Three‐Regime Asymmetric STAR Modeling and Exchange Rate Reversion

The breakdown of the Bretton Woods system and the adoption of generalized floating exchange rates ushered in a new era of exchange rate volatility and uncertainty. This increased volatility leads economists to search for economic models able to describe observed exchange rate behavior. In the present paper, we propose more general STAR transition functions that encompass both threshold nonlinearity and asymmetric effects. Our framework allows for a gradual adjustment from one regime to another and considers threshold effects by encompassing other existing models, such as TAR models. We apply our methodology to three different exchange rate data sets: one for developing countries and official nominal exchange rates, the second for emerging market economies using black market exchange rates, and the third for OECD economies.     

AN EMPIRICAL STUDY OF A NEW CLASS OF NO-ARBITRAGE-BASED DISCRETE MODELS OF THE TERM STRUCTURE

We present empirical tests of the new no-arbitrage-based term structure paradigm in discrete time. We derive and test empirical specifications for deterministic one-factor forward rate volatility models and examine the compatibility of these forward rate volatility functions using term structure dynamics. Our estimation technique uses the generalized method of moments and is based on forward bond price deviations. We do not impose restrictions on the market price of risk, and we incorporate all available term structure information. Our data consist of four sets of pure discount bonds derived from the CRSP bond files and U.S. Treasury bill quotes.     

Finite Dimensional Affine Realisations of HJM Models in Terms of Forward Rates and Yields

Finite dimensional Markovian HJM term structure models provide ideal settings for the study of term structure dynamics and interest rate derivatives where the flexibility of the HJM framework and the tractability of Markovian models coexist. Consequently, these models became the focus of a series of papers including Carverhill (1994), Ritchken and Sankarasubramanian (1995), Bhar and Chiarella (1997), Inui and Kijima (1998), de Jong and Santa-Clara (1999), Björk and Svensson (2001) and Chiarella and Kwon (2001a). However, these models usually required the introduction of a large number of state variables which, at first sight, did not appear to have clear links to the market observed quantities, and the explicit realisations of the forward rate curve in terms of the state variables were unclear. In this paper, it is shown that the forward rate curves for these models are affine functions of the state variables, and conversely that the state variables in these models can be expressed as affine functions of a finite number of forward rates or yields. This property is useful, for example, in the estimation of model parameters. The paper also provides explicit formulae for the bond prices in terms of the state variables that generalise the formulae given in Inui and Kijima (1998), and applies the framework to obtain affine representations for a number of popular interest rate models.     

Alternative asset-price dynamics and volatility smile

Abstract

We review the general class of analytically tractable asset-price models that was introduced by Brigo and Mercurio (2001a Mathematical Finance—Bachelier Congr. 2000 (Springer Finance) ed H Geman, D B Madan, S R Pliska and A C F Vorst (Berlin: Springer) pp 151–74), where the considered asset can be an exchange rate, a stock index or even a forward Libor rate. The class is based on an explicit SDE under a given forward measure and includes models featuring (i) explicit asset-price dynamics, (ii) a virtually unlimited number of parameters and (iii) analytical formulae for European options.

We also review the fundamental case where the asset-price density is given, at every time, by a mixture of log-normal densities with equal means. We then introduce two other cases: the first is still based on log-normal densities, but it allows for different means in the distributions; the second is based on processes of hyperbolic-sine type.

Finally, we test the goodness of calibration to real market data of the considered models, choosing a particularly asymmetric volatility surface. As expected, the model based on hyperbolic-sine density mixtures achieves the lowest calibration error.     

Efficient Calibration of Trinomial Trees for One-Factor Short Rate Models

In this paper we propose a computationally efficient implementation of general one factor short rate models with a trinomial tree. We improve the Hull–Whites procedure to calibrate the tree to bond prices by circumventing the forward rate induction and numerical root search algorithms. Our calibration procedure is based on forward measure changes and is as general as the Hull–White procedure, but it offers a more efficient and flexible method of constructing a trinomial term structure model. It can be easily implemented and calibrated to both prices and volatilities. JEL classification G13, C6     

Simulating the Evolution of the Implied Distribution

Motivated by the implied stochastic volatility literature (Britten–Jones and Neuberger, forthcoming; Derman and Kani, 1997; Ledoit and Santa–Clara, 1998) this paper proposes a new and general method for constructing smile–consistent stochastic volatility models. The method is developed by recognising that option pricing and hedging can be accomplished via the simulation of the implied risk neutral distribution. We devise an algorithm for the simulation of the implied distribution, when the first two moments change over time. The algorithm can be implemented easily, and it is based on an economic interpretation of the concept of mixture of distributions. It can also be generalised to cases where more complicated forms for the mixture are assumed.     

