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Financial Mathematics
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Financial Mathematics and Applied Probability Seminars 2009-2010

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 loan-portfolio 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 non-uniform 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.

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 old-fashioned problem, option pricing in a Black-Scholes world, and specifically at the differences that occur between models in which dividends are paid, or averages are sampled, in a continous-time 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.

Tuesday 27 October, 2009

No seminar this day.

Tuesday 3 November, 2009
5:30 pm
Professor Igor Evstingeev
University of Manchester
Von Neumann-Gale Dynamical Systems and their Applications in Finance

Von Neumann-Gale 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 one-step 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 Neumann-Gale 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.

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
Put-call symmetry

The pricing formulae for put and call options in the Black--Scholes model satisfy a certain symmetry relationship. There has been growing interest in asset price models that exhibit this put-call symmetry since, in the context of such models, certain barrier options can be replicated by a semi-static trading strategy in the underlying stock. This talk will survey these results as well as recent results on characterizing models that exhibit put-call symmetry.

Tuesday 1 December, 2009
5:30 pm
Dr. Dorje Brody
Department of Mathematics, Imperial College
Hazardous information on credit risk

An information-based 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.

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 B-spline 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 B-spline. We further show that the well known variance-gamma (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 B-spline 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.

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: self-organization 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 cause-effect 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 self-organized criticality. The analysis is therefore shifted from the cause-effect 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.

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.

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 non-Markovian approach

We present and further develop the non-Markovian 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 non-Markovian stochastic process that allows for modeling spikes directly as self-reversing 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 multi-layered 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 risk-neutral 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 American-type call options, which include the perpetual American, Russian and lookback-American call options as special cases. We solve this genuinely two-dimensional optimal stopping problem by means of an explicit construction of its value function. In particular, we fully characterise the free-boundary that provides the optimal strategy, and which involves the analysis of a highly non-linear 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 so-called principle of smooth fit.

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 well-understood 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.

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 multi-dimensional jump-diffusion 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.

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 Barndorff-Nielsen and Shephard model. Under mild conditions on the jump measures, we derive (semi) closed-form formulae for the implied volatility as the maturity gets large or small.

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