Summer term 2011

Title: Some non-Markovian control problems arising in order execution
Alexander Schied (University of Mannheim)
Wednesday, 11 May, 5:30pm, Room 139, Huxley Building

Abstract: We discuss several finite-fuel control problems that arise in connection with the optimal execution of orders under a risk criterion. These problems can be non-Markovian either through the price process, which may be a general semimartingale, or directly through the cost criterion, which may involve delay terms. In some cases, these problems can be solved explicitly via verification techniques. 

Title: An infinitely divisible distribution function useful for the cosmological many-body problem and (?) the financial many-body problem
Bill Saslaw (University of Virginia & Cambridge)
Wednesday, 18 May, 5:30pm, Room 139, Huxley Building

Title: Local volatility pricing models for long-dated derivatives in finance and insurance
Griselda Deelstra (Université Libre de Bruxelles) 
Wednesday, 25 May, 5:30pm, Room 139, Huxley Building 

Abstract: We study the local volatility function in the Foreign Exchange market where both domestic and foreign interest rates are stochastic. This model is suitable to price long-dated FX derivatives. We derive the local volatility function and obtain several results that can be used for the calibration of this local volatility on the FX option's market. We further concentrate upon Guaranteed Annuity Options (GAO), which give the right at the policyholder to convert his accumulated funds to a life annuity either at a rate based on the current market rates or at a fixed rate called the Guaranteed Annuity rate. We derive prices of GAO’s in the settings of a two-factor pricing model where the equity is locally governed by a geometric Brownian motion with a local volatility, while the interest rate follows a Hull-White one-factor Gaussian model.

Title: American option pricing via a probabilistic penalty method
Jan Palczewski (University of Warsaw and University of Leeds)
Wednesday, 1 June, 5:30pm, Clore LT (Level 2), Huxley Building

Abstract: In my talk I will present results concerning optimal stopping of functionals which exhibit discontinuity of the reward function at the boundary of an open (possibly unbounded) set. This type of functionals naturally arises when pricing American style options whose payoff changes abruptly when stock prices leave a given region. Of particular interest to me will be the continuity of the value function and the existence and form of optimal stopping times. These properties will be explored using the penalty method, an approach widely used in proving smoothness of solutions to QVI's. I will, however, apply penalization at the level of functionals, a method dating back to Robin (1978) and later mostly forgotten. Unjustly, as it works under weak assumptions (weakly Feller underlying processes) and gives rise to approximations of both the value function and an optimal stopping time. I will conclude with a discussion of numerical methods. This talk is partly based on a joint work with Lukasz Stettner.

Autumn term 2010-2011

Title: Zeta processes and their financial applications
Lane Hughston (Imperial College London)
Wednesday, 13 October, 5:30pm, Room 139, Huxley Building

Abstract: 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 behaviour. It makes sense therefore to consider financial contracts for which the payoff is represented by a random variable of that type. This talk will present an overview of some of the properties of the zeta distribution and the associated multipicative Lévy process, which we shall call the zeta process, with a view to financial applications. 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. (Based on work with D. Brody, S. Lyons and M. Pistorius.)

Title: Stochastic differential games and applications to energy consumer goods market
Ronnie Sircar (Princeton)
Wednesday, 20 October, 5:30pm, Room 139, Huxley Building

Abstract: We discuss Cournot and Bertrand models of oligopolies, first in the context of static games and then in dynamic models. The static games, involving firms with different costs, lead to questions of how many competitors actively participate in a Nash equilibrium and how many are sidelined or blockaded from entry. The dynamic games lead to systems of nonlinear partial differential equations for which we discuss asymptotic and numerical approximations. Applications include competition between energy producers in the face of exhaustible resources such as oil (Cournot); and markets for substitutable consumer goods (Bertrand). Joint work with Chris Harris, Sam Howison and Andrew Ledvina.

Title: Calibration of chaotic models for interest rates
Matheus Grasselli (McMaster, Ontario, Canada)
Wednesday, 27 October, 5:30pm, Room 139, Huxley Building

Abstract: The Wiener chaos approach to interest rates was introduced a few years ago by Hughston and Rafailidis as an axiomatic framework to model positive interest rates, continuing a line of research started by the seminal Flesaker and Hughston model and including the elegant potential approach of Rogers and others. Apart from ensuring positivity, one appealing feature of the chaotic approach is its hierarchical way t o introduce randomness into a model through different orders of chaos expansions. We propose a systematic way to calibrate Wiener chaos models to market data, and compare the performance of chaos expansions of different orders& nbsp;with popular interest rate models in the presence of interest rate derivatives of increased complexity. This is joint work with Tsunehiro Tsujimoto. 