The empirical similarity approach for volatility prediction

In this paper we adapt the empirical similarity (ES) concept for the purpose of combining volatility forecasts originating from different models. Our ES approach is suitable for situations where a decision maker refrains from evaluating success probabilities of forecasting models but prefers to think by analogy. It allows to determine weights of the forecasting combination by quantifying distances between model predictions and corresponding realizations of the process of interest as they are perceived by decision makers. The proposed ES approach is applied for combining models in order to forecast daily volatility of the major stock market indices.     

A Class of Jump-Diffusion Bond Pricing Models within the HJM Framework

This paper considers a class of term structure models that is a parameterisation of the Shirakawa (1991) extension of the Heath et al. (1992) model to the case of jump-diffusions. We consider specific forward rate volatility structures that incorporate state dependent Wiener volatility functions and time dependent Poisson volatility functions. Within this framework, we discuss the Markovianisation issue, and obtain the corresponding affine term structure of interest rates. As a result we are able to obtain a broad tractable class of jump-diffusion term structure models. We relate our approach to the existing class of jump-diffusion term structure models whose starting point is a jump-diffusion process for the spot rate. In particular we obtain natural jump-diffusion versions of the Hull and White (1990, 1994) one-factor and two-factor models and the Ritchken and Sankarasubramanian (1995) model within the HJM framework. We also give some numerical simulations to gauge the effect of the jump-component on yield curves and the implications of various volatility specifications for the spot rate distribution.     

Maximum likelihood estimation of stochastic volatility models

We develop and implement a method for maximum likelihood estimation in closed-form of stochastic volatility models. Using Monte Carlo simulations, we compare a full likelihood procedure, where an option price is inverted into the unobservable volatility state, to an approximate likelihood procedure where the volatility state is replaced by proxies based on the implied volatility of a short-dated at-the-money option. The approximation results in a small loss of accuracy relative to the standard errors due to sampling noise. We apply this method to market prices of index options for several stochastic volatility models, and compare the characteristics of the estimated models. The evidence for a general CEV model, which nests both the affine Heston model and a GARCH model, suggests that the elasticity of variance of volatility lies between that assumed by the two nested models.     

Dynamic factor long memory volatility

In this paper, we develop a long memory orthogonal factor (LMOF) multivariate volatility model for forecasting the covariance matrix of financial asset returns. We evaluate the LMOF model using the volatility timing framework of Fleming et al. [J. Finance, 2001, 56, 329–352] and compare its performance with that of both a static investment strategy based on the unconditional covariance matrix and a range of dynamic investment strategies based on existing short memory and long memory multivariate conditional volatility models. We show that investors should be willing to pay to switch from the static strategy to a dynamic volatility timing strategy and that, among the dynamic strategies, the LMOF model consistently produces forecasts of the covariance matrix that are economically more useful than those produced by the other multivariate conditional volatility models, both short memory and long memory. Moreover, we show that combining long memory volatility with the factor structure yields better results than employing either long memory volatility or the factor structure alone. The factor structure also significantly reduces transaction costs, thus increasing the feasibility of dynamic volatility timing strategies in practice. Our results are robust to estimation error in expected returns, the choice of risk aversion coefficient, the estimation window length and sub-period analysis.     

Kolmogorov’s forward PIDE and forward transition rates in life insurance

We consider a doubly stochastic Markov chain, where the transition intensities are modelled as diffusion processes. Here we present a forward partial integro-differential equation (PIDE) for the transition probabilities. This is a generalisation of Kolmogorov’s forward differential equation. In this set-up, we define forward transition rates, generalising the concept of forward rates, e.g. the forward mortality rate. These models are applicable in e.g. life insurance mathematics, which is treated in the paper. The results presented follow from the general forward PIDE for stochastic processes, of which the Fokker–Planck differential equation and Kolmogorov’s forward differential equation are the two most known special cases. We end the paper by considering the semi-Markov case, which can also be considered a special case of a general forward partial integro-differential equation.     

Yield-factor volatility models

The term structure of interest rates is often summarized using a handful of yield factors that capture shifts in the shape of the yield curve. In this paper, we develop a comprehensive model for volatility dynamics in the level, slope, and curvature of the yield curve that simultaneously includes level and GARCH effects along with regime shifts. We show that the level of the short rate is useful in modeling the volatility of the three yield factors and that there are significant GARCH effects present even after including a level effect. Further, we find that allowing for regime shifts in the factor volatilities dramatically improves the model’s fit and strengthens the level effect. We also show that a regime-switching model with level and GARCH effects provides the best out-of-sample forecasting performance of yield volatility. We argue that the auxiliary models often used to estimate term structure models with simulation-based estimation techniques should be consistent with the main features of the yield curve that are identified by our model.     

