Talk Titles and Abstracts
Lorenzo Bergomi (Societe Generale)
Smile Dynamics IV
Static and dynamic properties of stochastic volatility models:
a structural connection
Stochastic volatility models do two jobs at once: they produce
a smile and generate a dynamics for implied volatilities. For
general stochastic volatility models, working at order one in
the volatility of volatility we establish the structural connection
between both aspects of a model.
The derivation calls for the introduction of the Skew Stickiness
Ratio, a dimensionless number that quantifies the amount by which
the ATM volatility moves when the spot moves, in units of the
ATM skew. We derive lower and higher bounds for the SSR and relate
the SSR to the decay of the ATM skew as a function of maturity,
which leads to a natural partition of stochastic volatility models
into two classes.
We then consider the historical joint dynamics of spot and implied
volatilites, assess whether our generic results hold in practice
and introduce the notion of realized skew.
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Tomasz Bielecki (IIT)
Hedging of counterparty risk
Counterparty risk is one of the fundamental forms of risks underlying
financial transactions. Thus, assessment and mitigation of this
risk is of primary importance to financial institutions. In this
talk we shall look at counterparty risk as the risk associated
with certain complex financial derivative, known as CCDS (contingent
CDS). We first discuss the issue of valuation of CCDS, where the
corresponding price process is called the CVA (credit valuation
adjustment). We shall then discuss the issue of hedging of the
CCDS. We will specify our general results to the case of so called
Markovian copula model. In this context, specific formulae for
self-financing hedging strategies will be given, when hedging
portfolio is created from so called rolling CDS contracts, and
perhaps some other contracts as well. This is a joint work with
Monique Jeanblanc and Stephane Crepey.
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Nicholas Bingham (Imperial College)
Multivariate elliptic processes
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Dorje C. Brody (Imperial College London)
Rational Term Structure Models with Geometric Lévy Martingales
In the positive interest models of Flesaker & Hughston (1996)
the nominal discount bond system is represented by a one-parameter
family of positive martingales. In the present paper we extend
the analysis to include a variety of distributions for the martingale
family, parameterised by a function ?(x) that determines the behaviour
of the market risk premium. These distributions include jump and
diffusion characteristics that generate various interesting properties
for discount bond returns. For example, one can generate skewness
and excess kurtosis in the discount bond returns by choosing the
martingale family to be given by (a) exponential gamma processes,
or (b) exponential variance-gamma processes. The models are “rational”
in the sense that the discount bond price process is given by
the ratio of a pair of sums of positive martingales. Our findings
lead to semi-analytical and Fourier-inversion style solutions
for the prices of European options on discount bonds, foreign
exchange rates, and foreign discount bonds. The paper is motivated
in part by the results of Filipovic, Tappe & Teichmann (2009),
who demonstrated that the term structure density approach of Brody
& Hughston (2001) admits a natural extension to general positive
term-structure models driven by a class of Lévy processes.
(Based on joint work with L.P. Hughston and E. Mackie.)
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Rama Cont (Paris VI-VII)
Functional Ito calculus and the pricing and hedging of path-dependent
derivatives
We develop a non-anticipative calculus for path-dependent functionals
of a semimartingale, using a notion of pathwise functional derivative
proposed by B. Dupire. The key ingredient is a functional extension
of the Ito formula, which is used to derive a martingale representation
formula for square integrable martingales. Regular functionals
of a semimartingale S which have the local martingale property
are characterized as solutions of a functional di erential equation,
for which a uniqueness result is given.
This result is used to derive a universal pricing equation for
the price of path-dependent derivatives with underlying asset
S: this pricing equation is shown to be a functional equation
whose coefficients involve the local characteristics of S. Using
these results we derive a general formula for the hedging strategy
of a path-dependent contingent
claim and present a numerical method for computing this hedging
strategy. By contrast with methods based on Malliavin calculus,
this representation is based on non-anticipative quantities which
may be computed pathwise and leads to simple simulation-based
estimators for computing hedging strategies for path-dependent
options.
