This page contains downloadable versions of a selection of recent discussion papers that have not yet appeared in print.   For a full list of my discussion papers since 2000 see the ICMA Centre Discussion Papers in Finance

Generalized Beta-Generated Distributions
Co-author: Jose Maria Sarabia

This paper introduces a new class of generalized beta-generated distributions that have very flexible shapes and tractable properties. Their quantiles and moments have a simple closed form and they are maximum entropy distributions under three simple conditions. Two special cases are the classical beta-generated and the Kumaraswamy-generated distributions. An attractive feature of generalized beta-normal distributions is that the three generalized beta parameters afford greater control over the weights in both tails and in the centre of the generated distribution, compared with the classical beta-normal distribution.

Endogenizing Model Risk in Quantile Estimates
Co-author: Jose Maria Sarabia

We quantify and endogenize the model risk associated with quantile estimates using a maximum entropy distribution (MED) as benchmark. Moment-based MEDs cannot have heavy tails, however generalized beta generated distributions have attractive properties for popular applications of quantiles. These are MEDs under three simple constraints on the parameters that explicitly control tail weight and peakness. Model risk arises because analysts are constrained to use a model distribution that is not the MED. Then the model's alpha quantile differs from the alpha quantile of the MED so the tail probability under the MED associated with the model's alpha quantile is not alpha, it is a random variable. Model risk is endogenized by parameterizing the uncertainty about this random variable, whence the model's alpha quantile becomes a generated random variable. To obtain a point model-risk-adjusted quantile, the generated distribution is used to adjust the model's alpha quantile for any systematic bias and uncertainty due to model risk. An illustration based on Value-at-Risk (VaR) computes a model-risk-adjusted VaR for risk capital reserves which encompass both portfolio and VaR model risk.

Stochastic Volatility Jump-Diffusions for Equity Index Dynamics
Co-author: Andreas Kaeck

This paper examines the ability of twelve different continuous-time two-factor models with mean-reverting stochastic volatility to capture the dynamics of the S&P 500 and three European equity indices. The stochastic volatility models are the square root variance, GARCH, and log volatility diffusions, and each is augmented with price and volatility jump extensions. Parameter estimation is by Markov Chain Monte Carlo using daily spot index returns from 1987 to 2010. For each index we find that GARCH diffusions augmented with correlated price and volatility jumps outperform other specifications with respect to all the tests we perform. The European indices have similar dynamics, which are relatively easy to capture using several of our specifications, but the S&P 500 index has different dynamics and here the GARCH-jump specification is very clearly superior.

Does model fit matter for hedging? Evidence from FTSE 100 options
Co-author: Andreas Kaeck

This paper implements a variety of different calibration methods applied to the Heston model and examines their effect on the performance of standard and minimum-variance hedging of vanilla options on the FTSE 100 index. Simple adjustments to the Black-Scholes-Merton model are used as a benchmark. Our empirical findings apply to delta, delta-gamma or delta-vega hedging and they are robust to varying the option maturities and moneyness, and to different market regimes. On the methodological side, an efficient technique for simultaneous calibration to option price and implied volatility index data is introduced.

Exact Moment Simulation using Random Orthogonal Matrices
Co-authors: Walter Ledermann, Daniel Ledermann

This paper introduces a method for simulating multivariate samples that have exact means, covariances, skewness and kurtosis. A new class of rectangular orthogonal matrices is fundamental to the methodology, and these “L-matrices” can be deterministic, parametric or data specific in nature. The target moments determine an L-matrix, then infinitely many random samples with the same exact moments may be generated by multiplying the L-matrix by arbitrary random orthogonal matrices. The methodology is thus termed “ROM simulation”. We discuss certain classes of random orthogonal matrices and show how each class produces samples with difference characteristics. ROM simulation has applications to many problems that are resolved using standard Monte Carlo methods. But since no parametric assumptions are required there is no sampling error caused by the discrete approximation of a continuous distribution, which is a major source of error in standard Monte Carlo simulations. For illustration, we apply ROM simulation to determine the value-at-risk of a stock portfolio.