Consistent Variance Curve Models

We introduce a general approach to model a joint market of stock price and a term structure of variance swaps in an HJM-type framework. In such a model, strongly volatility-dependent contracts can be priced and risk-managed in terms of the observed stock and variance swap prices. To this end, we introduce equity forward variance term structure models and derive the respective HJM-type arbitrage conditions. We then discuss finite-dimensional Markovian representations of the fixed time-to-maturity forward variance swap curve and derive consistency results for both the standard case and for variance curves with values in a Hilbert space. For the latter, our representation also ensures non-negativity of the process. We then give a few examples of such variance curve functionals and briefly discuss completeness and hedging in such models. As a further application, we show that the speed of mean reversion in some standard stochastic volatility models should be kept constant when the model is recalibrated.     

Score-driven copula models for portfolios of two risky assets

ABSTRACT

The precise measurement of the association between asset returns is important for financial investors and risk managers. In this paper, we focus on a recent class of association models: Dynamic Conditional Score (DCS) copula models. Our contributions are the following: (i) We compare the statistical performance of several DCS copulas for several portfolios. We study the Clayton, rotated Clayton, Frank, Gaussian, Gumbel, rotated Gumbel, Plackett and Student's t copulas. We find that the DCS model with the Student's t copula is the most parsimonious model. (ii) We demonstrate that the copula score function discounts extreme observations. (iii) We jointly estimate the marginal distributions and the copula, by using the Maximum Likelihood method. We use DCS models for mean, volatility and association of asset returns. (iv) We estimate robust DCS copula models, for which the probability of a zero return observation is not necessarily zero. (v) We compare different patterns of association in different regions of the distribution for different DCS copulas, by using density contour plots and Monte Carlo (MC) experiments. (vi) We undertake a portfolio performance study with the estimation and backtesting of MC Value-at-Risk for the DCS model with the Student's t copula.     

On pricing kernels and finite-state variable Heath Jarrow Morton models

Once a pricing kernel is established, bond prices and all other interest rate claims can be computed. Alternatively, the pricing kernel can be deduced from observed prices of bonds and selected interest rate claims. Examples of the former approach include the celebrated Cox, Ingersoll, and Ross (1985b) model and the more recent model of Constantinides (1992). Examples of the latter include the Black, Derman, and Toy (1990) model and the Heath, Jarrow, and Morton paradigm (1992) (hereafter HJM). In general, these latter models are not Markov. Fortunately, when suitable restrictions are imposed on the class of volatility structures of forward rates, then finite-state variable HJM models do emerge. This article provides a linkage between the finite-state variable HJM models, which use observables to induce a pricing kernel, and the alternative approach, which proceeds directly to price after a complete specification of a pricing kernel. Given such linkages, we are able to explicitly reveal the relationship between state-variable models, such as Cox, Ingersoll, and Ross, and the finite-state variable HJM models. In particular, our analysis identifies the unique map between the set of investor forecasts about future levels of the drift of the pricing kernel and the manner by which these forecasts are revised, to the shape of the term structure and its volatility. For an economy with square root innovations, the exact mapping is made transparent.     

Alternative Defaultable Term Structure Models

The objective of this paper is to consider defaultable term structure models in a general setting beyond standard risk-neutral models. Using as numeraire the growth optimal portfolio, defaultable interest rate derivatives are priced under the real-world probability measure. Therefore, the existence of an equivalent risk-neutral probability measure is not required. In particular, the real-world dynamics of the instantaneous defaultable forward rates under a jump-diffusion extension of a HJM type framework are derived. Thus, by establishing a modelling framework fully under the real-world probability measure, the challenge of reconciling real-world and risk-neutral probabilities of default is deliberately avoided, which provides significant extra modelling freedom. In addition, for certain volatility specifications, finite dimensional Markovian defaultable term structure models are derived. The paper also demonstrates an alternative defaultable term structure model. It provides tractable expressions for the prices of defaultable derivatives under the assumption of independence between the discounted growth optimal portfolio and the default-adjusted short rate. These expressions are then used in a more general model as control variates for Monte Carlo simulations of credit derivatives. Nicola Bruti-Liberati: In memory of our beloved friend and colleague.     

Double-jump diffusion model for VIX: evidence from VVIX

This paper studies the continuous-time dynamics of VIX with stochastic volatility and jumps in VIX and volatility. Built on the general parametric affine model with stochastic volatility and jumps in the logarithm of VIX, we derive a linear relationship between the stochastic volatility factor and the VVIX index. We detect the existence of a co-jump of VIX and VVIX and put forward a double-jump stochastic volatility model for VIX through its joint property with VVIX. Using the VVIX index as a proxy for stochastic volatility, we use the MCMC method to estimate the dynamics of VIX. Comparing nested models of VIX, we show that the jump in VIX and the volatility factor are statistically significant. The jump intensity is also stochastic. We analyse the impact of the jump factor on VIX dynamics.