Title: Portfolio optimisation for general investor risk-return objectives and distributions
William Shaw (King's College London)
Wednesday, 3 November, 5:3 0pm, Room 139, Huxley Building

Abstract: We consider the problem of optimizing a gene ral investor objective (MV, Sharpe, VaR, CVaR , Utility, Omega, Behavourial Prospect....) with no restrictions on the termin al distributions of the asse ts comprising a portfolio. The sol ution proposed, initially for long-only portfolios of small to modest dimension, is based on introducing an efficient random sampling of the simplicial structures characterizing portfolio configurations. The sample may be optimized in combination with a treatment of risk functions that are either simple analytical objects or entities also requiring Monte Carlo simulation of the return distribution. Examples will be given. Further details are available at: ssrn.com/abstract=1680224 

Title: Option valuation in a general stochastic volatility model
Nick Webber (Warwick Business School)
Wednesday, 10 November, 5:30pm, Room 139, Huxley Building

Abstract: Stochastic volatility models are frequently used in the markets to model the implied volatility surface. These models have several failings. Firstly, although improvements on a basic Black-Scholes model, they nevertheless fail to fit the entire surface adequately. Secondly, the improvements they offer are usually at the cost of greatly reduced tractability. Thirdly,  these models still fail to fit to market prices of non-vanilla securities. This paper addresses the second of these three issues. A general stochastic volatility model is described, nesting both the Heston model and a Sabr-related model. A control variate Monte Carlo valuation method for this model is presented that, when it can be applied, is shown to be a significant improvement over existing simulation methods; when applied to barrier op tion pricing, it out-performs importance sampling methods. By providing a plausible simulation method for this general model, the paper opens the possibility of exploring calibration to non-vanilla, as well as vanilla, instruments.

Title: Optimal exercise of portfolios of American options
Vicky Henderson (Mathematical Institute, Oxford)
Wednesday, 17 November, 5:30pm, Room 139, Huxley Building

Abstract: We consider the optimal exercise of a portfolio of American call options in an incomplete market. Options are written on a single underlying asset but may have different characteristics of strikes, maturities and vesting dates. Our motivation is to model the decision faced by an e mployee who is granted options periodically on the stock of her company, a nd who is not permitted to trade this stock. We first show how the exercise of a single option depends upon the option's characteristics. The main result is that options with co-monotonic strike, maturity and vesting date should be exercised in order of increasing strike. Our results are model free (we do not specify a stock price process) and require only weak assumptions on preferences. Utility indifference pricing is a particular example and we solve the resulting dynamic programming problem under CARA to illustrate the portfolio exercise ordering results. Joint work with Jia Sun and Elizabeth Whalley (WBS).

Title: BSDEs and nonlinear expectations in general probability spaces
Samuel Cohen (Mathematical Institute and St John's College, Oxford)
Wednesday, 24 November, 5:30pm, Room 139, Huxley Building

Abstract: Since the first work in the early 1990's, the theory of nonlinear Backward Stochastic Differential Equations (BSDEs) has found numerous applications in Mathematical Finance and Optimal Control. Typically, this is done in the context of a Brownian filtration. We consider Backward Stochastic Differential Equations in probability spaces with a general filtration, in either discrete or continuous time. We show existence and uniqueness of solutions to these equations, and prove a comparison theorem. Using these results, we can construct filtration consistent nonlinear expectations (for any filtration), or equ ivalently, time-consistent dynamic risk measures. In discrete time, we show that every nonlinear expectation must be of this form.

Title: Complexity, concentration and contagion
Sujit Kapadia (Bank of England)
Wednesday, 8 December, 5:30pm, Room 139, Huxley Building

Abstract: This paper develops a model of the interbank network in which unsecured claims and obligations, repo activity and shocks to the haircuts applied to collateral assume centre stage. We show how systemic liquidity crises of the kind associated with the interbank market collapse of 2007-8 can arise within such a framework, with contagion spreading widely through the web of interlinkages. And we illustrate how greater complexity and concentration in the financial network may contribute to fragility. We then suggest how a range of policy measures - including tougher liquidity regulation, macro-prudential policy, and surcharges for systemically important financial institutions - may make the financial system more resilient.