Analytic Approximations for Multi-Asset Option Pricing (v3) (updated)
Co-author: Aanand Venkatramanan

We derive general analytic approximations for pricing European basket and rainbow options on N assets. The key idea is to express the option’s price as a sum of prices of various compound exchange options, each with different pairs of sub-ordinate multi- or single-asset options. For some multi-asset options a strong condition holds, whereby each compound exchange option is equivalent to a standard single-asset option under a modified measure, and in such cases an almost exact analytic price exists for the multi-asset option. The underlying asset prices are assumed to follow lognormal processes, although the strong condition can be extended to certain other price processes for the underlying. More generally, approximate analytic prices for multi-asset options are derived using a weak lognormality condition, where the approximation stems from making constant volatility assumptions on the price processes that drive the prices of the sub-ordinate basket options. The analytic formulae for multi-asset option prices, and their Greeks, are defined in a recursive framework. For instance, the option delta is defined in terms of the delta relative to sub-ordinate multi-asset options, and the deltas of these sub-ordinate options with respect to the underlying assets. Simulations test the accuracy of our approximations, given some assumed values for the asset volatilities and correlations, and we demonstrate how to calibrate these parameters to market data so that multi-asset option prices are consistent with the implied volatility and correlation skews of the assets.

Analytic Approximations for Spread Options
Co-author: Aanand Venkatramanan

This paper expresses the price of a spread option as the sum of the prices of two compound options. One compound option is to exchange vanilla call options on the two underlying assets and the other is to exchange the corresponding put options. This way we derive a new analytic approximation for the price of a European spread option, and a corresponding approximation for each of its price, volatilty and correlation hedge ratios. Our approach has many advantages over existing analytic approximations, which have limited validity and an indeterminacy that renders them of little practical use. The compound exchange option approximation for European spread options is then extended to American spread options on assets that pay dividends or incur carry costs. Simulations quantify the accuracy of our approach; we also present an empirical application, to the American crack spread options that are traded on NYMEX. For illustration, we compare our results with those obtained using the approximation attributed to Kirk [1996], which is commonly used by traders.

Stochastic Local Volatility
Co-author: Leonardo Nogueira

Abstract: There are two unique volatility surfaces associated with any arbitrage-free set of standard European option prices, the implied volatility surface and the local volatility surface. Several papers have discussed the stochastic differential equations for implied volatilities that are consistent with these option prices but the static and dynamic no-arbitrage conditions are complex, mainly due to the large (or even infinite) dimensions of the state probability space. These no-arbitrage conditions are also instrument-specific and have been specified for some simple classes of options. However, the problem is easier to resolve when we specify stochastic differential equations for local volatilities instead. And the option prices and hedge ratios that are obtained by making local volatility stochastic are identical to those obtained by making instantaneous volatility or implied volatility stochastic. After proving that there is a one-to-one correspondence between the stochastic implied volatility and stochastic local volatility approaches, we derive a simple dynamic no-arbitrage condition for the stochastic local volatility model that is model-specific. The condition is very easy to check in local volatility models having only a few stochastic parameters.

Markov Switching GARCH Diffusion
Co-author: Emese Lazar

Abstract: GARCH option pricing models have the advantage of a well-established econometric foundation. However, multiple states need to be introduced as single state GARCH and even Lévy processes are unable to explain the term structure of the moments of financial data. We show that the continuous time version of the Markov switching GARCH(1,1) process is a stochastic model where the volatility follows a switching process. The continuous time switching GARCH model derived in this paper, where the variance process jumps between two or more GARCH volatility states, is able to capture the features of implied volatilities in an intuitive and tractable framework.

Weak GARCH Diffusion
Co-author: Emese Lazar

Abstract:Discrete time volatility analysis has focussed almost exclusively on GARCH processes, which are very flexible models for time varying conditional variance. The continuous limit of these processes is therefore of considerable interest for continuous time volatility modelling. Unfortunately, progress in this area has been hampered by conflicting results. The limit of the symmetric normal GARCH model is fundamental for limits of other GARCH processes, yet even this has been the point of much debate amongst econometricians. Nelson (1990) derived the limit of the strong GARCH model as a stochastic volatility process that is uncorrelated with the price process, so this limit has limited applicability. However, since the strong GARCH process is not time aggregating one should question whether it is sensible to derive its continuous limit at all. This paper derives the continuous limit of the weak GARCH process, which is time aggregating. Moreover the limit model is a stochastic volatility model with non-zero price volatility correlation in which both the variance diffusion coefficient and the price-volatility correlation are related to the skewness and kurtosis of the physical returns density. When returns are normally distributed this limit model reduces to Nelson’s strong GARCH diffusion, however, more generally it has the flexibility to fit most short term volatility skews without adding jumps in either the price or the volatility process.

A Primer on the Orthogonal GARCH Model

The orthogonal GARCH model is a multivariate, factor GARCH model based on principal components analysis. It is my preferred approach for constructing large dimensional GARCH covariance matrices for highly correlated systems, such as term structures of interest rates or commodity futures. I wrote this primer, more than 10 years ago, as a simple introduction to the model, aimed at inexerienced users. Although I have since published many more advanced articles on the subject, I have made this little primer available on my website by popular demand.
Examples for the OGARCH Primer