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Financial Mathematics
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Financial Mathematics and Applied Probability Seminars 2010-2011

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 12 October, 2010
5:30 pm
Dr Paul McCloud
Symmetry methods for quadratic Gaussian models of interest rate and FX processes

The quadratic Gaussian model has a long history as a smile model for low-factor interest rate structured products. It is significantly enhanced, both in scope and implementation, by the introduction of the quadratic Gaussian symmetry groups. These groups represent measure change and conditional expectation as purely algebraic operations, thereby circumventing tricky and potentially unstable numerical quadratures. Core financial variables and option prices are generated as closed-form expressions of simple state processes, and this facilitates efficient algorithms for calibration and pricing. When used to transform between the equivalent pricing measures associated with distinct numeraires, the group actions extend the quadratic Gaussian model to a multi-factor cross-asset setting, and this brings a substantially wider range of hybrid structures within its domain of applicability. In particular, this enables consistent modelling of the smile dynamics of interest rates and FX indexes for long-dated FX structures.

Tuesday 19 October, 2010
5:30 pm
Dr Giuseppe Di Graziano
Deutsche Bank
Target volatility option pricing

In this talk, I shall present two 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, scaled 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 popular with investors and hedgers because they are typically cheaper than their vanilla equivalent. I will present two approaches for the pricing of TVOs: a power series expansion and a Laplace transform method. The pricing methodologies have been tested numerically and results will be provided.

Tuesday 26 October, 2010
Dr Takashi Shibata
Tokyo Metropolitan University
Optimal investment timing under financing constraint

This paper examines the optimal investment timing decision subject to financing constraint. In particular, we extend the optimal investment and capital structure decision problem in a real options model by incorporating financing constraint. We show that financing constraint may accelerate the investment although it always decreases the firm value.

Tuesday 2 November, 2010
5:30 pm
Dr Peter England
General insurance stochastic claims reserving and internal capital models for Solvency II

Article 101 of the Solvency II Directive for insurance companies states that ''The Solvency Capital Requirement (SCR) shall... correspond to the Value-at-Risk of the basic own funds of an insurance or reinsurance undertaking subject to a confidence level of 99.5% over a one-year period.'' So it seems straightforward to estimate the SCR using a simulation-based model: simply create a simulated distribution of the basic own funds over 1 year, then calculate the VaR at 99.5%. The basic own funds is simply the net assets on a ''Solvency II'' balance sheet, so it is necessary to project the assets and liabilities on a Solvency II basis over a one year time horizon. The largest component of the liabilities is the outstanding claims reserves. In this presentation, an example simulation based capital model will be presented for a non-life company, but with emphasis on the reserve risk elements.
Primary Keywords: GLM, Bootstrap, MCMC. Secondary Keywords: EVT, Copula, ESG

Tuesday 9 November, 2010

No seminar

Tuesday 16 November, 2010
Professor Lane Hughston
Imperial College London
Financial Applications of the Zipf Distribution and the Zeta Process

The zeta distribution, sometimes also called the Zipf distribution, is the discrete analogue of the so-called Pareto distribution, and has been used to model a variety of interesting phenomena with fat-tailed power-law behaviour. Examples include word frequency, corporate income, citations of scientific papers, web hits, copies of books sold, frequency of telephone calls, magnitudes of earthquakes, diameters of moon craters, intensities of solar flares, intensities of wars, personal wealth, frequencies of family names, frequencies of given names, populations of cities. It makes sense therefore to consider financial contracts for which the payoff is represented by a random variable of this type. This talk will present an overview of some of the properties of the zeta distribution and the associated multiplicative Levy process, which we shall call the zeta process, with a view to financial applications. The objective here is in part to link some ideas in econophysics to mainstream mathematical finance. The material under consideration can be regarded more generally as part of an ongoing program, being pursued by a number of authors, devoted to various aspects of the relationship between probability and number theory. (Work with Dorje Brody, Martijn Pistorius and Simon Lyons.)

Tuesday 23 November, 2010
5:30 pm
Dr Luca Taschini
London School of Economics
Flexibility Premium in Marketable Permits

We study the market for emission permits in the presence of reversible abatement measures characterized by delay in implementation. We assume that the new operating profits follow a one-dimensional geometric Brownian motion and that the company is risk-neutral. The policy for reversible abatement options is evaluated under both instantaneous and Parisian criteria, nesting the model of Bar-Ilan and Strange (1996). By taking the difference between these two values at their respective optima, we derive an analytic solution of the premium for flexibility embedded in marketable permits. This extends the findings in Chao and Wilson (1993) and Zhao (2003). Numerical results are presented to illustrate the likely magnitude of the premium and how it is affected by uncertainty and delays in implementation.

