Financial Mathematics and Applied
Probability Seminars 20092010
Unless otherwise indicated, all seminars take place at Lecture Theatre K2.31 (formerly known as 2C),
King's College London, The Strand, London WC2R 2LS.
Tuesday 6 October, 2009 5:30 pm 
Dr Reimer Kühn
Department of Mathematics, King's College London
Statistical Physics Approach to Models of Risk
We look at the problem of estimating risk (Operational Risk, Credit Risk and Market
Risk), taking a systemic point of view. We argue that risk elements, such as processes
in an organization, credits in a loanportfolio or share prices in an investment portfolio
cannot be regarded as independent. This naturally leads to formulating risk models as
dynamical models of interacting degrees of freedom (particles). The operational risk
and credit risk problems can be cast into a language describing heterogeneous lattice
gasses, in which interaction parameters and nonuniform chemical potentials have an
interpretation in terms of unconditional and conditional failure probabilities. For
the market risk problem, a minimal interacting generalization of the classical Geometric
Brownian Motion model leads to a formulation of market dynamics that is formally similar
to the dynamics of graded response neurons. We describe elements of the statistical
mechanical analysis of these models to reveal their macroscopic properties. The dominant
role of interaction is to allow avalanches (bursts) of risk events, leading to significant
fattening of tails in loss distributions.
Presentation

Tuesday 13 October, 2009

No seminar this day.

Tuesday 20 October, 2009 5:30 pm 
Professor Sam Howison
Mathematical Institute, University of Oxford
European, Asian and American options with discrete dividend payments
This talk looks at a rather oldfashioned problem, option pricing in a
BlackScholes world, and specifically at the differences that occur
between models in which dividends are paid, or averages are
sampled, in a continoustime way, and those in which the payment or
sampling is discrete. Some intricate phenomena appear, and they will be
explored using asymptotic methods and in particular the method of
multiple scales.
Presentation

Tuesday 27 October, 2009

No seminar this day.

Tuesday 3 November, 2009 5:30 pm 
Professor Igor Evstingeev
University of Manchester
Von NeumannGale Dynamical Systems and their Applications in Finance
Von NeumannGale dynamical systems are defined in terms of multivalued operators possessing certain
properties of convexity and homogeneity. These operators assign to each element of a given cone a
convex subset of the cone describing possible onestep transitions from one state of the system to another.
The classical, deterministic theory of such dynamics was originally aimed at the modelling of economic
growth (von Neumann 1937 and Gale 1956). First attempts to build a stochastic generalization of this
theory were undertaken in the 1970s by Dynkin, Radner and their research groups. However, the initial
attack on the problem left many questions unanswered. Substantial progress was made only in the late
1990s, and final solutions to the main open problems were obtained only in the last two or three years.
Recently it has been observed that stochastic analogues of von NeumannGale systems provide a
natural and convenient framework for financial modelling (asset pricing and hedging under transaction
costs, capital growth theory). This observation gave a new momentum to studies in the field and posed
new interesting questions. The talk will give an introduction into the theory, review recent progress
and discuss applications.
Presentation

Tuesday 10 November, 2009

No seminar this day.

Tuesday 17 November, 2009

No seminar this day. (MSc project talks).

Tuesday 24 November, 2009 5:30 pm 
Dr. Mike Tehranchi
Department of Pure Mathematics and Mathematical Statistics,
University of Cambridge
Putcall symmetry
The pricing formulae for put and call options in the BlackScholes
model
satisfy a certain symmetry relationship. There has been growing interest
in asset price models that exhibit this putcall symmetry since, in the
context of such models, certain barrier options can be replicated by a
semistatic trading strategy in the underlying stock. This talk will
survey these results as well as recent results on characterizing models
that exhibit putcall symmetry.
Presentation

Tuesday 1 December, 2009 5:30 pm 
Dr. Dorje Brody
Department of Mathematics, Imperial College
Hazardous information on credit risk
An informationbased model for the dynamics of the hazard process
associated with a default event is introduced. This is used to derive
models for defaultable bonds, and applied to price credit derivatives.
The work is carried out in collaboration with L. P. Hughston and A. Macrina.
Presentation

