King's College London
Financial Mathematics



Financial Mathematics and Applied Probability Seminars 2007-2008

Unless otherwise indicated, all seminars take place at Lecture Theatre K2.31 (the usual room, formerly known as 2C), King's College London, The Strand, London WC2R 2LS.

Tuesday 16 October,
5:30 pm

Dr Umut Cetin
Department of Statistics, London School of Economics

Pricing and hedging in carbon emissions markets

Joint work with M. Verschuere, Electrabel.

We propose a model in order to study EU markets for carbon emissions. Starting with a no-arbitrage relationship between the prices of carbon allowances belonging to subsequent phases, we solve for the price and optimal hedging strategies of derivatives in this market using a local risk minimisation approach under complete and incomplete information regarding the market's collective position in carbon credits.

Dr Cetin’s talk is available as a PDF.


Tuesday 23 October,

Prof Michael Giles
Computer Laboratory, University of Oxford

Monte Carlo evaluation of Greeks

In finance, we need to know not only the correct prices for financial derivative options, but also their sensitivity to changes in various parameters, such as the current asset price, interest rate, exchange rate and level of volatility - derivatives known collectively as the "Greeks". In this talk I will discuss various aspects of computing Greeks through Monte Carlo simulation. I will start by presenting the three main approaches: finite differences, likelihood ratio method (LRM) and pathwise sensitivity calculation. The last of these leads very naturally to an adjoint implementation which makes it possible to compute the sensitivity to a large number of input parameters at a very low cost, little more than the cost of evaluating the price itself. The practical development of adjoint codes is greatly assisted by using Automatic Differentiation (AD) tools. I will explain the underlying ideas and discuss the ise of the FADBAD++ software package which is based on templates and operator overloading within C++. The pathwise approach is not applicable when the payoff is not differentiable. Even when the payoff is differentiable, the use of scripting in real-world implementations means it can be very difficult in practice to evaluate the derivative in very complex financial products. To address these limitations, I will present a new idea to combine the adjoint pathwise approach for the stochastic path evolution with LRM for the payoff evaluation.

Prof. Giles’ talk is available as a PDF.


Tuesday 30 October,

Prof. William Shaw
Department of Mathematics King's College, London

Quantile mechanics and dependency without copulas

Joint work with G. Steinbrecher

This talk will begin by discussing the representation of quantile functions for Monte Carlo simulation as solutions of certain non-linear ordinary and partial differential equations, in particular providing an efficient method for convering Gaussian samples to fat-tailed samples. The PDE representation leads us to a natural generalization to a collection of multivariate distributions in which quite exotic combinations of marginal distributions are coupled together in a natural way. In this way we generate a natural alternative to the copula philosophy, where dependency is generated via suitably coupled stochastic differential equations.

Prof. Shaw’s talk is available as a PDF.


Tuesday 6 November,

No seminar today
MSc Project Week & Presentations

Tuesday 13th November,
5.30 pm

Dr Arun Verma
Bloomberg, New York

Break-even volatility surfaces - Historical vols conditional on Moneyness and Maturity

Joint work with Dr Bruno Dupire

Usually, the historical volatility of a time series is computed as the annualized standard deviation of the log returns. It provides an estimate of the volatility parameter to be input in the Black-Scholes formula to price options of various strikes and maturities. However, most markets exhibit a strong dependency of implied volatility on strikes, named skew or smile, which the historical volatility estimate misses as it provides a single number for all strikes. We propose a methodology to extract from a time series of prices not only one historical volatility but a whole volatility surface, corresponding to the implied volatilities that should have been used in the past to properly price and hedge options of various strikes and maturities. It is based on the notion of break-even volatilities and exploits delta hedging principles that lie at the heart of option pricing theory and reflect trading practice. This approach has two main applications: 1) in the case of an underlying where no options are available, it gives an indication of where implied volatilities should be and 2) when options are available, it provides a rich/cheap analysis tool.

Dr Verma’s talk is available as a Microsoft PowerPoint file.


Tuesday 20 November,
5.30 pm

Dr Christoph Reisinger
Department of Mathematics, University of Oxford

Modelling and numerical aspects of basket credit derivatives pricing

Based on joint work with Helen Haworth, William Shaw, and Ben Hambly.


The simulation of multi-name credit derivatives raises significant challenges, among others from the perspective of dependence modelling, calibration, and computational complexity. Structural models are based on the evolution of firm values, often modelled by market and idiosyncratic factors, to create a rich correlation structure. In addition to this, we will allow for contagious effects, to account for the important scenarios where the default of a number of companies has a time-decaying impact on the credit quality of others. If any further evidence for the importance of this was needed, the recent developments in the credit markets have furnished it, in spades. We will give illustrations for small n-th-to-default baskets, and propose extensions to large basket credit derivatives by exploring the limit for an increasing number of firms.


Dr. Reisinger’s talk is available as a PDF.


Tuesday 27 November,
5:30 pm

Dr Peter Jaeckel
Head of Credit, Hybrid, Inflation, and Commodity Derivative Analytics, ABN Amro, London

“Hyp Hyp Hooray”

A new stochastic-local volatility model is introduced. The new model's structural features are carefully selected to accommodate economic principles, financial markets' reality, mathematical consistency, and ease of numerical tractability when used for the pricing and hedging of exotic derivative contracts. Also, we present a generic analytical approximation for Black volatilities for plain vanilla options implied by any parametric-local-and-stochastic-volatility model, apply it to the new model, and demonstrate its accuracy.


