Q Group Australia 2000 Colloquium

Agenda

Abstracts

Ron Bird and/or John McKinnon

Change in the Behaviour of Earnings Surprise: Evidence and Implications

Alan Brace

Exact Fit to the Swaption Volatility Matrix using Semidefinite Programming Alan Brace & Rob Womersley

This paper address the problem of parametrizing jointly to all swaptions in the lognormal libor market model. It is first shown that swaprates are nearly lognormal with a determinisitc volatility. We then work directly with covariance matrices rather than then the usual two variable volatility function. Semidefinite programming, in which the variables are positive semidefinite matrices (covariance matrices), minimizes a linear function of the variables subject to linear constraints on the variables. Exact fits to the swaption volatility matrix are imposed through linear constraints on the variables. Various objectives can be used to get close to a historical covariance matrix. The method is illustrated on the pricing of Bermudan options.

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Carl Chiarella

Asset Price Dynamics with Heterogeneous Beliefs, Risk and Learning
Chiarella Carl and He Xue-Zhong (Tony)

Dynamic economic theory has long been dominated by the paradigm of the rational representative agent and homogeneous beliefs. It is really only in the last decade that economic theorists have started to come to grips with models populated by heterogeneous agents with differing attitudes to risk, differing beliefs about possible future outcomes and who are trying to learn about their economic environment. This talk will focus on models that seek to incorporate heterogeneity, beliefs, risk and learning into the paradigmatic asset pricing model of finance theory. The talk will initially seek to give some overview on the various models that have been developed as well as the technical apparatus used to analyse them. Broadly speaking these may be categorised as computational or analytical though many analyses involve both elements, as does the appear on which this talk is based. The talk will then focus on the ongoing research of the authors that seeks to characterise the stability properties of the asset price dynamics model with heterogeneous agents having differing attitudes to risk and using various learning schemes. There will also be some discussion about the effect of different mechanisms for arriving at the market price in each time period. The model that is finally developed displays the essential characteristics of the standard asset price dynamics model assumed in continuous time finance in that the asset price is fluctuating around an exponentially growing trend.

Chiarella Carl, School of Finance and Economics, University of Technology, Sydney, Australia
He Xue-Zhong (Tony), School of Finance and Economics, University of Technology, Sydney, Australia

Michael Kelly

Single Parameter Semi-Analytical Solutions to the American Put Option

The most common of the Exotic Options is the American Put which allows early exercise. The property of early exercise makes the option path dependent. This dependency corresponds to the problem of identifying the free or moving boundary of the associated second order PDE which describes the option and its constraints. It has been shown that within the Black-Scholes framework it is possible to identify the free boundary as the solution to a particular saltus problem (Kolodner, 1956), and that either by Fourier transform methods (McKean, 1965) or by the Volume Potential Method (Jacka, 1989;Jamshidian, 1990) this results in a recursive second order Volterra integral equation.

There are no known methods for solving such integral equations unless one has a model for the structure of the free boundary. Using the near expiry models suggested by Carr et al (1992), Barles (1995), Wilmott et al (1997) and Kruske and Keller (1998) we propose a single parameter family of curves for the moving boundary. It will be shown that this is capable of a quick (~1 second) numerical solution using either Newton-Raphson or interpolation methods and that the results are within penny accuracy of the binomial tree and finite difference methods, although the latter require on average 100 seconds for similar accuracy.

Using the work of William Shaw (1998) it will also be shown that the Carr and Jamshidian integral equations for the free boundary can be rewritten as recursive algebraic equations in terms of a free parameter related to the Black and Scholes k1 parameter. All solutions are done using the equation solving functions of Mathematica. Unlike the intensive numerical approaches currently favoured, these solutions afford an insight into the underlying structure of the free boundary which is the heart of the American Put problem.

Wendy McTainish

Measuring market risk in electricity markets - translating financial market concepts

Value at Risk (VaR) as a concept was introduced into financial markets in the mid-90's and popularised by the JP Morgan risk-metrics approach. It essentially aims to measure how much a trading book can lose in a worst case scenario. Major banks world-wide have introduced sophisticated modelling tools to measure VaR and assign capital for this risk.

