Events 2004-2005

Professor Steven Kou

Columbia University

Credit Spreads, Optimal Capital Structure and Implied Volatility with Endogenous Default and Jump Risk

We propose a two-sided jump model for credit risk by extending the Leland-Toft endogenous default model based on pure diffusion. The model shows that jump risk and endogenous default can have significant impacts on credit spreads, optimal capital structure, and implied volatility of equity options: (1) The jump and endogenous default can produce a variety of non-zero credit spreads, including upward, humped, and downward shapes; interesting enough, the model can even produce, consistent with empirical findings, upward credit spreads for speculative grade bonds. (2) The jump risk leads to much lower optimal debt/equity ratio; in fact, with jump risk highly risky firms tend to have very little debt. (3) The two-sided jumps lead to a variety of shapes for implied volatility curves of equity options, even for long maturity options; and in general credit spreads and implied volatility tend to move in the same direction with jump risk and exogenous default, but this may not be true in presence of both jump risk and endogenous default. In terms of mathematical contribution, we give a proof of a version of "smooth fitting" principle for the endogenous boundary for the jump model, justifying a conjecture first suggested by Leland and Toft in the setting of the Brownian model.

Professor David Hobson

University of Bath

The Curious Incident of the Investment in the Market: Real Options and a Fair Gamble

"Is there any point to which you would wish to draw my attention?"
"To the curious incident of the investment in the market."
"The agent did nothing in the market."
" "That was the curious incident." (with apologies to Sir Arthur Conan-Doyle.)
In this paper we study an optimal timing problem for the sale of a non-traded real asset. We solve this problem for a utility maximizing, risk averse manager under two scenarios: firstly when the manager has access to no other investment opportunities, and secondly when they may also invest in a continuously traded financial asset. We construct the model such that the financial asset is uncorrelated with the real asset, so that it cannot be used for hedging, and such that the financial asset has zero risk premium. In the absence of the real asset, the manager would not include the financial asset in her optimal portfolio. The traded asset represents a fair gamble. Although the problem is designed such that naive intuition would imply that the optimal strategy and value functions are the same irrespective of whether the manager is allowed to invest in the financial asset, we find that curiously, for certain parameter values this is not the case.

Professor Steven E Shreve

Department of Mathematical Sciences, Carnegie Mellon University

A Two-Person Game for Pricing Convertible Bonds

A firm issues a convertible bond. At each subsequent time, the bondholder must decide whether to continue to hold the bond, thereby collecting coupons, or to convert it to stock. The bondholder wishes to choose a conversion strategy to maximize the bond value. Subject to some restrictions, the bond can be called by the issuing firm, which presumably acts to maximize the equity value of the firm by minimizing the bond value. This creates a two-person game. We show that if the coupon rate is below the interest rate times the call price, then conversion should precede call. On the other hand, if the dividend rate times the call price is below the coupon rate, call should precede conversion. In either case, the game reduces to a problem of optimal stopping. This is joint work with Mihai Sirbu.

Professor Professor Klaus Reiner Schenk-Hoppé

Leeds University Business School
School of Mathematics, University of Leeds

Evolutionary Finance

The evolutionary finance strand of research explores financial markets from a Darwinian perspective. From this viewpoint assets are traded in an ongoing competition for wealth. The talk introduces our evolutionary finance model and presents results on the long-run behavior of the stochastic wealth dynamics. The study uses local and global methods from random dynamical systems theory. I will discuss why seemingly rational strategies can do very poorly against seemingly irrational strategies, the interaction of strategies can lead to stochastic time series of asset prices that do not converge, and evolutionary stable stock markets are those in which assets are valued in terms of expected relative dividends. This is joint work with Igor V. Evstigneev (Manchester) and Thorsten Hens (Zurich).

Professor Sjur D Flam

Visiting Professor, School of Economics, University of Manchester
Professor of Economics, University of Bergen

Portfolio management without probabilities or statistics

Considered is on-line financial management, aimed at maximal long-run growth of wealth. Suppose statistical information and methods are not available - or deemed too difficult. On that assumption an adaptive procedure may nonetheless solve the problem over time.