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Mark Davis (Imperial College)
On SDEs with state-dependent jump measure
Virtually all textbook treatments of jump-diffusion SDEs assume
that the driving processes are a Brownian motion and an independent
homogeneous Poisson random measure. In many applications, for
example modelling of credit-risky securities, it seems that solution-dependence
of the compensator of the random measure should be allowed. The
reason for not including this goes back to the 1972 book of Gihman
and Skorohod, where it is shown how a problem with state-dependent
compensator can be 'reduced' to an equivalent one with homogeneous
random measure. There may however be good reasons for not doing
this transformation: for example the homogeneous random measure
may have infinite activity even if the jump rate in the original
model is a.s. finite. These questions are discussed and some general
results about existence and uniqueness with state-dependent jump
measure are given.
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Giuseppe Di Graziano (Deutsche Bank London)
Target Volatility Option Pricing
In this talk we shall present three approximation methods for
the pricing of Target Volatility Options (TVOs), a recent market
innovation in the field of volatility derivatives. TVOs allow
investors to take a joint view on the future price of a given
underlying (e.g. stocks, commodities, etc) and its realized volatility.
For example, a target volatility call pays at maturity the terminal
value of the underlying minus the strike, floored at zero, rescaled
by the ratio of a given Target Volatility (an arbitrary constant)
and the realized volatility of the underlying over the life of
the option. TVOs are typically used by investors and hedgers to
cheapen the price of an option or to leverage their exposure to
the underlying.
We present three approaches for the pricing of TVOs: a power series
expansion, a Laplace transform method and approximations based
on Bernstein polynomials. The three approximations have been tested
numerically and results are provided.
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Rudiger Frey (Leipzig)
Portfolio optimization under partial information with expert
opinions (joint with R. Wunderlich and A.Gabih)
We investigates optimal portfolio strategies for utility maximizing
investors in a market with partial information on the drift. The
drift is modelled by a continuous-time Markov chain with finitely
many states which is not directly observable. Information on the
drift is obtained from the observation of stock prices. Moreover,
and this is the novel feature of this paper, expert opinions are
included in the analysis. This additional information is modeled
by a marked point process with jump-size distribution depending
on the current state of the hidden Markov chain. We derive the
filtering equation for the return process and incorporate the
filter into the state variables of the
optimization problem. For this reformulated completely observable
problem we investigate for the case of power utility the associated
Hamilton-Jacobi-Bellman equation. Since this equation contains
non-linearities in a jump part we adopt a policy improvement method
to obtain an approximation of the optimal strategy. Numerical
results are presented.
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Pavel Gapeev (London School of Economics)
Pricing and filtering in a two-dimensional dividend switching
model
In our recent joint paper with Monique Jeanblanc, we have studied
a model of a financial market in which the dividend rates of two
risky assets change their initial values to other constant ones
at the times at which certain unobservable external events occur.
The asset price dynamics were described by geometric Brownian
motions with random drift rates switching at exponential random
times that are independent of each other and the constantly correlated
driving Brownian motions. We have obtained closed form expressions
for the rational values of European contingent claims through
the filtering estimates of occurrence of the switching times and
their conditional probability density derived given the filtration
generated by the underlying asset price processes.
Building on the results described above, we consider the model
in which two underlying assets are driven by dependent (compound)
Poisson processes belonging to exponential families. We obtain
closed form expressions for the prices in the case in which the
parameters of the asset price dynamics change one constants to
other at the times of occurrence of unobservable external events
and derive stochastic differential equations for the filtering
estimates. We also discuss the solution to the problem of pricing
of perpetual American options in a one-dimensional continuous
diffusion model for the asset price with switching dividend rates
under partial information.
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Jim Gatheral (Merrill Lynch, NY)
Implied Volatility from Local Volatility
There is a well-known simple formula for computing local volatility
given implied volatility as a function of strike and expiration.
Given local volatilities, implied volatilities may be computed
numerically using numerical PDE techniques. However, such computations
are typically too time-consuming to permit fast calibration of
local volatilities to option prices. In this talk, we review various
methods that have been proposed for computing implied volatility
from local volatility including heat kernel-based expansions and
parameter averaging. We focus in particular on the most-likely-path
approximation showing by specific example that it tends to perform
better in practice than competing approximations.
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David Hobson (Warwick)
Model independent bounds for variance swaps
Under an assumption of continuity on the price process, and under
an assumption that a continuum of calls on the underlying are
traded, the work of Neuberger and Dupire gives that the price
for the variance swap is equal to twice the price of a log contract.