Dr Jaeckel’s talk is available as a PDF.


Tuesday 4 December,
5:30 pm

No seminar today

Tuesday 22 January 2008 5.30pm

Prof Xunyu Zhou
Nomura Professor of Mathematical Finance, University of Oxford

“Prospect Theory - A New Paradigm for Portfolio Choice”

In this talk I shall report recent progress on continuous-time behavioural portfolio choice under Kahneman and Tversky's (cumulative) prospect theory, featuring S-shaped utility functions and probability distortions. It is shown that the model well-posedness becomes a prominent issue in such a behavioural model. The optimal terminal wealth positions, derived in fairly explicit forms, possess surprisingly simple structure reminiscent of a gambling policy betting on a good state of the world while accepting a fixed, known loss in case of a bad one. If time permits I will also discuss on the incomplete market and single-period models as well as the equity premium puzzle.


Prof. Zhou’s talk is available as a PDF.


Tuesday 29 January 2008 5.30pm


Prof Helyette Geman
Birkbeck, London

“Seasonal and Stochastic Effects in Commodity Forward Curves”

Joint work with Svetlana Borovkova, Free university of Amsterdam

In this paper we develop an original model for the dynamics of commodity forward curves exhibiting seasonality such as natural gas, electricity or agricultural commodities. In the existing literature on the subject, the first state variable in multi-factor models is the commodity price, which mixes seasonal and stochastic features and may be unobservable. We propose to use instead the average forward price, which is devoid of seasonality and conveys a more robust representation of the current forward curve level. The second factor in our model is a quantity analogous to the stochastic convenience yield, which accounts for the random changes in the forward curve shape. The well-known cost-of-carry relationship is significantly improved by introducing a deterministic seasonal premium within the convenience yield. We develop model estimation procedures and apply them to a number of energy markets, such as natural gas and electricity.

The paper associated with Prof. Geman’s talk is available as a PDF.


Tuesday 5 February 2008
5:30 pm

Dr Martijn Pistorius (Mathematics Department, King’s College London) and Marc Jeannin (Nomura International plc)

On the pricing and hedging of barrier options driven by additive processes

Two ingredients encountered in  many option-pricing models are (a) stochastic volatility, and (b) jumps, where the latter are needed to capture short maturity option prices while the former is present to enable the model to fit simultaneously options of multiple maturities. The class of Levy processes has been successfully employed to model option prices at single maturities. However, it has been observed by many authors that, due to their rigid term structure of marginal distributions, Levy models are not in general capable of calibrating simultaneously observed option prices across maturities. Here we consider the generalisation of a Levy model by allowing the Levy characteristics to be deterministically time dependent. Restricting ourselves to a specific sub-class of additive processes we present an algorithm for valuing barrier options consistent with a given set of call and put prices. We illustrate the algorithm by simultaneously calibrating the model to Stoxx50E options at four different maturities and then calculating the values and Greeks of down-and-in call options and comparing the outcomes with Monte Carlo simulation results.  

Tuesday 12 February 2008
5:30 pm

Prof Paul Embrechts
ETH, Zurich

Quantitative Modelling of Operational Risk: facts and fantasies

Tuesday 26 February 2008,
5:30 pm

Benjamin Bruder
University Paris 7 (Denis Diderot), and SG Asset Management, Paris

Option pricing with uncertain volatilty and tolerance against losses

Tuesday 4 March 2008,
5:30 pm

Dr Alexander McNeil
Heriot-Watt University, Edinburgh

Tuesday 18 March

No seminar today
(MSc presentations week)


Tuesday 6 May 2008, 5:30pm

Dr Peter Friz

Statistical Laboratory, DPMMS, University of Cambridge

Tuesday 27 May 2008, 5:30pm

Dr Violetta Bernyk

Statistical Laboratory, DPMMS, University of Cambridge
An Optimal Selling Strategy Based on Predicting the Ultimate Maximum Price

Abtract:In this talk I will present an optimal selling strategy for an asset in the following sense. Suppose that an investor has a long position in one financial asset, whose price is modelled by some stochastic process. The investor's objective is to determine a ``best moment'' at which to close out the position before a given time and to sell the asset at the highest possible price, i.e. as close as possible to the ultimate maximum price over the whole period. This optimal decision must be based on continuous observations of the asset price performance and only on the information accumulated to date. Hence, the investor should use a prediction of the future evolution of the price of the financial security.

In the case where the asset price is modelled by a spectrally positive stable Levy process, we describe explicitly the optimal strategy under certain conditions on the model parameters. The optimal strategy is of the following form: the investor must stop the observation of the price process and sell the asset as soon as the associated reflected process crosses for the first time a particular stopping boundary. To this connection we need to derive also the law of the associated supremum process and the latter problem dates back to 1973. Finally we provide numerical estimates and simulation examples of the results obtained by using this strategy.

This is a joint work with Prof. R.C.Dalang, EPFL, Lausanne and Prof. G.Peskir, Manchester University.

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