FINANCIAL MATHEMATICS ACTIVITIES

November 27, 2014

New Directions in Financial Risk Management

Titles and Abstracts


Andrew Aziz,
Vice President, Professional Services, Algorithmics Inc.

Title: A Unified Framework for Enterprise Risk and Reward

Abstract: It is clear that it makes no sense for financial institutions to price, hedge, measure risk, allocate capital and compute asset allocations using different methods based on different assumptions.. Yet that is what happens today. There is no mathematical consistency in most financial institutions. We believe that consistency can be achieved and if implemented, has great value. This talk is about our Mark-to-Future, the culmination of our efforts in this direction.

Phelim Boyle, University of Waterloo

Title: Pricing new securities in an incomplete market: The catch 22 of derivative pricing

Abstract: There are two radically different approaches to the valuation of a new security in an incomplete market. The first approach is common in the mathematical finance literature. This method takes the prices of the existing traded securities as fixed and uses no arbitrage arguments to derive the set of equivalent martingale measures that are consistent with the initial prices of the traded securities. The price of the new security is obtained by invoking some preference assumption or appealing to certain arbitrary criteria. The second method prices the new secturity using a general equilibrium approach. In this case the prices of all the existing securities will normally change. This talk clarifies the distinction between the two approaches and also outlines a proof that the introduction of the new security will typically change the prices of all the existing securities.


Hélyette Geman, Université Paris IX Dauphine and ESSEC

Title: Stochastic Time Changes, Lévy Processes and Asset Price Modelling

Abstract: We investigate the relative importance of diffusion and jumps in a new jump diffusion model for asset returns. In contrast to the standard modelling of jumps for asset returns, the jump component of our process can display finite or infinite activity, and finite or infinite variation. Empirical investigations of time series indicate that index dynamics are essentially devoid of a diffusion component, while this component may be present in the dynamics of individual stocks. This result leads to the conjecture that the risk-neutral process should be free of a diffusion component for both indices and individual stocks. Empirical investigation of options data tends to confirm this conjecture. We conclude that the statistical and risk-neutral processes for indices and stocks tend to be pure jump processes of infinite activity and finite variation.


Lane P. Hughston, King's College, London

Title: Entropy and Information in the Interest Rate Term Structure

Abstract: Associated with every positive interest term structure there is a probability density function over the positive half line. This fact can be used to turn the problem of term structure analysis into a problem in the comparison of probability distributions, an area well developed in statistics, known as information geometry. The key idea is to take the square-root of the density function, which embeds the space of densities into a Hilbert space. As a consequence, Hilbert space operations can be employed to study the structure of interest rate models. Some of the information-theoretic and geometric aspects of term structures thus arising will be illustrated. In particular, we introduce a new term structure calibration methodology based on maximisation of entropy, and also present some new families of interest rate models arising naturally in this context. (Joint work with D. C. Brody, Imperial College, London)


Dilip B. Madan, Robert H. Smith School of Business, University of Maryland

Title: Stochastic Volatility for Levy Processes

Abstract: Three processes reflecting persistence of volatility are formulated by evaluating three Levy processes at a time change given by the integral of a square root process. Stock price models are considered by exponentiating and mean correcting these processes or by stochastically exponentiating these processes. The resulting chracteristic functions for the log price yield option pricing systems via the fast Fourier transform. Results on index options and single name options suggest advantages to employing higher dimensional Levy systems for index options and lower dimensional structures for single names. In general mean corrected exponentiation performs better than employing the stochastic exponential. Martingale laws for the mean corrected exponential are also studied and two new concepts termed Levy and martingale marginals are introduced.


Moshe Milevsky, Schulich School of Business, York University

Title: Personal Value-at-Risk: An Overview of the Real Options in Your Life.

Abstract: In this presentation, I will describe how many of the quantitative techniques originally developed in the area of corporate risk management, can be applied to the individual vis a vis their portfolio of human and financial capital. I argue that consumers are naturally endowed with Real Options to 'time' various irreversible 'investments' that share a striking resemblance to classical decisions in corporate finance. Whether or not these options should be priced risk-neutrally is quite independent of the fact that they exist. I will provide some examples as they relate to insurance and asset allocation decisions over the life-cycle, and discuss some of the interesting stochastic modeling problems that emerge....


Marcel Rindisbacher, Joseph L. Rotman School of Management, University of Toronto

Title: A Monte-Carlo Method for Optimal Portfolios

Abstract: This paper provides (i) simulation-based approaches for the computation of asset allocation rules,(ii) economic insights on the behavior of the hedging components and (iii) a comparison of numerical methods.

For general utility functions with wealth-dependent risk aversion and diffusion state variable processes, hedging demands are conditional expectations of random variables depending on the parameters of the model, which can be estimated using standard simulation methods. We propose a modified simulation approach which relies on a simple transformation of the underlying state variables an improves the performance of Monte Carlo estimators of portfolio rules. Our approach is flexible and applies to arbitrary utility functions, any finite number of state variables, general diffusion processes for state variables and any finite number of assets. The procedure is implemented for a class of multivariate nonlinear diffusions for the market price of risk (MPR), the interest rate (IR) and other factors (such as dividends). After calibrating the models to the data we document the portfolio behavior. Intertemporal hedging demands are found to (i) significantly increase the demand for stocks and (ii) exhibit low volatility. (iii) Non-linearities are shown to be important: significant biases in allocation rules are documented if one uses typical (affine/or square root) processes calibrated to the data. (iv) Dividend predictability is found to have negligible effect. Portfolio policies are also computed and examined for (v) HARA utility functions and (vi) in markets with a large number of state variables and mutual funds.


Luis Seco, Mathematics, University of Toronto

Title: Mark-to-Future in non-gaussian markets.

Abstract: After the initial success of the RiskMetrics methodology, effective risk management practices in non-gaussian markets leads to a number of interesting mathematical questions. This talk will review a number of business situations and the associated mathematical tools relevant to each of them.