricity markets will illustrate this
 Hélyette Geman, Pure Jump Lévy Processes for Asset
Price Modelling, published in the Journal of Banking and Finance 26 (7), 1297-1316 (2002)
 Peter Carr,
Dilip B. Madan,
Marc Yor, Stochastic Volatility for Lévy
Paul Glasserman: Monte Carlo Methods in Risk Management
These lectures will review some recent developments in the
application of Monte Carlo methods in finance.
Credit Risk: Here the challenge lies in accurate estimation
of small probabilities of large losses (and associated risk measures).
Importance sampling (IS) is a natural candidate for improving precision,
but the application of IS is complicated by the types of dependence
structures (e.g., normal copula) typically used in factor models
of credit risk. We present two-part IS methods that change the
distribution of the factors and increase default probabilities
conditional on the factors in order to produce more scenarios with
large losses. The methods are supported through asymptotics as
both the portfolio size and loss threshold increase. We also discuss
interactions between Monte Carlo and other computational methods.
Market Risk: The problem of estimating the value at risk of a
large portfolio over a relatively short horizon can also be
addressed using importance sampling, in this case based on a
delta-gamma (or other quadratic) approximation to portfolio value.
Extensions to heavy-tailed distributions are of special importance
but present particular challenges to traditional IS strategies.
We present methods to address these problems and discuss their
asymptotic optimality properties. We also discuss the combination
of IS with other techniques.
American Options: The pricing of American options by simulation
is made difficult by the embedded optimal stopping problem.
We give 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. We also discuss recent results on the
convergence of these methods.
 Paul Glasserman, Monte Carlo Methods in Financial
Engineering, Applications of Mathematics, Vol. 53, Springer, 2003
 P. Glasserman, Tail Approximations for Portfolio Credit Risk
 P. Glasserman and Bin Yu,
Number of Paths Versus Number of Basis Functions in American Option
 P.Glasserman and Bin Yu, Simulation for American Options: Regression Now or Regression
Later?, in Monte Carlo and Quasi-Monte Carlo Methods 2002,
(H. Niederreiter, ed.), Springer, Berlin, 213-226.
 P. Glasserman and Jingyi Li,
Importance Sampling for a Mixed Poisson Model of Portfolio Credit Risk,
to appear in the Proceedings of the Winter Simulation Conference 2003
 Paul Glasserman, Philip Heidelberger and Perwez Shahabuddin, Portfolio value-at-risk with heavy-tailed
Special invited lectures
On Dynamic Models for Portfolio Credit Risk and Credit Contagion
It is by now well known that the performance of models for
portfolio credit risk is
very sensitive to the modelling of dependence between defaults of different
In this talk we will be concerned with dynamic models for portfolios of
defaults. After a survey of existing approaches, we concentrate on models
contagion, i.e. models where the default of one company has a direct impact
default intensity of other firms. We introduce a Markovian model and discuss
various types of interaction. Finally we present limit results for large
a homogeneous model with mean-field interaction and analyze the impact of
contagion on the portfolio loss distribution.
 Slides of the lecture
 Rüdiger Frey and Jochen Backhaus, Interacting Defaults and Counterparty Risk:
a Markovian Approach
Wolfgang Runggaldier: Estimation via stochastic filtering in
financial market models
When specifying a financial market model one needs to specify also the model coefficients.
The latter may be only partially known and so, in order to solve problems
related to financial markets, it becomes important to exploit all the information
coming from the market itself in order to update the knowledge of the not fully
known quantities and this is where stochastic filtering becomes useful.
The information from the market is not only given by the prices of the underlying primary
assets, but also by the prices of the liquidly traded derivatives. A major problem in
this context is that derivative prices are given as expectations under a martingale
measure, while the actual observations take place under the real world probability
measure. We shall discuss various ways to deal with this problem.
 Slides of the lecture
 Wolfgang J. Runggaldier, Estimation via stochastic filtering in financial market
Uwe Wystup: FX exotics and the relevance of computational methods in their pricing
and risk management
Starting with an overview of the current FX derivatives industry we
take a look at a few examples where computational methods are crucial to run
the daily business. The examples will include installment contracts,
accumulative forward contracts and the efficient computation of option price
 The market price of one-touch options in foreign
exchange markets, appeared in Derivatives Week
 Jürgen Hakala, Ghislain Perissé and Tino Senge, The pricing of second generation exotics,
chapter 7 of Jürgen Hakala and Uwe Wystup, Foreign Exchange Risk
 Hatem Ben-Ameur, Michèle Breton and Pascal François, Pricing Installment Options with an Application
to ASX Installment Warrants
 Uwe Wystup, How the Greeks would have hedged correleation risk of foreign
exchange option, chapter 14 of Jürgen Hakala and Uwe Wystup, Foreign Exchange
of the lecture
Structural Breaks in the Time Change
of the S&P 500 index
Recent results indicate that financial processes constitute, in financial time, a Brownian motion.