With the introduction of the competitive electricty markets, a market where price volatility can be extreme, the concept was adopted by almost all major players in the markets. Its use has had varied success as a management or trading tool and the reasons for this is the mathematical complexity involved. This paper will outline what has been done in the electricity markets, what has and has not been successful, and outline some practical alternatives for going forward. These issues present some interesting challenges to the quantitative community.

Marek Rutkowski

Multiple rating model of defaultable term structure
(Tomasz Bielecki and Marek Rutkowski)

A new approach to modelling of credit risk, to valuation of defaultable debt, and to pricing of credit derivatives is developed. Our approach, based on the HJM methodology, uses the available information about the credit spreads combined with the available information about the recovery rates to model the intensities of credit migrations between various credit ratings classes. This results in a conditionally Markovian model of credit risk. We then combine our model of credit risk with an HJM based model of interest rate risk in order to derive an arbitrage-free model of defaultable bonds. As expected, the market price processes of interest rate risk and credit risk provide a natural connection between the actual and the martingale probabilities.

References
Bielecki, T. and Rutkowski, M. (1999) \Defaultable Term Structure: Conditionally Markov Approach." Working paper.
Bielecki, T. and Rutkowski, M. (2000) \Multiple Ratings Model of Defaultable Term Structure." Mathematical Finance 10, 125{139.
Bielecki, T. and Rutkowski, M. (2000) \HJM with Multiples." Risk 13(4), 95{97.

The full writeup is available in three versions (partially overlapping) of the joint research with Tom Bielecki. Paper 1 Paper 2 Paper 3.

These are available in PDF format. Download the Adobe reader from the internet.

Erik Schloegl

A multi-currency extension of the lognormal interest rate market models

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Glenn Solomon and Heather Ward

After-tax performance - Assessing the the real performance of Investment Managers

Most investors pay tax, and certainly all superannuation funds are subject to a tax on earnings - normally at 15%. Currently much of the focus on performance is on before tax and fees calculations. The primary reason for this is comparability, although it might also be attributed, at least in part, to convenience. Current trends and recent tax changes in the industry, however, have served to strengthen the call to measure investment performance after tax.

The P Group has created a sub-committee, chaired by Heather Ward, to look at the issues involved and develop a standard approach suitable for widespread use in the industry. A first draft of their work is nearing completion and will be summarised and discussed with the Q Group in this session.

The P-Group (Performance Analyst Group) officially became part of IFSA, as a forum under the Investment Committee. The P-Group is chaired by Glenn Solomon from AMP Investment Administration and the Vice-Chair is Heather Ward from Equitilink.

The charter of the P-Group is a forum for investment professionals to discuss and develop ideas, concepts and theories relating to performance measurement and attribution, and produce working papers for submission to industry associations and regulators.

Joe Winsen

Statistical Analysis of NSW Electricity Prices
(L. Wilkinson and J. Winsen, Macquarie Generation)

Electricity is now traded with prices set every half-hour. The behaviour of these prices is of interest to those wanting to price electricity derivatives such as caps and swaptions. In NSW the industry has traditionally grouped electricity pricing into 4 day of the week buckets (Mondays, Tuesdays to Fridays, Saturdays, Sundays and Public holidays) and into 2 or 3 time of day buckets - off-peak, peak, and sometimes shoulder (more recently shoulder has been combined with peak). Peak is working weekdays 7am to 9am and 5pm to 8pm; shoulder is working weekdays 9am to 5pm; off-peak is all other times, including all times at weekends and public holidays.

In this paper, we evaluate these traditional classifications by examining data for the 12 months 1/4/98 to 31/3/99 (over 17,500 prices) and for the two "seasons" 1/4/98 to 30/9/98 and 1/10/98 to 31/3/99. While there are two shoulder seasons, these vary from year to year (it is difficult to specify in advance the start of summer and winter). While demand data appears to fall neatly into the traditional classifications, the introduction of an electricity market (the NEM) has created uncertainty about the supply side response to demand, so that while prices often follow demand there are frequent exceptions.

We also examine the term structure of volatility (mean reversion creates serial correlation so that volatility no longer grows with the square root of time) and test for lognormality of the data.

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Prepared by Chris Condon