Tuesday 8 December, 2009 5:30 pm 
Professor Vladimir Kaishev
Cass Business School, City University, London
Stochastic processes induced by Dirichlet (B) splines
with applications in finance and insurance
A new stochastic process, called LG process, is defined as a linear
combination of independent gamma processes. Its distribution and related
properties are established, by following its relation to polynomial and
Dirichlet (B) splines. By considering Bspline processes, it is shown
that the density of an LG process can be expressed through Dirichlet
(B) splines, introduced independently by Ignatov and Kaishev (1985,
1987, 1989) and Karlin et al. (1986). It is also shown that, for certain
time epochs, the LG density is expressed through a classical polynomial
Bspline. We further show that the well known variancegamma (VG)
process, introduced by Madan and Seneta (1990) is a special case of an
LG process and derive a new (alternative) expression for the VG density.
The LG process has two sets of parameters, the Bspline knots and their
multiplicities, and offers more flexibility than the VG in controlling
the shape of the Levy density. Multivariate LG processes are further
introduced and shown to provide a competitive alternative to the
multivariate asymmetric VG process considered by Cont and Tankov (2004)
and Luciano and Schoutens (2006), and to its recent generalization by
Semeraro (2008) and Luciano and Semeraro (2007). Applications of these
new Dirichlet (B) spline related processes in finance and insurance,
such as modelling the joint dynamics of multiple exchange rates, valuing
exotic options and participating life insurance contracts are also
considered.
Presentation

Tuesday 19 January, 2010 5:30 pm 
Professor Luciano Pietronero
Institute of Complex Systems, CNR Rome and
Department of Physics, University of Rome La Sapienza, Italy
Agents in the global network: selforganization and instabilities
After the subprime crisis there have been many conjectures for the
possible origin of this instability. Most suggestions focus on concepts
like collective behavior, contagion, network domino effect, coherent
portfolios, lack of trust, liquidity crisis, and, in general,
psychological components in the traders' behavior. These properties are
usually neglected in the standard risk analysis which is based on a
linear analysis within a causeeffect relation. These new concepts
require a novel approach to the risk problem, which could profit from
the general ideas of complex systems theory. This corresponds to the
introduction of suitable models with heterogeneous agents and a
different perspective in which the interaction between agents (direct or
indirect) is explicitly considered together with the idea that the
system may become globally unstable in the sense of selforganized
criticality. The analysis is therefore shifted from the causeeffect
relation to the study of the possibile intrinsic instabilities. We
discuss some steps towards a systematic analysis of these ideas based
on agent models and order book models, together with the statistical
analysis of experimental data. The final objective of these studies
would be to define the characteristic properties of each of the above
concepts from the models, and then to identify their role and importance
in the real financial markets.
Presentation

Tuesday 26 January, 2010 5:30 pm 
Professor Robert M. May (Lord May of Oxford, OM AC
FRS)
Zoology Department, Oxford University
Systemic risk: dynamics of banking systems
The recent banking crises have made it clear that increasingly complex
strategies for managing risk in individual banks and investment funds
(pension funds, etc) has not been matched by corresponding attention to
overall systemic risks. Simple mathematical caricatures of
“banking ecosystems”, which capture some of the essential
dynamics and which have
some parallels (along with significant differences) with earlier work on
stability and complexity in ecological food webs, have interesting
implications. In particular, strategies that tend to minimise risk for
individual banks can  under certain circumstances  maximise the
probability of systemic failure. This talk will first sketch these
models and then discuss some of the ensuing conclusions.
Presentation

Tuesday 2 February, 2010 5:30 pm 
Dr. Valery A. Kholodnyi
Verbund Austrian Power Trading AG, Vienna, Austria
Modelling energy American options in the framework of
the nonMarkovian approach
We present and further develop the nonMarkovian approach to modeling
energy prices with spikes proposed earlier by the author. In contrast to
other approaches, we model energy prices with spikes as a nonMarkovian
stochastic process that allows for modeling spikes directly as
selfreversing jumps. We use this approach to value American options on
energy spots, forwards and swaps, one of the most popular types of
energy options. The valuation is based on the semilinear evolution
equation for American options and the multilayered tree methods both
introduced earlier by the author. We consider a practically important
example of the crude oil American options and show that the extracted
riskneutral probability distributions allowed to conclude as early as
at the end of March 2008 that the crude oil prices were at an upward
spike and, due to the detected possible downward spikes, were highly
likely to fall to the levels below $40 dollars by the end of 2008, well
before they even picked later that summer.