Tuesday 30 November, 2010
5:30 pm
Dr Anke Wiese
Heriot-Watt University
Positive Volatility Simulation in the Heston Model

In the Heston stochastic volatility model, the variance process is given by a mean-reverting square-root process. It is known that its transition probability density can be represented by a non-central chi-square density. There are fundamental differences in the behaviour of the variance process depending on the number of degrees of freedom: if the number of degrees of freedom is larger or equal to 2, the zero boundary is unattainable; if it is smaller than 2, the zero boundary is attracting and attainable. We focus on the attainable zero boundary case and in particular the case when the number of degrees of freedom is smaller than 1, typical in foreign exchange markets. We prove a new representation for the density based on powers of generalized Gaussian random variables. Further we prove that Marsaglia's polar method extends to the generalized Gaussian distribution, providing an exact and efficient method for generalized Gaussian sampling. Thus, we establish a new exact, unbiased and efficient method for simulating the Cox--Ingersoll--Ross process for an attracting and attainable zero boundary, and thus establish a new simple method for simulating the Heston model. We demonstrate our method in the computation of option prices for parameter cases that are described in the literature as challenging and practically relevant.

Tuesday 7 December, 2010
5:30 pm
Dr Enrico Biffis
Imperial College Business School
Optimal insurance with counterparty default risk

We study the design of optimal insurance contracts when the insurer can default on its obligations. In our model default arises endogenously from the interaction of the insurance premium, the indemnity schedule, and the insurer's assets. This allows us to understand the joint effect of insolvency risk and background risk on efficient contracts. The results may shed light on the aggregate risk retention sched- ules observed in catastrophe reinsurance markets, and can assist in the design of (re)insurance programs and guarantee funds.

Tuesday 14 December, 2010
5:30 pm
Professor Raymond Brummelhuis
Birkbeck, University of London
Analytical Approaches to Non-linear Value-at-Risk

We discuss analytical approaches (as opposed to Monte Carlo method- ology) for computing probability distributions and associated quantiles of portfolios whose value depend non-linearly on a given, typically large, set of risk factors. The typical example would be that of a portfolio of derivatives. A common approach is to replace the full non-linear portfolio by its second order Taylor approximation. We will give a rigorous theorem showing that the resulting quadratic VaR pro- vides a good approximation for the full non-linear VaR when the risk-factors have small variance-covariance matrix. Concentrating next on quadratic value-at-risk, we discuss asymptotic approximations (saddle-point or complex stationary phase methods) with explicit analytic error-bounds, both for Gaussian and non-Gaussian GED-distributed risk-factors. Even for Gaussian risk-factors, finding exact analyt- ical expressions for the probability distribution of a quadratic portfolio is a non- trivial problem, which has attracted a lot of attention in the statistical literature. We will touch on intriguing connections with certain classes of special functions of many variables (ordinary and confluent Lauricella functions, Carlson R- and S- functions) which do not seem to have been previously noted in the literature, and which may provide a new approach to the subject. Different parts of this talk rep- resent joint work with Jules Sadefo Kamdem (University of Montpellier) and with Brad Baxter (Birkbeck).

Tuesday 18 January, 2011
5:30 pm
Professor Emilio Barucci
Politecnico di Milano
Portfolio choices and VaR constraint with a defaultable asset

Considering a Constant Elasticity Variance model for the asset price, that is a defaultable asset showing the so called leverage effect (high volatility when the asset price is low), a VaR constraint reevaluated over time induces an agent with a CRRA utility more risk averse than a logarithmic utility to take more risk than in the uncostrained setting. The result shows that the risk of default joined by a VaR limit on the portfolio may induce the agent to take more risk.

Tuesday 25 January, 2011

No seminar

Tuesday 1 February, 2011
5:30 pm
Professor Damiano Brigo
King's College London
Credit models pre- and in- crisis: The importance of properly accounting for extreme scenarios in Valuation

We present three examples of credit products whose valuation poses challenging modeling problems related to armageddon scenarios and extreme losses, analyzing their behaviour pre- and in-crisis. The products are Credit Index Options (CIOs), Collateralized Debt Obligations (CDOs), and Credit Valuation Adjustment (CVA) related products. We show that poor mathematical treatment of possibly vanishing numeraires in CIOs and lack of modes in the tail of the loss distribution in CDOs may lead to inaccurate valuation, both pre- and especially in crisis. We finally enlarge the picture and comment on a number of common biases in the public perception of modeling in relationship with the crisis.

