A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk

This article presents a technique for nonparametrically estimating continuous-time diffusion processes that are observed at discrete intervals. We illustrate the methodology by using daily three and six month Treasury Bill data, from January 1965 to July 1995, to estimate the drift and diffusion of the short rate, and the market price of interest rate risk. While the estimated diffusion is similar to that estimated by Chan, Karolyi, Longstaff, and Sanders (1992) , there is evidence of substantial nonlinearity in the drift. This is close to zero for low and medium interest rates, but mean reversion increases sharply at higher interest rates.     

Estimation of continuous-time models with an application to equity volatility dynamics

The treatment of this article renders closed-form density approximation feasible for univariate continuous-time models. Implementation methodology depends directly on the parametric-form of the drift and the diffusion of the primitive process and not on its transformation to a unit-variance process. Offering methodological convenience, the approximation method relies on numerically evaluating one-dimensional integrals and circumvents existing dependence on intractable multidimensional integrals. Density-based inferences can now be drawn for a broader set of models of equity volatility. Our empirical results provide insights on crucial outstanding issues related to the rank-ordering of continuous-time stochastic volatility models, the absence or presence of nonlinearities in the drift function, and the desirability of pursuing more flexible diffusion function specifications.     

Inference Methods for Discretely Observed Continuous-Time Stochastic Volatility Models: A Commented Overview

In this paper an overview of inference methods for continuous-time stochastic volatility models observed at discrete times is presented. It includes estimation methods for both parametric and nonparametric models that are completely or partially observed in a variety of situations where the data might be nonlinear functions of the components of the model and/or contaminated with observation noise. In each case, the main reported methods are presented, making emphasis on underlying ideas, theoretical properties of the estimators (bias, consistency, efficient, etc.), and the viability of their implementation to solve actual problems in finance.     

Drift Estimation of Generalized Security Price Processes from High Frequency Derivative Prices

This paper presents a framework for using high frequency derivative prices to estimate the drift of generalized security price processes. This work may be seen more generally as a quasi-likelihood approach to estimating continuous-time parameters of derivative pricing models using discrete option data. We develop a generalized derivative-based estimator for the drift where the underlying security price process follows any arbitrary state-time separable diffusion process (including arithmetic and geometric Brownian motion as special cases). The framework provides a method to measure premia in derivative prices, test for risk-neutral pricing and leads to a new empirical approach to pricing derivative contingent claims. A sufficient condition for the asymptotic consistency of the generalized estimator is also obtained. A study based on generating the S&P500 index and calls shows that the estimator can correctly estimate the drift parameter. This revised version was published online in November 2006 with corrections to the Cover Date.     

Likelihood-based specification analysis of continuous-time models of the short-term interest rate

An extensive collection of continuous-time models of the short-term interest rate is evaluated over data sets that have appeared previously in the literature. The analysis, which uses the simulated maximum likelihood procedure proposed by Durham and Gallant (2002), provides new insights regarding several previously unresolved questions. For single factor models, I find that the volatility, not the drift, is the critical component in model specification. Allowing for additional flexibility beyond a constant term in the drift provides negligible benefit. While constant drift would appear to imply that the short rate is nonstationary, in fact, stationarity is volatility-induced. The simple constant elasticity of volatility model fits weekly observations of the three-month Treasury bill rate remarkably well but is easily rejected when compared with more flexible volatility specifications over daily data. The methodology of Durham and Gallant can also be used to estimate stochastic volatility models. While adding the latent volatility component provides a large improvement in the likelihood for the physical process, it does little to improve bond-pricing performance.     

A statistical comparison of the short-term interest rate models for Japan,U.S., and Germany

We propose a simple and practical model selection method for continuous time models. We apply the method to several continuous time short-term interest rate models using discrete time series data of Japan, U.S. and Germany. All the models can be easily estimated from discrete observations, and their performances can be evaluated in a uniform statistical framework. The models that allow dependence of volatility on the level of interest rates tend to perform well empirically. The degree of volatility dependence on the interest rate levels seems to be different across the countries. For the German data, we observe that a model with nonlinear drift performs better than the best linear drift model.     

Reduced form modeling of limit order markets

This paper proposes a parametric approach for stochastic modeling of limit order markets. The models are obtained by augmenting classical perfectly liquid market models with a few additional risk factors that describe liquidity properties of the order book. The resulting models are easy to calibrate and to analyse using standard techniques for multivariate stochastic processes. Despite their simplicity, the models are able to capture several properties that have been found in microstructural analysis of limit order markets. Calibration of a continuous-time three-factor model to Copenhagen Stock Exchange data exhibits, for example, mean reversion in liquidity as well as the so-called crowding out effect, which influences subsequent mid-price moves. Our dynamic models are also well suited for analysing market resilience after liquidity shocks.     