Spring term 2011

Title: Variance derivatives and estimating realised variance from high-frequency data
John Crosby (UBS and Glasgow University)
Wednesday, 19 January, 5:30pm, Room 139, Huxley Building

Abstract: We introduce generalised variance swaps, which generalise and include, as special cases, popular derivatives such as variance swaps and gamma swaps. We also consider other types of volatility-related derivatives such as proportional variance swaps and skewness swaps. We obtain pricing formulae for both discretely monitored swaps and continuously monitored swaps and discuss the convergence of the former to the latter as the number of monitoring dates is increased. We discuss the implications for trading and hedging these instruments. The use of high-frequency data to estimate realised variance (and, more generally, to estimate model parameters) has been a “hot” topic in finance in recent years. We discuss the implications of our work on this topic focussing, in particular, upon the “leverage effect”. This is joint work with Mark Davis in the Mathematics Department at Imperial College London.

Title: The uniqueness of solutions and the martingale property of asset prices
Hao Xing (LSE)
Wednesday, 9 February, 5:30pm, Room 139, Huxley Building

Abstract: We report a natural connection between the uniqueness of solutions to valuation equations for European options and the martingale property of asset prices: the option value function is the unique solution if and only if the asset price is a martingale under a risk-neutral measure. This connection is demonstrated in local volatility models first, then in a general framework of stochastic volatility models, which allows various model behaviour, for example, the volatility process potentially  reaches zero and either stay there or instantaneously reflect. Combining with necessary and sufficient conditions on the martingale property of asset prices, this connection generalizes classical result s in PDE onthe uniqueness of classical solutions. It also links several different& n bsp;aspects: stock price bubbles, local martingales, multiple solutions for PDEs, and boundary point c lassification for diffusions. This is a joint work with Erhan Bayraktar and Kostas Kardaras.

Title: Utility theory front to back --- inferring utility from agents' choices
Jan Obloj (Oxford)
Wednesday, 23 February, 5:30pm, Room 139, Huxley Building

Abstract: We pursue an inverse approach to utility theory and consumption & investment problems. Instead of specifying agents' utility function and deriving their actions, we assume we observe their actions (i.e. the consumption and investment strategies) and ask if it is possible to derive a utility function for which the observed behaviour is optimal. We work in continuous time both in a deterministic and stochastic setting. In a deterministic setup, we find that there are infinitely many utility functions generating a given consumption pattern. In the stochastic setting of the Black-Scholes complete market it turns out that the consumption and investment strategies have to satisfy a consistency condition (PDE) if they come from a classical utility maximisation problem. We further show that agent's important characteristics such as attitude towards risk (e.g.\ DARA) can be directly deduced from her consumption/investment choices.

Special Lecture: Counterexamples in Probability
Jordan Stoyanov (Newcastle)
Tuesday, 1 March, 5:00pm, Room 139, Huxley Building

Title: Some results and open questions for distributions used in stochastic finance
Jordan Stoyanov (Newcastle)
Wednesday, 2 March, 5:30pm, Room 139, Huxley Building

Title: Heat kernel approach in finance and its applications
Takahiro Tsuchiya (Ritsumeikan University, Kyoto)
Wednesday, 9 March, 4:00pm, Room 140, Huxley Building

Title: In search of the Minsky moment: credit dynamics, asset price bubbles, and financial fragility
Matheus Grasselli (McMaster, Ontario, Canada)
Wednesday, 9 March, 5:30pm, Room 139, Huxley Building

Abstract: Hyman Minsky's main contribution to economics - the financial insta b ility hypothesis - links the expansion of credit to fund new investment to the increase in asset prices and the inherent fragility of an over-leveraged financial system. In this talk I first briefly review the economic literature on asset price bubbles, in particular a model where investors overbid when buying assets with borrowed funds. In the second part, I describe an agent-based model for the emergence of a banking system in a society with random liquidity preferences. Finally in the third part of the talk, I discuss a three-dimensional dynamical system for wages, employment and debt exhibiting locally stable but globally unstable equilibria. Put together, these three ingredients constitute a first pass at a comprehensive dynamic model for endogenous formation and crash of asset price bubbles.

Special Seminar in Mathematical Physics
Prof. Sir Michael Berry (Bristol)
Wednesday, 23 March, 5:00pm, Room 140, Huxley Building