Mr Dimitris Melas

Global Head of Quantitative Research, HSBC Asset Management

The Merits of Absolute Return Quantitative Investment Strategies

● Current trends in hedge fund investing and role of quantitative strategies
● Subjective taxonomy and performance record of quantitative strategies
● Why do these strategies work: the quantitative investment paradox
● Using quantitative strategies to build investable hedge fund indices
● Using quantitative strategies to exploit HF return predictability
● Potential risks associated with quantitative strategies
● Ideal market conditions for quantitative strategies
● How to build effective quantitative strategies

Dr Chris Jones and Dr Toby Goodworth

CIO and Head of Risk Management, respectively, Key Asset Management

Hedge Funds and Risk

In this talk we take a look at the poential risks in hedge fund investing and suggest a framework to measure and manage such risks.

Professor Nick Webber

Warwick Business School and The Newton Institute

Option valuation, Monte Carlo methods and Levy processes: Using fast bridge techniques

Increasingly sophisticated option markets require increasingly sophisticated pricing and calibration methods. In an attempt to better fit to implied volatility surfaces much research has recently been devoted to using Levy processes to model asset returns or interest rate processes. Problems arise when using Levy driven models to value path dependent options. Monte Carlo methods must be used but these are slow unless speedup methods are used. We show how stratified sampling methods may be used to value path dependent options in the VG and NIG Levy models. We write down the bridge distributions of the subordinators for these processes and demonstrate how effective sampling methods can be used. Numerical results are presented demonstrating that speedups of factors of several hundred are easily obtained.

Dr Alvaro Cartea

Lecturer in Financial Mathematics, Birbeck College, London

Hedging under Non-Gaussian Processes: a fractional calculus approach

We propose a new dynamic hedging strategy based on tools of fractional calculus. We compare the profit and loss (P&L) resulting from hedging vanilla options when the classical approach of Delta- and Gamma-neutrality is employed to the results delivered by what we label Delta- and Fractional-Gamma-hedging. For particular cases such as the FMLS of Carr and Wu 2003 and Merton's Jump-Diffusion the volatility of the P&L is at least 25% of that resulting from Delta- and Gamma-neutrality. Moreover, we show that the pricing equation satisfied by European-style options written on securities that follow some of the most widely used jump process, like for example CGMY, satisfy a fractional PDE.

Professor Andrew Cairns

Isaac Newton Institute and Heriot-Watt University

Optimal Investment for Defined Contribution Pension Plans

In this talk we look at asset-allocation strategies for DC pension plan members. We consider a simple model that incorporates three sources of risk: asset risk, labour-income risk and interest-rate risk. We propose a new form of terminal utility function incorporating a form of habit formation that uses final salary as a numeraire. The paper discusses various properties and characteristics of the optimal asset-allocation strategy. We compare the performance of this optimum with some typical `default' strategies offered by pension providers: specifically static strategies and the deterministic lifestyle strategy (which involves a gradual switch from equities into bonds according to preset rules).

Professor Dilip Madan

University of Maryland and Morgan Stanley, New York

Asset Allocation for CARA Utility with Multivariate Levy Returns

We apply a signal processing technique known as independent component analysis (ICA) to multivariate financial time series. The main idea of ICA is to decompose the observed time series into statistically independent components (ICs). We further assume that the ICs follow the variance gamma (VG) process. The VG process is Brownian motion evaluated with drift at a random time given by a gamma process. We build a multivariate VG portfolio model and analyze empirical results of the investment.

Dr Dorje Brody

Research Associate, Imperial College, London

Entropic Calibration Revisited

The entropic calibration of the risk-neutral density function is effective in recovering the strike dependence of options, but encounters difficulties in determining the relevant greeks. In the first part of the talk, by use of put-call reversal we apply the entropic method to the time reversed economy, which allows us to obtain the spot price dependence of options and the relevant greeks. In the second part of the talk, the use of Renyi entropy is applied in the forward time economy to determine the tail distribution of the risk-neutral density function implied by the traded option prices.

Dr Stephen Blyth

Managing Director and Head of European Arbitrage Trading, Deutsche Bank, London

Practical Relative Value Volatility Trading

We present a framework for the identification and extraction of value within the interest-rate options markets. We outline in detail our modeling paradigm, in particular stressing the importance of a consistent approach avoiding the "proceduralism" which often afflicts quantitative financial modeling. We implement one possible modeling framework and apply it to the US dollar and euro interest-rate markets. The framework adapts the market-model methodology of Brace-Gatarek-Musiela. Parametric and nonparametric forward volatility surfaces are constructed and both are used to inform trading decisions. We present a series of examples in the US dollar and euro markets, identifying current opportunities and describing historical case studies. We discuss additional dimensions of volatility markets where a similar modeling approach can be applied.

