Common Correlation and Calibrating the Lognormal Forward Rate Model

1997 three papers that introduced very similar lognormal diffusion processes for interest rates appeared virtuously simultaneously. These models, now commonly called the 'LIBOR models' are based on either lognormal diffusions of forward rates as in Brace, Gatarek & Musiela (1997) and Miltersen, Sandermann & Sondermann (1997) or lognormal diffusions of swap rates, as in Jamshidian (1997). The consequent research interest in the calibration of the LIBOR models has engendered a growing empirical literature, including many papers by Brigo and Mercurio, and Riccardo Rebonato (www.fabiomercurio.it and www.damianobrigo.it and www.rebonato.com). The art of model calibration requires a reasonable knowledge of option pricing and a thorough background in statistics - techniques that are quite different to those required to design no-arbitrage pricing models. Researchers will find the book by Brigo and Mercurio (2001) and the forthcoming book by Rebonato (2002) invaluable aids to their understanding.

The Present, Future and Imperfect of Financial Risk Management

Current research on financial risk management applications of econometrics centres on the accurate assessment of individual market and credit risks with relatively little theoretical or applied econometric research on other types of risk, aggregation risk, data incompleteness and optimal risk control. We argue that consideration of the model risk arising from crude aggregation rules and inadequate data could lead to a new class of reduced form Bayesian risk assessment models. Logically, these models should be set within a common factor framework that allows proper risk aggregation methods to be developed. We explain how such a framework could also provide the essential links between risk control, risk assessments and the optimal allocation of resources.Financial risk assessment; risk control, RAROC, economic capital; regulatory capital; optimal allocation of resources

Bivariate Normal Mixture Spread Option Valuation

This paper explores the properties of a European spread option valuation method for correlated assets when the marginal distribution each asset return is assumed to be a mixture of normal distributions. In this ‘bivariate normal mixture’ (BNM) approach no-arbitrage option values are just weighted sums of different 2GBM values based on two correlated lognormal diffusions, and likewise for their sensitivities. The main advantage of this approach is that BNM option values are consistent with the volatility smiles for each asset and an implied correlation ‘frown’, both of which are often observed when spread options are priced under the 2GBM assumptions. It is simple to perform an extensive consideration of model values for varying strike, and for different asset volatility and correlation structures. We compare BNM valuations with those based on the ‘2GBM’ assumption of two correlated lognormal diffusions and explain the differences between the BNM values and the 2GBM values of spread options as a weighted sum of six second order 2GBM value sensitivities. We also investigate the BNM sensitivities and these, like the option values, can sometimes be significantly different from those obtained under the 2GBM model. Finally, we show how the correlation frown that is implied by this model is affected as we change the parameters in the bivariate normal mixture density of the asset returns.spread option, implied correlation, bivariate normal mixture density

The Equity Index Skew, Market Crashes and Asymmetric Normal Mixture GARCH

The skewness in physical distributions of equity index returns and the implied volatility skew in the risk-neutral measure are subjects of extensive academic research. Much attention is now being focused on models that are able to capture time-varying conditional skewness and kurtosis. For this reason normal mixture GARCH(1,1) models have become very popular in financial econometrics. We introduce further asymmetries into this class of models by modifying the GARCH(1,1) variance processes to skewed variance processes with leverage effects. These asymmetric normal mixture GARCH models can differentiate between two different sources of asymmetry: a persistent asymmetry due to the different means in the conditional normal mixture distributions, and a dynamic asymmetry (the leverage effect) due to the skewed GARCH processes. Empirical results on five major equity indices first employ many statistical criteria to determine whether asymmetric (GJR and AGARCH) normal mixture GARCH models can improve on asymmetric normal and Student’s-t GARCH specifications. These models were also used to simulate implied volatility smiles for the S&P index, and we find that much the most realistic skews are obtained from a GARCH model with a mixture of two GJR variance components.GARCH process, normal misture, equity skew, market crash, skew persistence, leverage effect

Asymmetries and Volatility Regimes in the European Equity Markets

This paper provides and empirical examination of four European equity indices between 1991 and 2005. We investigate the ability of fifteen different GARCH models to capture the characteristics of historical daily returns effectively and generate realistic implied volatility skews. Using many different model selection criteria we conclude that a normal mixture GARCH model with two volatility components, two sources of asymmetry and endogenous time-varying conditional higher moments provides the best fit overall. Since this model is relatively new in the literature we discuss the theoretical and empirical properties of such models. Examining the estimated parameters we show that they provide information on the likelihood of a crash and they specify the return and volatility behaviour, the leverage effect and the persistence of volatility during the two regimes (‘normal’ and ‘crash’). We also find that asymmetric normal mixture GARCH models, even without a volatility risk premium, afford a sufficiently rich structure to match the empirical characteristics of implied volatility skew surfaces, whereas single-state GARCH models give unrealistic shapes for the equity index skew.equity skew, market cras, GARCH process, normal mixture, skey peristence, leverage effect, volatility regimes

