Modeling volatility and changes in the swap spread

We investigate the determinants of changes in U.S. interest rate swap spreads using a model that explicitly allows for volatility interactions between swaps of different terms to maturity. Changes in the swap spread are found to be positively related to interest rate volatility, to changes in the default risk premium in the corporate bond market, and to changes in the liquidity premium for government securities. Swap spread changes are negatively related to changes in the level of interest rates and changes in the slope of the term structure. We also find that there is a strong and significant volatility interaction among spreads for swaps of different maturities and that the process for the conditional variance of the spread is highly persistent across all maturities.

Reality check for volatility models

Asset allocation decisions and value at risk calculations rely strongly on volatility estimates. Volatility measures such as rolling window, EWMA, GARCH and stochastic volatility are used in practice. GARCH and EWMA type models that incorporate the dynamic structure of volatility and are capable of forecasting future behavior of risk should perform better than constant, rolling window volatility models. For the same asset the model that is the ‘best’ according to some criterion can change from period to period. We use the reality check test∗ to verify if one model out-performs others over a class of re-sampled time-series data. The test is based on re-sampling the data using stationary bootstrapping. For each re-sample we check the ‘best’ model according to two criteria and analyze the distribution of the performance statistics. We compare constant volatility, EWMA and GARCH models using a quadratic utility function and a risk management measurement as comparison criteria. No model consistently out-performs the benchmark.

Forward implied volatility expansions in LSV models

In this work we address the problem of finding formulas for efficient and reliable analytical approximation for the calculation of forward implied volatility in LSV models, a problem which is reduced to the calculation of option prices as an expansion of the price of the same financial asset in a Black-Scholes dynamic. Our approach involves an expansion of the differential operator, whose solution represents the price in local stochastic volatility dynamics. Further calculations then allow to obtain an expansion of the implied volatility without the aid of any special function or expensive from the computational point of view, in order to obtain explicit formulas fast to calculate but also as accurate as possible.

Testing co-volatility spillovers for natural gas spot, futures and ETF spot using dynamic conditional covariances

There is substantial empirical evidence that energy and financial markets are closely connected. As one of the most widely-used energy resources worldwide, natural gas has a large daily trading volume. In order to hedge the risk of natural gas spot markets, a large number of hedging strategies can be used, especially with the rapid development of natural gas derivatives markets. These hedging instruments include natural gas futures and options, as well as Exchange Traded Fund (ETF) prices that are related to natural gas stock prices. The volatility spillover effect is the delayed effect of a returns shock in one physical, biological or financial asset on the subsequent volatility or co-volatility of another physical, biological or financial asset. Investigating volatility spillovers within and across energy and financial markets is a crucial aspect of constructing optimal dynamic hedging strategies. The paper tests and calculates spillover effects among natural gas spot, futures and ETF markets using the multivariate conditional volatility diagonal BEKK model. The data used include natural gas spot and futures returns data from two major international natural gas derivatives markets, namely NYMEX (USA) and ICE (UK), as well as ETF data of natural gas companies from the stock markets in the USA and UK. The empirical results show that there are significant spillover effects in natural gas spot, futures and ETF markets for both USA and UK. Such a result suggests that both natural gas futures and ETF products within and beyond the country might be considered when constructing optimal dynamic hedging strategies for natural gas spot prices.

Volatility spillovers for spot, futures, and ETF prices in energy and agriculture

The agricultural and energy industries are closely related, both biologically and financially. The paper discusses the relationship and the interactions on price and volatility, with special focus on the covolatility spillover effects for these two industries. The interaction and covolatility spillovers or the delayed effect of a returns shock in one asset on the subsequent volatility or covolatility in another asset, between the energy and agricultural industries is the primary emphasis of the paper. Although there has already been significant research on biofuel and biofuel-related crops, much of the previous research has sought to find a relationship among commodity prices. Only a few published papers have been concerned with volatility spillovers. However, it must be emphasized that there have been numerous technical errors in the theoretical and empirical research, which needs to be corrected. The paper not only considers futures prices as a widely-used hedging instrument, but also takes an interesting new hedging instrument, ETF, into account. ETF is regarded as index futures when investors manage their portfolios, so it is possible to calculate an optimal dynamic hedging ratio. This is a very useful and interesting application for the estimation and testing of volatility spillovers. In the empirical analysis, multivariate conditional volatility diagonal BEKK models are estimated for comparing patterns of covolatility spillovers. The paper provides a new way of analyzing and describing the patterns of covolatility spillovers, which should be useful for the future empirical analysis of estimating and testing covolatility spillover effects.