The time change, which maps physical time points to financial time points, is determined by the quadratic variation
in the case of a continuous local martingale.
We expose the time change of the S&P 500 future index by using large amounts intraday data.
Analyzing this time change we observe several
structural breaks. Periods at which the speed of
financial time is high alternate with periods at which the speed of financial time is low.
Since volatility is intimately related to the time change this indicates that
that volatility switches from one regime, at which the volatility is stochastic but constant at its first moment,
to another. We shall approximate the time change by a piecewise linear function and discuss the location
of the breakpoints.
 Remco T. Peters, Shocks in the S&P500
Antoine van der Ploeg: A State Space Approach to the Estimation of Multi-Factor
Affine Stochastic Volatility Option Pricing Models
We propose a class of stochastic volatility (SV) option pricing models that is more flexible than
the more conventional models in different ways. We assume the conditional variance of the
stock returns to be driven by an affine function of an arbitrary number of latent factors, which
follow mean-reverting Markov diffusions. This set-up, for which we got the inspiration from the
literature on the term structure of interest rates, allows us to empirically investigate if
volatilities are driven by more than one factor. We derive a call pricing formula for this class.
Next, we propose a method to estimate the parameters of such models based on the Kalman
filter and smoother, exploiting both the time series and cross-section information inherent in
the options and the underlying simultaneously. We argue that this method may be considered
an attractive alternative to the efficient method of moments (EMM). We use data on the
FTSE100 index to illustrate the method. We provide promising estimation results for a 1-factor
model in which the factor follows an Ornstein-Uhlenbeck process. The results indicate that the
method seems to work well. Diagnostic checks show evidence of there being more than one
factor that drives the volatility, indicate the existence of level-dependent volatility, and provide
an incentive to consider realized volatility in our empirical analysis. Further research is clearly
needed and is being worked on.
 Antoine P.C. van der Ploega, H. Peter Boswijk and Frank de Jong, A State Space Approach to the Estimation of Multi-Factor
Affine Stochastic Volatility Option Pricing Models
Raoul Pietersz: Projection Iteration Calibration of the Libor BGM Model
This paper tackles the problem of calibrating the Libor BGM model to cap
volatility and interest rate correlation by applying the projection
iteration algorithm of Dykstra (1983) and Han (1988). The latter
algorithm efficiently finds the nearest point over the intersection of
closed convex sets by successive projections onto the individual sets.
We show that the BGM cap volatility calibration is of this form when
minimizing the distance of the model future cap volatility curves to
today's curve. The low rank approximation of a correlation matrix
however may not be cast in this form because the set of n by n matrices
that have rank d < n is not convex. Nonetheless applying the algorithm
surprisingly leads to a deficient version of the Lagrange multiplier
algorithm of Wu (2002). A slight modification then leads to a converging
algorithm that is easier to implement than Wu's. The methods are
compared in terms of computational efficiency. (Joint work with Igor Grubisic, Universiteit
 Igor Grubisic and Raoul Pietersz, Efficient rank reduction of correlation
Alessandro Sbuelz: Equilibrium Asset Pricing with Time-Varying Pessimism
We will study the pricing effects of pessimism, as modelled by Knightian model uncertainty
aversion for a neighborhood of indistinguishable model specifications that
are constrained in their relative entropy from a given reference model. We will fully
characterise the equilibrium of a pessimistic, representative agent, exchange economy
with intertemporal consumption, stochastic opportunity set, and a relative entropy
constraint that can depend on the state of the economy. We will find that Knightian
pessimism yields outstanding First Order Risk Aversion (FORA) excess equity
returns. On the other hand, equity dynamics are virtually unaffected. Precisely,
riskfree rates and equity premia feel a direct impact of pessimism whereas equity
returns and worst case equity premia feel an indirect impact only, which completely
disappears with log utility. We will compute and calibrate explicit equilibrium examples
of a pessimistic economy whith an amount of pessimism associated to an
11% upper probability of making confusion between the worst case model and the
reference model. Relative entropy is the key in fixing such a realistic amount of pessimism
in our calibrations. Even for log utility, we will find that such small amount
of pessimism generates some 55 basis points more of unconditional equity premium
by pushing riskfree rates down. Thus, we will conclude that Knightian pessimism
provides an economically and observationally different description of excess equity returns.
Its good theoretical and empirical properties could contribute to solve some of
the long-standing macro finance puzzles. (Joint work with Fabio Trojani, University
of Southern Switzerland.)
 Alessandro Sbuelz and Fabio Trojani, Equilibrium Asset Pricing with Time-Varying Pessimism
 Slides of the lecture