On The Continuous Limit of GARCH

GARCH processes constitute the major area of time series variance analysis hence the limit of these processes is of considerable interest for continuous time volatility modelling. The limit of the GARCH(1,1) model is fundamental for limits of other GARCH processes yet it has been the subject of much debate. The seminal work of Nelson (1990) derived this limit as a stochastic volatility process that is uncorrelated with the price process but a subsequent paper of Corradi (2000) derived the limit as a deterministic volatility process and several other contradictory papers followed. In this paper we reconsider this continuous limit, arguing that because the strong GARCH model is not aggregating in time it is incorrect to examine its limit. Instead it is legitimate to use the weak definition of GARCH that is time aggregating. We prove that its continuous limit is a stochastic volatility model that reduces to Nelson’s GARCH diffusion only under certain assumptions. In general, the weak GARCH limit has correlated Brownian motions in which both the variance diffusion coefficient and the price-volatility correlation are related to the skewness and kurtosis of the physical returns density.GARCH, stochastic volatility, time agtregation, continuous limit

Symmetric Normal Mixture GARCH

Normal mixture (NM) GARCH models are better able to account for leptokurtosis in financial data and offer a more intuitive and tractable framework for risk analysis and option pricing than student’s t-GARCH models. We present a general, symmetric parameterisation for NM-GARCH(1,1) models, derive the analytic derivatives for the maximum likelihood estimation of the model parameters and their standard errors and compute the moments of the error term. Also, we formulate specific conditions on the model parameters to ensure positive, finite conditional and unconditional second and fourth moments. Simulations quantify the potential bias and inefficiency of parameter estimates as a function of the mixing law. We show that there is a serious bias on parameter estimates for volatility components having very low weight in the mixing law. An empirical application uses moment specification tests and information criteria to determine the optimal number of normal densities in the mixture. For daily returns on three US Dollar foreign exchange rates (British pound, euro and Japanese yen) we find that, whilst normal GARCH(1,1) models fail the moment tests, a simple mixture of two normal densities is sufficient to capture the conditional excess kurtosis in the data. According to our chosen criteria, and given our simulation results, we conclude that a two regime symmetric NM-GARCH model, which quantifies volatility corresponding to ‘normal’ and ‘exceptional’ market circumstances, is optimal for these exchange rate data.Volatility regimes, conditional excess kurtosis, normal mixture, heavy trails, exchange rates, conditional heteroscedasticity, GARCH models.

The Continuous Limit of GARCH Processess

Contrary to popular belief, the diffusion limit of a GARCH variance process is not a diffusion model unless one makes a very specific assumption that cannot be generalized. In fact, the normal GARCH(1,1) prices of European call and puts are identical to the Black-Scholes prices based on the average of a deterministic variance process. In the case of GARCH models with several normal components – and these are more realistic representations of option prices and returns behaviour – the continuous limit is a stochastic model with uncertainty over which deterministic local volatility governs the return. The risk neutral model prices of European options are weighted averages of Black-Scholes prices based on the integrated forward variances in each state. An interesting area to be considered for application of this model is path dependent options. Since both marginal and transition price densities are lognormal mixtures the mixture GARCH option pricing model is not equivalent to the mixture option pricing models that have previously been discussed by several authors.GARCH diffusion, normal mixture, stochastic volatility, time aggregation

The Cointegration Alpha: Enchanced Index Tracking and Long-Short Equity Market Neutral Stragies

This paper presents two applications of cointegration based trading strategies: a classic index tracking strategy and a long-short equity market neutral strategy. As opposed to other traditional index tracking or long-short equity strategies, the portfolio optimisation is based on cointegration rather than correlation. The first strategy aims to replicate a benchmark accurately in terms of returns and volatility, while the other seeks to minimise volatility and generate steady returns under all market circumstances. Additionally, several combinations of these two strategies are explored. To validate the applicability of the cointegration technique to asset allocation, pioneered by Lucas (1997) and Alexander (1999), and explain how and why it works, we have employed a panel data on DJIA and its constituent stocks. When applied to constructing trading strategies in the DJIA, the cointegration technique produces encouraging results. For example, between January 1995 and December 2001 the most successful self-financing statistical arbitrage strategies returned (net of transaction and repo costs) approximately 10% with roughly 2% annual volatility and negligible correlation with the market. The comprehensive set of back-test results reported is meant to offer a detailed picture of the cointegration mechanism, and to emphasise its practical implementation issues. Its key characteristics, i.e. mean reverting tracking error, enhanced weights stability and better use of the information contained in stock prices, allow a flexible design of various funded and self-financing trading strategies, from index and enhanced index tracking, to long-short market neutral and alpha transfer techniques. Further enhancement of the strategy should target first, the identification of successful stock selection rules to supplement the simple cointegration results and second, the investigation of the potential benefits of applying optimal rebalancing rules.cointegration, enchanced index tracking, long-short equity, market neutral, hedge fund, alpha strategy

An Uncertain Volatility Explanation for Delayed Calls of Convertible Bonds

Arbitrage-free price bounds for convertible bonds are obtained assuming a stochastic volatility process for the common stock that lies within a band but makes few other assumptions about volatility dynamics. Equity-linked hazard rates, stochastic interest rates and different assumptions about default and recovery behavior are accommodated within this approach. A non-linear multi-factor reduced-form equity-linked default model leads to a set of non-linear partial differential complementarity equations that are governed by the volatility path. Empirical results focus on call notice period effects, showing that uncertain volatility can capture the call premia so often observed in issuer’s call policies. Increasingly pessimistic values for the issuer’s substitution asset obtain as we introduce more uncertainty during the notice period. Volatility uncertainty is thus a useful mechanism to explain issuers delayed call policies.call notice period, call premium, convertible bond, delayed calls, equity-linked default, stochastic interest rates, volatility uncertainty