Asymptotic Theory for Extended Asymmetric Multivariate GARCH Processes

The paper considers various extended asymmetric multivariate conditional volatility models, and derives appropriate regularity conditions and associated asymptotic theory. This enables checking of internal consistency and allows valid statistical inferences to be drawn based on empirical estimation. For this purpose, we use an underlying vector random coefficient autoregressive process, for which we show the equivalent representation for the asymmetric multivariate conditional volatility model, to derive asymptotic theory for the quasi-maximum likelihood estimator. As an extension, we develop a new multivariate asymmetric long memory volatility model, and discuss the associated asymptotic properties.

Pseudo-marginal algorithms with multiple CPUs

The emergence of pseudo-marginal algorithms has led to improved computational efficiency for dealing with complex Bayesian models with latent variables. Here an unbiased estimator of the likelihood replaces the true likelihood in order to produce a Bayesian algorithm that remains on the marginal space of the model parameter (with latent variables integrated out), with a target distribution that is still the correct posterior distribution. Very efficient proposal distributions can be developed on the marginal space relative to the joint space of model parameter and latent variables. Thus psuedo-marginal algorithms tend to have substantially better mixing properties. However, for pseudo-marginal approaches to perform well, the likelihood has to be estimated rather precisely. This can be difficult to achieve in complex applications. In this paper we propose to take advantage of multiple central processing units (CPUs), that are readily available on most standard desktop computers. Here the likelihood is estimated independently on the multiple CPUs, with the ultimate estimate of the likelihood being the average of the estimates obtained from the multiple CPUs. The estimate remains unbiased, but the variability is reduced. We compare and contrast two different technologies that allow the implementation of this idea, both of which require a negligible amount of extra programming effort. The superior performance of this idea over the standard approach is demonstrated on simulated data from a stochastic volatility model.

On the efficacy of Fourier series approximations for pricing European options

This paper investigates several competing procedures for computing the prices of vanilla European options, such as puts, calls and binaries, in which the underlying model has a characteristic function that is known in semi-closed form. The algorithms investigated here are the half-range Fourier cosine series, the half-range Fourier sine series and the full-range Fourier series. Their performance is assessed in simulation experiments in which an analytical solution is available and also for a simple affine model of stochastic volatility in which there is no closed-form solution. The results suggest that the half-range sine series approximation is the least effective of the three proposed algorithms. It is rather more difficult to distinguish between the performance of the halfrange cosine series and the full-range Fourier series. However there are two clear differences. First, when the interval over which the density is approximated is relatively large, the full-range Fourier series is at least as good as the half-range Fourier cosine series, and outperforms the latter in pricing out-of-the-money call options, in particular with maturities of three months or less. Second, the computational time required by the half-range Fourier cosine series is uniformly longer than that required by the full-range Fourier series for an interval of fixed length. Taken together,these two conclusions make a case for pricing options using a full-range range Fourier series as opposed to a half-range Fourier cosine series if a large number of options are to be priced in as short a time as possible.

Accelerating Pseudo-Marginal MCMC using Gaussian Processes

Pseudo-marginal methods such as the grouped independence Metropolis-Hastings (GIMH) and Markov chain within Metropolis (MCWM) algorithms have been introduced in the literature as an approach to perform Bayesian inference in latent variable models. These methods replace intractable likelihood calculations with unbiased estimates within Markov chain Monte Carlo algorithms. The GIMH method has the posterior of interest as its limiting distribution, but suffers from poor mixing if it is too computationally intensive to obtain high-precision likelihood estimates. The MCWM algorithm has better mixing properties, but less theoretical support. In this paper we propose to use Gaussian processes (GP) to accelerate the GIMH method, whilst using a short pilot run of MCWM to train the GP. Our new method, GP-GIMH, is illustrated on simulated data from a stochastic volatility and a gene network model.