Option Pricing Under Autoregressive Random Variance Models

Abstract

The autoregressive random variance (ARV) model introduced by Taylor (1980, 1982, 1986) is a popular version of stochastic volatility (SV) models and a discrete-time simplification of the continuous-time diffusion SV models. This paper introduces a valuation model for options under a discrete-time ARV model with general stock and volatility innovations. It employs the discretetime version of the Esscher transform to determine an equivalent martingale measure under an incomplete market. Various parametric cases of the ARV models, are considered, namely, the log-normal ARV models, the jump-type Poisson ARV models, and the gamma ARV models, and more explicit pricing formulas of a European call option under these parametric cases are provided. A Monte Carlo experiment for some parametric cases is also conducted.     

Asset Pricing Using Trading Volumes in a Hidden Regime-Switching Environment

By utilizing information about prices and trading volumes, we discuss the pricing of European contingent claims in a continuous-time hidden regime-switching environment. Hidden market sentiments described by the states of a continuous-time, finite-state, hidden Markov chain represent a common factor for an asset’s drift and volatility, as well as its trading volumes. Using observations about trading volumes, we present a filtered estimate of the hidden common factor. The asset pricing problem is then considered in a filtered market, where the hidden drift and volatility are replaced by their filtered estimates. We adopt the Esscher transform to select an equivalent martingale measure for pricing and derive a partial-differential integral equation for the option price.     

Convergence from discrete- to continuous-time contingent claims prices

This article generalizes the Cox, Ross, and Rubinstein (1979)binomial option-pricing model, and establishes a convergencefrom discrete-time multivariate multinomial models to continuous-timemultidimensional diffusion models for contingent claims prices.The key to the approach is to approximate the N-dimensionaldiffusion price process by a sequence of N-variate, (N+1)-nomialprocess. It is shown that contingent claims prices and dynamicreplicating portfolio strategies derived from the discrete timemodels converge to their corresponding continuous-time limits.     

Multiperiod Optimal Investment-Consumption Strategies with Mortality Risk and Environment Uncertainty

Abstract

In this article we investigate three related investment-consumption problems for a risk-averse investor: (1) an investment-only problem that involves utility from only terminal wealth, (2) an investment-consumption problem that involves utility from only consumption, and (3) an extended investment-consumption problem that involves utility from both consumption and terminal wealth. Although these problems have been studied quite extensively in continuous-time frameworks, we focus on discrete time. Our contributions are (1) to model these investmentconsumption problems using a discrete model that incorporates the environment risk and mortality risk, in addition to the market risk that is typically considered, and (2) to derive explicit expressions of the optimal investment-consumption strategies to these modeled problems. Furthermore, economic implications of our results are presented. It is reassuring that many of our findings are consistent with the well-known results from the continuous-time models, even though our models have the additional features of modeling the environment uncertainty and the uncertain exit time.     

Semi-parametric modelling in finance: theoretical foundations

Abstract

The benchmark theory of mathematical finance is the Black-Scholes-Merton theory, based on Brownian motion as the driving noise process for asset prices. Here the distributions of returns of the assets in a portfolio are multivariate normal. The two most obvious limitations here concern symmetry and thin tails, neither being consistent with real data. The most common replacements for the multinormal are parametric—stable, generalized hyperbolic, variance gamma. In this paper we advocate the use of semi-parametric models for distributions, where the mean vector μ and covariance Σ are parametric components and the so-called density generator (function) g is the non-parametric component. We work mainly within the family of elliptically contoured distributions, focusing particularly on normal variance mixtures with self-decomposable mixing distributions. We show how the parametric cases can be treated in a unified, systematic way within the non-parametric framework and obtain the density generators for the most important cases.     

An efficient algorithm for pricing barrier options in arbitrage-free binomial models with calibrated drift terms

The interrelation between the drift coefficient of price processes on arbitrage-free financial markets and the corresponding transition probabilities induced by a martingale measure is analysed in a discrete setup. As a result, we obtain a flexible setting that encompasses most arbitrage-free binomial models. It is argued that knowledge of the link between drift and transition probabilities may be useful for pricing derivatives such as barrier options. The idea is illustrated in a simple example and later extended to a general numerical procedure. The results indicate that the option values in our fitted drift model converge much faster to closed-form solutions of continuous models for a wider range of contract specifications than those of conventional binomial models.     

Direct Derivation of Finite-Time Ruin Probabilities in the Discrete Risk Model with Exponential or Geometric Claims

Abstract

Growing research interest has been shown in finite-time ruin probabilities for discrete risk processes, even though the literature is not as extensive as for continuous-time models. The general approach is through the so-called Gerber-Shiu discounted penalty function, obtained for large families of claim severities and discrete risk models. This paper proposes another approach to deriving recursive and explicit formulas for finite-time ruin probabilities with exponential or geometric claim severities. The proposed method, as compared to the general Gerber-Shiu approach, is able to provide simpler derivation and straightforward expressions for these two special families of claims.     