Tuesday 8 February, 2011
5:30 pm
No seminar

Tuesday 15 February, 2011
5:30 pm
Professor Mark Davis
Imperial College
Arbitrage Bounds for Prices of Options on Realized Variance

This paper builds on earlier work by Davis and Hobson (Mathematical Finance, 2007) giving model-free---except for a 'frictionless markets' assumption---necessary and sufficient conditions for absence of arbitrage given a set of current-time put and call options on some underlying asset. Here we suppose that the prices of a set of put options, all maturing at the same time, are given and satisfy the conditions for consistency with absence of arbitrage. We now add a path-dependent option, specifically a weighted variance swap, to the set of traded assets and ask what are the conditions on its time-0 price under which consistency with absence of arbitrage is maintained. We assume that the underlying asset price process has continuous paths. It is well known that a vanilla variance swap is then equivalent to an option with log payoff plus trading in the underlying asset. Other variance swaps have a similar representation with different convex payoffs, and we obtain bounds by considering sub- and super-replicating portfolios, which turns out to be a problem in semi-infinite linear programming. This is joint work with Vimal Raval and Jan Oblój .

Tuesday 22 February, 2011
5:30 pm
Andrey Pogudin
Interest rates volatility markets and SABR model

We discuss interest rates volatility markets including market segmentation and dynamics of volatility surfaces. We also focus on one of the most widely used stochastic volatility models: Stochastic Alpha, Beta, Rho model (SABR) proposed by Hagan et al. The model allows fast calibration, a variety of skew shapes and the efficient calculation of market risks. We then discuss drawbacks and pitfalls of the model and potential ways to fix them. We also introduce LMM-SABR framework for joint pricing of caps/floors and swaptions.

Tuesday 1 March, 2011
5:30 pm
Professor Giulia Iori
City University
Herding Effects in Order Driven Markets: The Rise and Fall of Gurus

We introduce an order driver market model with heterogeneous traders that imitate each other on a dynamic network structure. The communication structure evolves endogenously via a fitness mechanism based on agents performance. We assess under which assumptions imitation among noise traders can give rise to the emergence of gurus and their rise and fall in popularity over time. We study the wealth distribution of gurus, followers and non followers and show that traders have an incentive to imitate and to be imitated since herding is profitable.

Tuesday 8 March, 2011
5:30 pm
Professor Terry Lyons
Oxford University
The expected signature of a stochastic process. Some new PDE's.

How can one describe a probability measure of paths? And how should one approximate to this measure so as to capture the effect of this randomly evolving system. Markovian measures were efficiently describes by Strook and Varadhan through the Martingale problem. But there are many measures on paths that are not Markovian and a new tool, the expected signature provides a systematic ways of describing such measures in terms of their effects. We explain how to calculate this expected signature I the case of the measure on paths corresponding to a Brownian motion started at a point x in the open set and run until it leaves the same set. A completely new (at least to the speaker) PDE is needed to characterise this expected signature. Joint work with Ni Hao.

Tuesday 15 March, 2011
5:30 pm
Dr Youssef Elouerkhaoui
Trading CVA: A New Development in Correlation Modelling

Since the beginning of the credit crisis, the modelling of counterparty risk and the correct pricing and hedging of CVA has become a critical issue for financial institutions. In this talk, we address the issue of valuing CVA for credit correlation books. We shall see that given the "Exotic" nature of the CVA derivative payoff, we need to use a variety of modelling techniques that were developed over the last few years. This includes: default correlation modelling, the pricing of credit options, dynamic credit modelling, and CDO-Squared pricing. First, we derive generic model-independent CVA formulas and construct the building blocks to evaluate CVA for CDO tranches. We introduce the Conditional Forward Annuity Measure and derive the CVA for funded and unfunded CDS contracts. And finally, we combine the CDO-Squared model, the Tranche Option model, and a Markovian model for Forward Tranches to generate the CVA for CDOs.

Tuesday 22 March, 2011
5:30 pm
Professor Steven Haberman
City University
Modelling Dynamics in Mortality Rates

The increases in lifetimes experienced in many developed and developing countries represent a (generally)positive change at the individual level. For those planning to manage the consequences of an ageing population, it is important to have available models that can be use to represent and forecast such future trends and the uncertainty in these trends. The seminar will look at some of the models that have been recently proposed in the actuarial literature. From an actuarial perspective, the focus is on the impact of these trends and the uncertainty on pension plans and insurance companies selling life annuities.

Tuesday 29 March, 2011
5:30 pm
Dr Michael Kupper
Humboldt University Berlin
Concave Stochastic Target Problems

We study stochastic target problems and provide existence, stability and lower semicontinuity results for drivers which are monotone in y and concave in z. Viewed from a BSDE perspective our results are closely related to the monotonic limit theorems obtained by Peng [1999]. Our approach allows in particular to address problems of robustification, which we illustrate by studying superhedging under model uncertainty. The talk is based on joint work with Samuel Drapeau and Gregor Heyne.

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