Dr Wim Schoutens

Department of Mathematics, K U Leuven - UCS Belgium

Model risk for exotic and moment derivatives

We show that several advanced equity option models incorporating stochastic volatility can be calibrated very nicely to a realistic option surface. More specifically, we focus on the Heston Stochastic Volatility model (with and without jumps in the stock price process), the Barndorff-Nielsen-Shephard model and Levy models with stochastic time. All these models are capable of accurately describing the marginal distribution of stock prices or indices and hence lead to almost identical European vanilla option prices. As such, we can hardly discriminate between the different processes on the basis of their smile-conform pricing characteristics. We therefore are tempted applying them to a range of exotics. However, due to the different structure in path -behaviour between these models, the resulting exotics prices can vary significantly. We subsequently introduce moment derivatives. These are derivatives that depend on the realized moments of (daily) logreturns. An already traded example of these derivatives is the Variance Swap. We finally show how to hedge these options and calculate their prices by Monte-Carlo simulation. A comparison of these moment derivatives premiums demonstrates an even bigger discrepancy between the aforementioned models. It motivates a further study on how to model the fine stochastic behaviour of assets over time.

Dr Dherminder Kainth

Senior Quantitative Analyst, Group Risk Management, Royal Bank of Scotland, London

Using importance sampling to accelerate the pricing of portfolio credit derivatives

● The Gaussian copula model
● Importance sampling for nth to default baskets
● Greeks
● Path-wise method for discontinuous pay-offs
● Importance sampling for large pools and numerical results

Professor Mark Davis

Mathematics, Imperial College

Fluid and Diffusion Limits of Finite-State Markov Processes and Applications to Portfolio Credit Risk (A Queueing Network Approach to Portfolio Credit Risk)

Predicting the default performance of large heterogeneous portfolios is a major topic in credit risk. One approach to this is to derive analytic or partly analytic approximations based on the law of large numbers and/or central limit theorem; examples are Vasicek's large homogeneous portfolio model or the saddle-point approximations used in CreditRisk+. Here we introduce an approach based on ideas from queuing networks. The portfolio members are thought of as particles that move around a number of credit risk states (credit ratings) before eventually defaulting. The transition rates are supposed to depend on an external 'environment' process, thus introducing dependence between the particles. We study the limiting behaviour of this system as the number of particles increases, obtaining conditional fluid and diffusion limits from which portfolio performance can be predicted.

Professor Michael Dempster

Centre for Financial Research
Cambridge Systems Associates Limited

Dynamic Stochastic Programming for Asset Liability Management

Experience with commercial applications of dynamic stochastic programmes to financial planning problems involving variously long term asset allocation, asset liability management, pricing and hedging strategies and risk management has lead to the development of sophisticated techniques for model validation and performance testing. This talk will give an overview of the technical lessons learned regarding complex scenario generation, tree shapes, stability testing, Monte Carlo testing and historical back testing. Consideration will be given to model generation, experimental design, sample size, statistical power and solution visualisation from the practical viewpoint of run-time and memory requirements.

Dr Christoph Reisinger

OCIAM, Mathematical Institute, University of Oxford

Computational Strategies for High Dimensional Option Pricing Problems

Many heavily traded derivative instruments depend on a large number of risk factors, such that PDE pricing of these contracts is widely believed to be computationally infeasible. Remarkably, this curse of dimensionality can often be overcome by exploiting the interplay of asymptotic expansions - revealing lower dimensional structures - and numerical ('sparse grid') techniques, which are known to have optimal approximation properties in high dimensions. Numerical examples for American, Bermudan and European contracts in up to thirty dimensions, priced by multigrid methods in optimal complexity on a parallel cluster, show striking accuracy.

Benjamin Carton de Wiart

Exotic Equity Derivatives, CIBC London
Centre for Financial Research, Judge Institute of Management

Overlay and Alpha Generation Models in Foreign Exchange

Wavelet methods for partial differential equations have been around for some time now, but most fail to show a real improvement where it matters for practitioners in mathematical finance : reduced CPU time for real-life problem. In this research, we introduce a simple but efficient method that uses wavelets for their advantages in compression and interpolation in order to define a sparse computational domain, but uses finite difference filters for differentiation which provides us with a simple and sparse stiffness matrix. Since the method only uses nodal basis, the application of non-constant terms, boundary conditions and free-boundary is straightforward. As a result, we can apply the method to financial products with American exercise features easily. We give empirical results for several products from the equity and fixed income markets and show a speed-up factor between 2 and 5 with no significant reduction of precision.