Affine Jump-diffusion Models Research Articles

Estimating Latent Variables and Jump Diffusion Models Using High-Frequency Data

This article proposes a new approach to exploit the information in high-frequency data for the statistical inference of continuous-time affine jump diffusion (AJD) models with latent variables. For this purpose, we construct unbiased estimators of the latent variables and their power functions on the basis of the observed state variables over extended horizons. With the estimates of the latent variables, we propose a generalized method of moments (GMM) procedure for the estimation of AJD models with the distinguishing feature that moments of both observed and latent state variables can be used without resorting to path simulation or discretization of the continuous-time process. Using high frequency return observations of the S&P 500 index, we implement our estimation approach to various continuous-time asset return models with stochastic volatility and random jumps.

A Bayesian Analysis of Return Dynamics with Lévy Jumps

We have developed Bayesian Markov chain Monte Carlo (MCMC) methods for inferences of continuous-time models with stochastic volatility and infinite-activity Levy jumps using discretely sampled data. Simulation studies show that (i) our methods provide accurate joint identification of diffusion, stochastic volatility, and Levy jumps, and (ii) the affine jump-diffusion (AJD) models fail to adequately approximate the behavior of infinite-activity jumps. In particular, the AJD models fail to capture the “infinitely many” small Levy jumps, which are too big for Brownian motion to model and too small for compound Poisson process to capture. Empirical studies show that infinite-activity Levy jumps are essential for modeling the S&P 500 index returns.

Estimating Latent Variables and Jump Diffusion Models Using High Frequency Data

This paper proposes a new approach to exploit the information in high frequency data for the statistical inference of continuous-time affine jump diffusion (AJD) models with latent variables. For this purpose, we construct unbiased estimators of the latent variables and their power functions based on the observed state variables over extended horizons. With the estimates of the latent variables, we propose a GMM procedure for the estimation of AJD models with the distinguishing feature that moments of both observed and latent state variables can be used without resorting to path simulation or discretization of the continuous-time process. Using high frequency return observations of the S&P 500 index, we implement our estimation approach to various continuous-time asset return models with stochastic volatility and random jumps.

Why the Rotation Count Algorithm Works

The characteristic functions of many affine jump-diffusion models, such as Heston’s stochastic volatility model and all of its extensions, involve multivalued functions such as the complex logarithm. If we restrict the logarithm to its principal branch, as is done in most software packages, the characteristic function can become discontinuous, leading to completely wrong option prices if options are priced by Fourier inversion. In this paper we prove under non-restrictive conditions on the parameters that the rotation count algorithm of Kahl and Jackel chooses the correct branch of the complex logarithm. Under the same restrictions we prove that in an alternative formulation of the characteristic function the principal branch is the correct one. Seen as this formulation is easier to implement and numerically more stable than Heston’s formulation, it should be the preferred one. The remainder of this paper shows how complex discontinuities can be avoided in the Schobel-Zhu model and the exact simulation algorithm of the Heston model, recently proposed by Broadie and Kaya. Finally, we show that Matytsin’s SVJJ model has a closed-form characteristic function, though the complex discontinuities that arise there due to the branch switching of the exponential integral cannot be avoided under all circumstances.

Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes

The stochastic differential equations for affine jump diffusion models do not yield exact solutions that can be directly simulated. Discretization methods can be used for simulating security prices under these models. However, discretization introduces bias into the simulation results, and a large number of time steps may be needed to reduce the discretization bias to an acceptable level. This paper suggests a method for the exact simulation of the stock price and variance under Heston’s stochastic volatility model and other affine jump diffusion processes. The sample stock price and variance from the exact distribution can then be used to generate an unbiased estimator of the price of a derivative security. We compare our method with the more conventional Euler discretization method and demonstrate the faster convergence rate of the error in our method. Specifically, our method achieves an O(s-1/2) convergence rate, where s is the total computational budget. The convergence rate for the Euler discretization method is O(s-1/3) or slower, depending on the model coefficients and option payoff function.

