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.