This price is model-free.
But what if we are not prepared to assume continuity? Then, given
call prices a range of possible prices is consistent with no-arbitrage.
In this talk we try to characterise this range.
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Lane Hughston (Imperial College)
Implied Density Models for Asset Pricing
In this paper we model the dynamics of asset prices and associated
derivatives by consideration of the dynamics of the conditional
probability density process for the value of an asset at some
specified time in the future. In the case where the asset is driven
by Brownian motion we derive an associated "master equation"
for the dynamics of the conditional probability density, and express
this equation in integral form. By a "model" for the
conditional density process we mean a solution to the master equation
along with a specification of the initial density and a specification
of the volatility structure for the density. The volatility structure
in particular is assumed at any given time and for each value
of the argument of the density function to take the form of a
functional that depends on the history of density up to that time.
The choice of this functional determines the particular model
for the conditional density, and in practice one specifies the
functional modulo sufficient parametric freedom to allow for the
input of additional option data apart from that already implicit
in the specification of the initial density. The scheme is sufficiently
flexible to allow for the input of various different types of
data depending on the nature of the options market under consideration
and the class of valuation problem being undertaken. Various specific
examples are studied in detail, with exact solutions provided
in some cases. (Co-authors: D. Filipovic, Ecole Polytechnique
Fédérale de Lausanne, Switzerland, and A. Macrina,
King's College London and Kyoto Institute of Economic Research.)
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Monique Jeanblanc (Evry)
Density models for credit risk
We present a model of default times based on the conditional
law of defaults. We show in particular that, in that general framework,
the intensity does not contain all the needed information. In
case of a single default, this model can be interpreted as an
extension of the Cox model, where the barrier depends on the reference
filtration. The extension of our study to several defaults can
be viewed as a dynamic copula approach.
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Yu Hang Kan (Columbia)
Default intensities implied by CDO spreads: inversion formula
and model calibration
We propose a simple computational method for constructing an
arbitrage-free CDO pricing model which matches a pre-specified
set of CDO tranche spreads. The key ingredient of the method is
an inversion formula for computing the aggregate default rate
in a portfolio, as a function of the number of defaults, from
its expected tranche notionals. This formula can be seen as an
analog of the Dupire formula for portfolio credit derivatives.
Together with a quadratic programming method for recovering expected
tranche notionals from CDO spreads, our inversion formula leads
to an efficient non-parametric method for calibrating CDO pricing
models. Contrarily to the base correlation method, our method
yields an arbitrage-free model.
Comparing this approach to other calibration methods, we find
that model-dependent quan- tities such as the forward starting
tranche spreads and jump-to-default ratios are quite sensitive
to the calibration method used, even within the same model class.
On the other hand, comparing the local default intensities implied
by different credit portfolio models reveals that apparently different
models, such as the static Student-t copula models and the reduced-form
affine jump- diffusion model, lead to similar marginal loss distributions
and tranche spreads.
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Thomas Kokholm (Aarhus)
A Consistent Pricing
Model for Index Options and Volatility Derivatives
We propose and study a flexible modeling framework for the joint
dynamics of an index and a set of forward variance swap rates
written on this index, allowing volatility derivatives and options
on the underlying index to be priced consistently. Our model reproduces
various empirically observed properties of variance swap dynamics
and allows for jumps in volatility and returns.
An affine specification using Lévy processes as building
blocks leads to analytically tractable pricing formulas for options
on the VIX as well as efficient numerical methods for pricing
of European options on the underlying asset. The model has the
convenient feature of decoupling the vanilla skews from spot/volatility
correlations and allowing for different conditional correlations
in large and small spot/volatility moves.
We show that our model can simultaneously fit prices of European
options on S&P 500 across strikes and maturities as well as
options on the VIX volatility index. The calibration of the model
is done in two steps, first by matching VIX option prices and
then by matching prices of options on the underlying.
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Roger Lee (Chicago)
Variation Swaps on Time-Changed Levy Processes
For a family of functions G, we define the G-variation, which
generalizes power variation; G-variation swaps, which pay the
G-variation of the returns on an underlying share price F; and
share-weighted G-variation swaps, which pay the integral of F
with respect to G-variation. For instance, the case G(x)=x^2 reduces
these notions to, respectively, quadratic variation, variance
swaps, and gamma swaps.