Approaches to forecasting volatility: Models and their performances for emerging equity markets

Abstract In this study, the performance of a number of well-known statistical and stochastic models is analyzed when applied to forecasting returns and volatility in some financial markets. A new affine jump-diffusion model is also introduced and it is showed that this model achieves better results than existing ones when used to forecast volatility. The bases for developing the new model are some results showing that jumps introduced both in the return and volatility process play an important role in forecasting volatility, particularly in highly volatile markets, such as emerging equity markets.

Linear-Quadratic Jump-Diffusion Modelling with Application to Stochastic Volatility

We aim at accommodating the existing affine jump-diffusion and quadratic models under the same roof, namely the linear-quadratic jump-diffusion (LQJD) class. We give a complete characterization of the dynamics underlying this class of models as well as identification constraints, and compute standard and extended transforms relevant to asset pricing. We also show that the LQJD class can be embedded into the affine class through use of an augmented state vector. We further establish that an equivalence relationship holds between both classes in terms of transform analysis. An option pricing application to multifactor stochastic volatility models reveals that adding nonlinearity into the model significantly reduces pricing errors, and further addition of a jump component in the stock price largely improves goodness-of-fit for in-the-money calls but less for out-of-the-money ones.

Arrow-Debreu Prices for Affine Models

We put forward a general methodology to price arbitrary payoffs linked to the realization of interest rates, asset prices, or other variables driven by the multivariate Affine Jump-Diffusion process of Duffie and Kan (1996). We attack and solve the basic problem of computing the Arrow-Debreu state prices or, equivalently, Green's functions associated with the process. Given the Arrow-Debreu state prices, one can price derivative instruments with payoffs of arbitrary complexity. Within this framework, we also develop a scheme to price derivatives with early exercise at intermediate dates. To derive Arrow-Debreu state prices we exploit the basic observation that the integral of the overnight interest rate is itself affine. We augment the state space to add the integral of the overnight rate and we use transform methods to compute the density of the augmented affine process to calculate Arrow-Debreu prices. The main goal of the paper is to provide a viable numerical implementation of the proposed methodology, and we illustrate with applications the concepts introduced below. Our primary interest lies in exploring the viability of the numerical implementation, and we will measure advantages and disadvantages of our approach in the associated metric. The method is well suited to price payoffs for which transform methods as, e.g., in Chacko and Das (1999) and Duffie, Pan, and Singleton (1998), cannot be applied. This is typically the case when payoffs are non-linear or non-loglinear in the underlying factors. While the techniques we exploit rely in essence on transform methods, this paper should be of interest also to researchers who prefer simulation or tree-based implementations. A scheme for improving the accuracy of tree-based methods is presented. In a similar vein, we suggest a simulation procedure for the general Affine Jump-Diffusion model, which recovers arbitrage-free prices regardless of the time step. In this context, the proposed methodology can serve as a tool to detect problems in alternative implementations. Consider the case of a jump for instance; our method suggests that the resulting distribution can be multimodal. It is difficult to envision that a tree-based implementation would easily recover the correct state prices without some form of tinkering with the implementation.

Comment on Merton and Samuelson

Affine jump-diffusion models have been the mainstream in options pricing because of their analytical tractability. Popular affine jump-diffusion models, however, are still unsatisfactory in describing the options data and the problem is often attributed to the diffusion term of the unobserved state variables. Using prices of variance-swaps (i.e., squared VIX) implied from options prices, we provide fresh evidence regarding the misspecification of affine jump-diffusion models, as variance-swap prices are affine functions of the state variables in a broader class of models that do not restrict the diffusion term of the state variables. We apply the nonparametric methodology used by Aït-Sahalia (1996b), supplemented with bootstrap tests and other parametric tests, to the S&P 500 index options data from January 1996 to September 2008. We find that, while the affine diffusion term of the state variables may contribute to the misspecification as the literature has suggested, the affine drift of the state variables, jump intensities, and risk premiums are also sources of misspecification.