Summer Term 2010

Artur Sepp (Bank of America Merrill Lynch)  

Martijn Pistorius (Imperial College London) 

Nicholas Westray (Deutsche Bank)
Title: A BSDE approach to the sensitivity of the utility maximization problem 

Abstract: Power utility maximization is a classical problem in mathematical finance that has been solved by both convex duality and BSDE methods. The first part of the presentation will revisit these approaches and demonstrate the exact interrelation between them along with some interesting and relevant counterexamples. Understanding this link allows us to attack the question of sensitivity by extending new results on the stability of quadratic semimartingale BSDEs.  In particular we show that the optimal wealth process and dual optimizer process are continuous with respect to the parameter inputs. If time permits we will discuss the extension to the case of power utility maximization under c on e constraints. Joint work with Markus Mocha Humboldt Universitaet Berlin. 

Jiang Lun Wu (Swansea)
Title: On stochastic differential equations and generalised Burgers equations

Abstract: We will discuss the link of Ito's stochastic differential equations to nonlinear partial differential equations of Burgers type. We will also give some economic interpretation of the result as well as the relevant conditions.

Spring Term 2010


Rama Cont (Columbia)
Title: A Consistent Pricing Model for Index Options and Volatility Derivatives 

Abstract: We propose and study a flexible modeling framework for the joint dynamics of an index and a set of forward variance swap rates written on this index, allowing options on forward variance swaps and options on the underlying index to be priced consistently. Our model reproduces various empirically observed properties of variance swap dynamics and allows for jumps in volatility and returns. An affine specification using Lévy processes as building blocks leads to analytically tractable pricing formulas for options on variance swaps as well as efficient numerical methods for pricing of European options on the underlying asset. The model has the convenient feature of decoupling the vanilla skews from spot/volatility correlations and allowing for different conditional correlations in large and small spot/volatility moves. We show that our model can simultaneously fit prices of European options on S&P 500 across strikes and maturities as well as options on the VIX volatility index. The calibration of the model is done in two steps, first by matching VIX option prices and then by matching prices of options on the underlying. We present examples of calibrations using VIX/ SP500 options using various specifications of the model. 


Lisa Goldberg (MSCI Barra and UC Berkeley)


Elena Boguslavskaya (LSE)
Title: Appell function: the way to solve optimal stopping problems for Levy processes in finite and infinite horizons

Abstract: We present a solution to the optimal stopping problem with finite (or infinite) horizon for a Levy process. We pay special attention to the case when the gain function is analytic and monotone. The main tool in our method is to construct the appropriate Appell function, and, consequently, find the optimal stopping boundary explicitly. Finally, we illustrate our method by several examples.  


Mike Tehranchi (Cambridge) 
Title: Put-call symmetry

Abstract: 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. 


Luciano Campi (Paris-Dauphine) 
Title: A structural risk-neutral model of electricity prices

Abstract: The objective of this paper is to present a model for electricity spot prices and the corresponding forward contracts, which relies on the underlying market of fuels, thus avoiding the electricity non-storability restriction. The structural aspect of our model comes from the fact that the electricity spot prices depend on the dynamics of the electricity demand at the maturity $T$, and on the random available capacity of each production means. Our model explains, in a stylized fact, how the prices of different fuels together with the demand combine to produce electricity prices. This modeling methodology allows one to transfer to electricity prices the risk- neutral probabilities of the market of fuels and under the hypothesis of independence between demand and outages on one hand, and prices of fuels on the other hand, it provides a regression- type relation between electricity forward prices and forward prices of fuels. Moreover, the model produces, by nature, the well-known peaks observed on electricity market data. In our model, spikes occur when the producer has to switch from one technology to the lowest cost available one. Numerical tests performed on a very crude approximation of the French electricity market using only two fuels (gas and oil) provide an illustration of the potential interest of this model. The talk is based on a joint work with René Aid (EDF), Adrien Nguyen Huu (EDF) and Nizar Touzi.

Autumn Term 2009


Damien Lamberton (Marne-La-Vallée)
Title: On the approximation of the supremum of a Lévy process

Abstract: This talk is based on joint work with El Hadj Aly Dia. After some remarks on the distribution of the supremum of a Lévy process, we will present some estimates for the difference between the supremum of a Lévy process and its discrete maximum. We will also discuss the truncation of small jumps. This study is motivated by lookback and barrier options in exponential Lévy models.