A semi-parametric approach to risk management

Abstract

The benchmark theory of mathematical finance is the Black–Scholes–Merton (BSM) theory, based on Brownian motion as the driving noise process for stock prices. Here the distributions of financial returns of the stocks in a portfolio are multivariate normal. Risk management based on BSM underestimates tails. Hence estimation of tail behaviour is often based on extreme value theory (EVT). Here we discuss a semi-parametric replacement for the multivariate normal involving normal variance–mean mixtures. This allows a more accurate modelling of tails, together with various degrees of tail dependence, while (unlike EVT) the whole return distribution can be modelled. We use a parametric component, incorporating the mean vector μ and covariance matrix Σ, and a non-parametric component, which we can think of as a density on [0,∞), modelling the shape (in particular the tail decay) of the distribution. We work mainly within the family of elliptically contoured distributions, focusing particularly on normal variance mixtures with self-decomposable mixing distributions. We discuss efficient methods to estimate the parametric and non-parametric components of our model and provide an algorithm for simulating from such a model. We fit our model to several financial data series. Finally, we calculate value at risk (VaR) quantities for several portfolios and compare these VaRs to those obtained from simple multivariate normal and parametric mixture models.     

Risk measures for derivatives with Markov-modulated pure jump processes

We consider a regime-switching HJB approach to evaluate risk measures for derivative securities when the price process of the underlying risky asset is governed by the exponential of a pure jump process with drift and a Markov switching compensator. The pure jump process is flexible enough to incorporate both the infinite, (small), jump activity and the finite, (large), jump activity. The drift and the compensator of the pure jump process switch over time according to the state of a continuous-time hidden Markov chain representing the state of an economy. The market described by our model is incomplete. Hence, there is more than one pricing kernel and there is no perfect hedging strategy for a derivative security. We derive the regime-switching HJB equations for coherent risk measures for the unhedged position of derivative securities, including standard European options and barrier options. For measuring risk inherent in the unhedged option position, we first need to mark the position into the market by valuing the option. We employ a well-known tool in actuarial science, namely, the Esscher transform to select a pricing kernel for valuation of an option and to generate a family of real-world probabilities for risk measurement. We also derive the regime-switching HJB-variational inequalities for coherent risk measures for American-style options.     

In- and out-of-sample specification analysis of spot rate models: Further evidence for the period 1982–2008

We review and construct consistent in-sample specification and out-of-sample model selection tests on conditional distributions and predictive densities associated with continuous multifactor (possibly with jumps) and (non)linear discrete models of the short term interest rate. The results of our empirical analysis are used to carry out a “horse-race” comparing discrete and continuous models across multiple sample periods, forecast horizons, and evaluation intervals. Our evaluation involves comparing models during two distinct historical periods, as well as across our entire weekly sample of Eurodollar deposit rates from 1982 to 2008. Interestingly, when our entire sample of data is used to estimate competing models, the “best” performer in terms of distributional “fit” as well as predictive density accuracy, both in-sample and out-of-sample, is the three factor Chen (Chen, 1996) model examined by Andersen, Benzoni and Lund (2004). Just as interestingly, a logistic type discrete smooth transition autoregression (STAR) model is preferred to the “best” continuous model (i.e. the one factor Cox, Ingersoll, and Ross (CIR: 1985) model) when comparing predictive accuracy for the “Stable 1990s” period that we examine. Moreover, an analogous result holds for the “Post 1990s” period that we examine, where the STAR model is preferred to a two factor stochastic mean model. Thus, when the STAR model is parameterized using only data corresponding to a particular sub-sample, it outperforms the “best” continuous alternative during that period. However, when models are estimated using the entire dataset, the continuous CHEN model is preferred, regardless of the variety of model specification (selection) test that is carried out. Given that it is very difficult to ascertain the particular future regime that will ensue when constructing ex ante predictions, thus, the CHEN model is our overall “winning” model, regardless of sample period.     

Imperfect Competition among Informed Traders

We analyze competition among informed traders in the continuous-time Kyle(1985) model, as Foster and Viswanathan (1996) do in discrete time. We explicitly describe the unique linear equilibrium when signals are imperfectly correlated and confirm the conjecture of Holden and Subrahmanyam (1992) that there is no linear equilibrium when signals are perfectly correlated. One result is that at some date, and at all dates thereafter, the market would have been more informationally efficient had there been a monopolist informed trader instead of competing traders. The relatively large amount of private information remaining near the end of trading causes the market to approach complete illiquidity.     

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.     

Attainability of European path-independent claims in incomplete markets

In this paper we consider the question which path-independent claims are attainable through self-financing trading strategies in an incomplete market. For continuous-time stochastic volatility models we show that only affine payoffs can be replicated. We provide a simple proof for this proposition based on the requirement that, for replication, the stock and the claim must be locally perfectly correlated, and based on the partial differential equation that any path-independent claim has to satisfy. Moreover, we show that this result does not carry over to discrete setups.