SCIENTIFIC PROGRAMS AND ACTIVITIES

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

March 16
Fields, Room 230

*Please note non-standard location

Tom Hurd, Department of Mathematics, McMaster University
Modelling financial networks and systemic risk

The study of "contagion" in financial systems, that is, the spread of defaults through a system of financial institutions, is very topical these days. In this talk I will address how mathematical models can help us understand systemic risk. After reviewing the basic economic picture of the financial system as a random graph, I will explore some useful "deliberately simplified models of systemic risk".

February 9

Sebastian Ferrando, Department of Mathematics, Ryerson University
Trajectory Based Pricing and Arbitrage Opportunities

Assuming as given a trajectory/path space, we define an associated market model and a notion of pricing interval and also describe how to obtain arbitrage free results for these models. These notions and results are purely analytical and do not depend on a probabilistic assumption. We indicate how one can also use the results to obtain arbitrage free results for non semi-martingale models. If time permits, we will present a simple, but practical, model that allow us to compute the pricing interval and to compare to market data. We report on preliminary numerical findings related to realizable arbitrage opportunities (as seen from our model) even when transaction costs are taken into consideration.

November 21

Elias Shiu, Department of Statistics and Actuarial Science, University of Iowa
Valuing Equity-Linked Death Benefits: Option Pricing Without Tears

Currently, a major segment of U.S. life insurance business is the variable annuities, which are investment products with (exotic) options and insurance guarantees. Many of these options and guarantees should be priced, hedged, and reserved using modern option-pricing theory, which involves sophisticated mathematical tools such as martingales, Brownian motion, stochastic differential equations, and so on. This talk will show that, if the guarantees or options are exercisable only at the moment of death of the policyholder and the underlying asset price is a geometric Brownian motion, the mathematics simplifies to an elementary calculus exercise. A key step behind this method is that the probability density function of the time-until-death random variable can be approximated by a combination of exponential densities.

This is joint work with Hans U. Gerber of the University of Lausanne and Hailiang Yang of the University of Hong Kong.

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Gordon Willmot, Department of Statistics and Actuarial Science, University of Waterloo
On mixing, compounding, and tail properties of a class of claim number distributions

The mathematical structure underlying a class of discrete claim count distributions is examined in some detail. In particular, the mixed Poisson nature of the class is shown to hold fairly generally. Using some ideas involving complete monotonicity, a discussion is provided on the structure of other class members which are well suited for use in aggregate claims analysis. The ideas are then extended to the analysis of the corresponding discrete tail probabilities, which arise in a variety of contexts including the analysis of the stop-loss premium.

Peter Forsyth, Cheriton School of Computer Science, University of Waterloo
Comparison between the Mean Variance and the Mean Quadratic Variation optimal trading strategies

We compare optimal stock liquidation policies in continuous time in the presence of trading impact using numerical solutions of Hamilton Jacobi Bellman (HJB)partial differential equations (PDE). In particular, we compare the time-consistent mean-quadratic-variation strategy (Almgren and Chriss) with the time-inconsistent (pre-commitment) mean-variance strategy. We show that the two different risk measures lead to very different strategies and liquidation profiles. In terms of the mean variance efficient frontier, the original Almgren/Chriss strategy is signficently sub-optimal compared to the (pre-commitment) mean-variance strategy.

This is joint work with Stephen Tse, Heath Windcliff and Shannon Kennedy.

Lane P. Hughston, Imperial College London
General Theory of Geometric Lévy Models for Dynamic Asset Pricing

The theory of Lévy models for asset pricing simplifies considerably if we take a pricing kernel approach, which enables one to bypass market incompleteness issues. The special case of a geometric Lévy model (GLM) with constant parameters can be regarded as a natural generalisation of the standard geometric Brownian motion model used in the Black-Scholes theory. In the one-dimensional situation, for any choice of the underlying Lévy process the associated GLM model is characterised by four parameters: the initial asset price, the interest rate, the volatility, and a risk aversion factor. The pricing kernel is given by the product of a discount factor and the Esscher martingale associated with the risk aversion parameter. The model is fixed by the requirement that for each asset the product of the asset price and the pricing kernel should be a martingale. In the GBM case, the risk aversion factor is the so-called market price of risk. In the GLM case, this interpretation is no longer valid as such, but instead one finds that the excess rate of return is given by a non-linear function of the the volatility and the risk aversion factor. We show that for positive values of the volatility and the risk aversion factor the excess rate of return above the interest rate is positive, and is monotonically increasing in the volatility and in the risk aversion factor. In the case of foreign exchange, we know from Siegel's paradox that it should be possible to construct FX models for which the excess rate of return (above the interest rate differential) is positive both for the exchange rate and the inverse exchange rate. We show that this condition holds for any GLM for which the volatility exceeds the risk aversion factor. Similar results are shown to hold for multiple-asset markets driven by vectorial Lévy processes, and for market models based on certain more general classes of Lévy martingales.

(Work with D. Brody, E. Mackie, F. Mina, and M. Pistorius.)

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Andrea Macrina, King’s College London
Randomised Mixture Models for Pricing Kernels

Numerous kinds of uncertainties may affect an economy, e.g. economic, political, and environmental ones. We model the aggregate impact of the uncertainties on a financial market by randomised mixtures of Levy processes. We assume that market participants observe the randomised mixtures only through best estimates based on noisy market information. The concept of incomplete information introduces an element of stochastic filtering theory in constructing what we term “filtered martingales”. We use this martingale family and apply the Flesaker-Hughston scheme to develop interest rate models. The proposed approach for pricing kernels is flexible enough to generate a variety of bond price models of which associated yield curves may change in level, slope, and shape. The choice of random mixtures has a significant effect on the model dynamics. Parameter sensitivity is analysed, and bond option price processes are derived. We extend the pricing kernel models by considering a weighted heat kernel approach, and establish the link to the interest rate models driven by the filtered martingales.

(In collaboration with P. A. Parbhoo)