We prove that a multiple of a log contract prices a G-variation
swap, and a multiple of an F log F contract prices a share-weighted
G-variation swap, under arbitrary exponential Levy dynamics, stochastically
time-changed by an arbitrary continuous clock having arbitrary
correlation with the Levy driver, under integrability conditions.
We solve for the multipliers, which depend only on the Levy process,
not on the clock.
In the case of quadratic G and continuity of the underlying paths,
each valuation multiplier is 2, recovering the standard no-jump
variance and gamma swap pricing results. In the presence of jump
risk, however, we show that the valuation multiplier differs from
2, in a way that relates (positively or negatively, depending
on the contract) to the Levy measure's skewness.
This work, joint with Peter Carr, extends Carr-Lee-Wu's treatment
of variance swaps, by generalizing from quadratic variation to
G-variation; and by encompassing not only unweighted but also
share-weighted payoffs.
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Andrea Macrina (King's College London)
Heat Kernels for Information-Sensitive Pricing Kernels
We consider a positive propagator that is driven by time-inhomogeneous
Markov processes. We multiply the propagator with a time-dependent,
decreasing positive weight function, and integrate the product
over time. The result is a so-called weighted heat kernel that
by construction is a supermartingale with respect to the filtration
generated by the time-inhomogeneous Markov processes. Such supermartingales
are suitable for the modelling of the pricing kernel in the case
where it is assumed to be given by a function of time and Markov
processes. This situation is encountered for example, if we assume
that the pricing kernel is sensitive to partial information about
economic factors, and the partial information is modelled by use
of time-inhomogeneous Markov processes. We show how closed-form
expressions for bond prices along with the associated interest-rate
and market price of risk models can be obtained, and indicate
the way towards the pricing of fixed-income derivatives within
this framework. (In collaboration with Jiro Akahori, Ritsumeikan
University)
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Gustavo Manso (MIT)
Information Percolation
We study the "percolation" of information of common
interest through a large market as agents encounter and reveal
information to each other over time. We provide an explicit solution
for the dynamics of the cross-sectional distribution of posterior
beliefs, and calculate its rate of convergence to a common posterior.
We also study how market segmentation, learning through public
signals, and endogenous search intensities affect information
percolation.
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Aleksandar Mijatovic (Imperial College)
Deterministic criteria for the absence
of arbitrage in one-dimensional diffusion models
In this talk we describe a deterministic characterisation of
the no free lunch with vanishing risk (NFLVR), the no generalised
arbitrage (NGA) and the no relative arbitrage (NRA) conditions
in the one-dimensional diffusion setting and examine how these
notions of no-arbitrage relate to each other. This is joint work
with Mikhail Urusov.
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Andreea Minca (Paris VI)
Resilience to contagion in financial networks
Given a macroeconomic stress scenario defined in terms of the
magnitude of common shocks across balance sheet, we perform an
asymptotic analysis of default contagion, using analytical methods,
and derive an expression for the fraction of defaulted nodes in
the limit where the number of nodes is large, in terms of the
empirical distribution of the in and out-degrees and the proportion
of weak links in the network. We show that the size of the default
cascade generated by the macroeconomic shock may exhibit a phase
transition when the macroeconomic shock affecting the financial
institutions reaches a certain threshold, beyond which the fraction
of defaults is close to one. This result is used obtain a criterion
for the resilience of a large network to macro-economic shocks
The asymptotic results are shown to be in good agreement with
simulations for networks whose sizes are realistic, showing the
relevance of the large network limit for macro-prudential regulation.
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Jan Obloj (Oxford)
On notion of arbitrage and robust pricing and hedging of variance
swaps
In robust pricing and hedging one does not assume any given model
but starts with market quoted prices of some options and deduces
no-arbitrage bounds on a given non-traded derivative, and further
specifies robust hedging strategies which enforce these bounds.
In this talk, we consider the case of a weighted variance swap
(e.g. a vanilla variance swap or a corridor variance swap) when
prices of finite number of co-maturing call/put options are given.