Mathematical Aspects of Financial Markets with Frictions

Frictions are factors that hinder trading of securities in financial markets. Typical frictions include limited market depth, transaction costs, lack of infinite divisibility of securities, and taxes. Conventional models used in mathematical finance often gloss over these issues, which affect almost all financial markets, by arguing that the impact of frictions is negligible and, consequently, the frictionless models are valid approximations. This dissertation consists of three research papers, which are related to the study of the validity of such approximations in two distinct modeling problems. Models of price dynamics that are based on diffusion processes, i.e., continuous strong Markov processes, are widely used in the frictionless scenario. The first paper establishes that diffusion models can indeed be understood as approximations of price dynamics in markets with frictions. This is achieved by introducing an agent-based model of a financial market where finitely many agents trade a financial security, the price of which evolves according to price impacts generated by trades. It is shown that, if the number of agents is large, then under certain assumptions the price process of security, which is a pure-jump process, can be approximated by a one-dimensional diffusion process. In a slightly extended model, in which agents may exhibit herd behavior, the approximating diffusion model turns out to be a stochastic volatility model. Finally, it is shown that when agents' tendency to herd is strong, logarithmic returns in the approximating stochastic volatility model are heavy-tailed. The remaining papers are related to no-arbitrage criteria and superhedging in continuous-time option pricing models under small-transaction-cost asymptotics. Guasoni, Rásonyi, and Schachermayer have recently shown that, in such a setting, any financial security admits no arbitrage opportunities and there exist no feasible superhedging strategies for European call and put options written on it, as long as its price process is continuous and has the so-called conditional full support (CFS) property. Motivated by this result, CFS is established for certain stochastic integrals and a subclass of Brownian semistationary processes in the two papers. As a consequence, a wide range of possibly non-Markovian local and stochastic volatility models have the CFS property.

Uniform stability of a particle approximation of the optimal filter derivative

Sequential Monte Carlo methods, also known as particle methods, are a widely used set of computational tools for inference in non-linear non-Gaussian state-space models. In many applications it may be necessary to compute the sensitivity, or derivative, of the optimal filter with respect to the static parameters of the state-space model; for instance, in order to obtain maximum likelihood model parameters of interest, or to compute the optimal controller in an optimal control problem. In Poyiadjis et al. [2011] an original particle algorithm to compute the filter derivative was proposed and it was shown using numerical examples that the particle estimate was numerically stable in the sense that it did not deteriorate over time. In this paper we substantiate this claim with a detailed theoretical study. Lp bounds and a central limit theorem for this particle approximation of the filter derivative are presented. It is further shown that under mixing conditions these Lp bounds and the asymptotic variance characterized by the central limit theorem are uniformly bounded with respect to the time index. We demon- strate the performance predicted by theory with several numerical examples. We also use the particle approximation of the filter derivative to perform online maximum likelihood parameter estimation for a stochastic volatility model.

An empirical comparison of the performance of alternative option pricing models

Published as an article in: Investigaciones Economicas, 2005, vol. 29, issue 3, pages 483-523.

Copula Processes

We define a copula process which describes the dependencies between arbitrarily many random variables independently of their marginal distributions. As an example, we develop a stochastic volatility model, Gaussian Copula Process Volatility (GCPV), to predict the latent standard deviations of a sequence of random variables. To make predictions we use Bayesian inference, with the Laplace approximation, and with Markov chain Monte Carlo as an alternative. We find both methods comparable. We also find our model can outperform GARCH on simulated and financial data. And unlike GARCH, GCPV can easily handle missing data, incorporate covariates other than time, and model a rich class of covariance structures.

Nonparametric estimation of structural models for high-frequency currency market data

Empirical modeling of high-frequency currency market data reveals substantial evidence for nonnormality, stochastic volatility, and other nonlinearities. This paper investigates whether an equilibrium monetary model can account for nonlinearities in weekly data. The model incorporates time-nonseparable preferences and a transaction cost technology. Simulated sample paths are generated using Marcet's parameterized expectations procedure. The paper also develops a new method for estimation of structural economic models. The method forces the model to match (under a GMM criterion) the score function of a nonparametric estimate of the conditional density of observed data. The estimation uses weekly U.S.-German currency market data, 1975-90. © 1995.