Goran Peskir (Manchester)
Title: Selling a Stock at the Ultimate Maximum 

Abstract: I will review recent results on the problem of predicting the maximum when the stock price follows a geometric Brownian motion.


Paul Schneider (Warwick)
Title: Estimation of Nonlinear Diffusion Processes and Applications in Finance

Abstract: We introduce a simple but general continuous-time asset pricing framework that combines semi-analytic pricing with a wealth of flexibility in time-series modelling. Our framework guarantees the existence of the processes used and gives a justification -- for empirical purposes -- to work with nonlinear specifications not considered in the literature so far. Additional flexibility turns out to be econometrically relevant: a nonlinear stochastic volatility diffusion model for the joint time-series of the S&P 100 and the VXO implied volatility index data shows superior forecasting power over standard specifications for implied--, and realized variance forecasting. Joint work with A. Mijatovic.


James Gleeson (Limerick)
Title: Cascades on random networks

Abstract: Network models may be applied to many complex systems, e.g. the Internet, the World Wide Web, inter-bank lending networks, etc. Cascade dynamics can occur when the (binary) state of a node is affected by the states of its neighbours in the network. Such models have been used to aid understanding of the spread of cultural fads and the diffusion of innovations, and can be generalized to include percolation problems, k-core sizes, and disease spread on networks. For this class of problems, I present recent results on the analytic determination of the expected size of cascades on networks of arbitrary degree distribution, and outline some extensions of this research. Application to models of contagion and systemic risk within banking networks will also be discussed (joint work with Sébastien Lleo and Tom Hurd). 


Umut Cetin (LSE)
Title: Dynamic Markov bridges motivated by models of insider trading

Abstract: Given a Markovian Brownian martingale Z, we build a process X which is a martingale in its own filtration and satisfies X(1) = Z(1). We call X a dynamic bridge, because its terminal value Z(1) is not known in advance. We compute explicitly its semimartingale decomposition under both its own filtration and the filtration generated jointly by X and Z. Our construction is heavily based on parabolic PDE's and filtering techniques. As an application, we explicitly solve an equilibrium model with insider trading, that can be viewed as a non-Gaussian generalization of Back and Pedersen, where insider's additional information evolves over time (joint work with L. Campi and A. Danilova). 


Jose Da Fonseca (Auckland)
Title: Riding on the Smiles

Abstract: This paper investigates the calibration performance of several multifactor stochastic volatility models. There is an empirical evidence that the dynamics of the implied volatility surface is driven by several factors. This leads to the extensions of the seminal Heston stochastic volatility model. Using a data set of derivatives on the major indices we study the calibration properties of these models using the FFT as the pricing methodology. We also study if adding jumps improves significantly the calibration accuracy of the models. We explain the advantages of the Wishart based stochastic volatility model when dealing with stochastic correlation risk. Then we focus on basket option pricing models and more precisely on the WASC model (Wishart Affine Stochastic Correlation) proposed by [Da Fonseca, Grasselli and Tebaldi, 2007]. We analyse the calibration property of this model and compare it with another model based on the Heston model. Finally, we provide some price approximations for vanilla options in the spirit of [Benabid, Bensusan and El Karoui, 2009] that are very useful to speed up the pricing process thus leading to reasonable calibration time. Also we provide some results on Malliavin calculus that allow for efficient computation of the sensitivities for derivative products in our models. 


Peter Carr (NYU & Bloomberg)
Title: What does an option price mean?

Abstract: It is well known that the market price of a standard option reflects the risk-neutral mean of its path-independent payoff. It is less well known that this same option price also reflects the risk-neutral mean of various path-dependent payoffs. We give several examples of such payoffs which together suggest that option prices convey much more information than one might initially expect. 


Vlad Bally (Marne-la-Vallée)
Title: Tube estimates for Itô processes and applications to stochastic volatility models

Abstract: We give lower bounds for the probability that an Itô process stays in a tube around a deterministic curve up to the time T. This in particular gives lower bounds for arriving in a ball around a specific point in the state-space at time T. We apply these estimates to a wide class of stochastic volatility models with local (i.e. non-constant) coefficients. Such models have become very common in financial markets because they are capable of calibrating perfectly to the implied volatility surface while retaining the desired dynamics of the spot process. We also prove that the moments of the stock in such a model can blow up in finite time (this is already known for the classical Heston model with constant coefficients). Finally we give estimates for the density of the law of the stock.