Tuesday 9 February, 2010 5:30 pm 
Professor Mihail Zervos
Department of Mathematics, London School of Economics
π options
We consider a discretionary stopping problem that arises in the context
of pricing a class of perpetual Americantype call options, which
include the perpetual American, Russian and lookbackAmerican call
options as special cases. We solve this genuinely twodimensional
optimal stopping problem by means of an explicit construction of its
value function. In particular, we fully characterise the freeboundary
that provides the optimal strategy, and which involves the analysis of a
highly nonlinear ordinary differential equation (ODE). In accordance
with other optimal stopping problems involving a running maximum process
that have been studied in the literature, it turns out that the
associated variational inequality has uncountable solutions that satisfy
the socalled principle of smooth fit.
Presentation

Tuesday 16 February, 2010 5:30 pm 
Professor David Spiegelhalter
Statistical Laboratory, Cambridge
Quantifying and visualising uncertainty
There has been a traditional division between ‘risk’,
which can be
quantified using probability distributions, and ‘uncertainty’,
which is
the surrounding mess of doubt, disagreement and ignorance. In
wellunderstood situations we may be happy to quote reasonable odds for
future events, and I shall look at ways in which these risks can be
communicated visually. When the problem is more complex, analysts may
use a mixture of judgement and historical data to construct a
mathematical model that can assess future risks, but deeper
uncertainties may be glossed over. I will use examples from swine flu to
climate change to illustrate different approaches to dealing with
uncertainty, from ignoring it to trying to fully quantify it, and
conclude that we should all try to be aware and open about the magnitude
and potential consequences of our ignorance.
Presentation

Tuesday 23 February, 2010 5:30 pm 
Dr. Albina Danilova
Department of Mathematics, London School of Economics
Stock market insider trading in continuous time with
imperfect dynamic information
This talk is based on the joint work with Luciano Campi and Umut Cetin.
I will present the solution to the insider trading model in the presence
of dynamic private information.
Differently from the previous literature, we assume that a) the
insider's signal is dynamic, i.e., rather then observing the final value
of the firm, the insider observes the stochastic
process of the firm's fundamental value, and b) that the signal received
by the insider is not necessarily Gaussian.
I will present the derivation of the optimal insider trading strategy
and its connection with the information drift. I will also demonstrate
that in the dynamic case, as in the static one,
the presence of the insider does not lead to arbitrage and increases
market efficiency.
Moreover, I will demonstrate existence of an equilibrium by deriving
solutions for the optimal order process of the
informed trader and the optimal pricing rule of the market maker.

Tuesday 2 March, 2010 5:30 pm 
Professor Alexander Lipton
Bank of America Merrill Lynch and Department of Mathematics,
Imperial College
CVA for CDS via SDM
We present a multidimensional jumpdiffusion version of a structural
default model (SDM) and show how to use it in order to value the credit
value adjustment (CVA) for a credit default swap (CDS). We also discuss
CVA problem in a broader context.
Presentation

Tuesday 11 May, 2010 5:30 pm 
Antoine Jacquier
Department of Mathematics,
Imperial College
Implied volatility asymptotics of affine stochastic volatility
models with jumps
Calibrating stochastic models is a huge issue in financial markets.
The aim of this talk is to propose a calibration methodology based on
the knowledge of the asymptotic behaviour of the implied volatility.
We focus on the general class of affine stochastic volatility models
with jumps, which encompasses the Heston (with jumps) model, exponential
Levy models, the BarndorffNielsen and Shephard model.
Under mild conditions on the jump measures, we derive (semi) closedform
formulae for the implied volatility as the maturity gets large or small.
Presentation