We analyse in some detail the arbitrage opportunities which may
arise when prices are mis-specified: model independent arbitrage,
weak arbitrage and weak free lunch with vanishing risk. These
new notions are necessary since do not have any pre-specified
probability space. Based on joint works with M. Davis and V. Raval
and with A. Cox.
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Goran Peskir (Manchester)
A Duality Principle for the Legendre Transform and the Valuation
of Financial Contracts
We present a duality principle for the Legendre transform that
yields the shortest path between the graphs of functions and embodies
the underlying Nash equilibrium. A useful feature of the algorithm
for the shortest path obtained in this way is that its implementation
has a local character in the sense that it is applicable at any
point in the domain with no reference to calculations made earlier
or elsewhere. The derived results are applied to the valuation
of financial contracts for Markov processes where the duality
principle corresponds to the semiharmonic characterisation of
the value function.
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Martijn Pistorius (Imperial College)
Continuously monitored barrier options under Markov processes
In this talk we present an algorithm for pricing barrier options
in one-dimensional Markov models. The approach rests on the construction
of an approximating continuous-time Markov chain that closely
follows the dynamics of the given Markov model. We illustrate
the method by implementing it for a range of models, including
a local Levy model and a local volatility jump-diffusion. We also
provide a convergence proof and error estimates for this algorithm.
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Marek Rutkowski (Sydney)
Market Models of Forward CDS Spreads
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Thorsten Schmidt (Leipzig)
Market Models for CDOs Driven by Time-Inhomogeneous Levy Processes
This paper considers a top-down approach for CDO valuation and
proposes a market model. We extend previous research on this topic
in two directions: on the one side, we use as driving process
for the interest rate dynamics a time-inhomogeneous Levy process,
and on the other side, we do not assume that all maturities are
available in the market. Only a discrete tenor structure is considered,
which is in the spirit of the classical Libor market model. We
create a general framework for market models based on multidimensional
semimartingales. This framework is able to capture dependence
between the default-free and the defaultable dynamics, as well
as contagion effects. Conditions for absence of arbitrage and
valuation formulas for tranches of CDOs are given.
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Steve Shreve (Carnegie Mellon)
Matching Statistics of an Ito Process by a Process of Diffusion
Type
Suppose we are given a multi-dimensional Ito process, which can
be regarded as a model for an underlying asset price together
with related stochastic processes, e.g., volatility. The drift
and diffusion terms for this Ito process are permitted to be arbitrary
adapted processes. We construct a weak solution to a diffusion-type
equation that matches the distribution of the Ito process at each
fixed time. Moreover, we show how to also match the distribution
at each fixed time of statistics of the Ito process, including
the running maximum and running average of one of the components
of the process. A consequence of this result is that a wide variety
of exotic derivative securities have the same prices when written
on the original Ito process as when written on the mimicking process.
This is joint work with Gerard Brunick.
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Stuart Turnbull (Houston)
Measuring and Managing Risk in Innovative
Financial Instruments
This paper discusses the difficult challenges of measuring and
managing risk of innovative financial products. To measure risk
requires the ability to first identify the
different dimensions of risk that an innovation introduces. The
list of possible factors is long: model restrictions, illiquidity,
limited ability to test models, product design,
counterparty risk and related managerial issues. For measuring
some of the different dimensions of risk the implications of limited
available data must be addressed. Given
the uncertainty about model valuation, how can risk managers respond?
All parties within a company - senior management, traders and
risk managers - have important
roles to play in assessing, measuring and managing risk of new
products.
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Johan Tysk (Uppsala)
Dupire's Equation for Bubbles
This is a report on a joint work with Erik Ekström. We study
Dupire's equation for local volatility models with bubbles, i.e.
for models in which the discounted underlying asset follows a
strict local martingale. If option prices are given by risk-neutral
valuation, then the discounted option price process is a true
martingale, and we show that the Dupire equation for call options
contains extra terms compared to the usual equation. Surprisingly
enough, however, the Dupire equation for put options takes the
usual form. Moreover, uniqueness of solutions to the Dupire equation
is lost in general, and we show how to single out the option price
among all possible solutions. The Dupire equation for models in
which the discounted derivative price process is merely a local
martingale is also studied.
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