Financial Mathematics and Applied
Probability Seminars 20102011
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
Nomura
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 lowfactor 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 closedform 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 multifactor crossasset 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 longdated FX structures.
Presentation

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.
Paper

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.
Presentation

Tuesday 2 November, 2010 5:30 pm 
Dr Peter England
EMB
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 ValueatRisk of the basic own funds of an insurance or reinsurance undertaking subject to a confidence level of 99.5% over a oneyear period.'' So it seems straightforward to estimate the SCR using a simulationbased 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 nonlife company, but with emphasis on the reserve risk elements.
Primary Keywords: GLM, Bootstrap, MCMC.
Secondary Keywords: EVT, Copula, ESG
Presentation

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 socalled Pareto distribution, and has been used to model a variety of interesting phenomena with fattailed powerlaw 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 onedimensional geometric Brownian motion and that the company is riskneutral. The policy for reversible abatement options is evaluated under both instantaneous and Parisian criteria, nesting the model of BarIlan 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.
Presentation

Tuesday 30 November, 2010 5:30 pm 
Dr Anke Wiese
HeriotWatt University
Positive Volatility Simulation in the Heston Model
In the Heston stochastic volatility model, the variance process is given
by a meanreverting squareroot
process. It is known that its transition probability density can be
represented by a noncentral chisquare 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 CoxIngersollRoss 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.
Presentation

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.
Presentation

Tuesday 14 December, 2010 5:30 pm 
Professor Raymond Brummelhuis
Birkbeck, University of London
Analytical Approaches to Nonlinear ValueatRisk
We discuss analytical approaches (as opposed to Monte Carlo method
ology) for computing probability distributions and associated quantiles of portfolios
whose value depend nonlinearly 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 nonlinear 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 nonlinear VaR when the riskfactors have
small variancecovariance matrix. Concentrating next on quadratic valueatrisk,
we discuss asymptotic approximations (saddlepoint or complex stationary phase
methods) with explicit analytic errorbounds, both for Gaussian and nonGaussian
GEDdistributed riskfactors. Even for Gaussian riskfactors, 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.
Presentation

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 incrisis. 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.
Presentation

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 modelfreeexcept for a 'frictionless markets' assumptionnecessary and sufficient conditions for absence of arbitrage given a set of currenttime 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 pathdependent option, specifically a weighted variance swap, to the set of traded assets and ask what are the conditions on its time0 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 superreplicating portfolios, which turns out to be a problem in semiinfinite linear programming. This is joint work with Vimal Raval and Jan Oblój .
Presentation

Tuesday 22 February, 2011 5:30 pm 
Andrey Pogudin
RBS
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 LMMSABR 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.
Presentation

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
Citigroup
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 CDOSquared pricing. First, we derive generic modelindependent 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 CDOSquared model, the Tranche Option model, and a Markovian model for Forward Tranches to generate the CVA for CDOs.
Presentation

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.
Presentation

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.


