Stochastic volatility: A tale of co-jumps,non-normality,GMM and high frequency data

In this article we introduce a linear–quadratic volatility model with co-jumps and show how to calibrate this model to a rich dataset. We apply GMM and more specifically match the moments of realized power and multi-power variations, which are obtained from high-frequency stock market data. Our model incorporates two salient features: the setting of simultaneous jumps in both return process and volatility process and the superposition structure of a continuous linear–quadratic volatility process and a Lévy-driven Ornstein–Uhlenbeck process. We compare the quality of fit for several models, and show that our model outperforms the conventional jump diffusion or Bates model. Besides that, we find evidence that the jump sizes are not normally distributed and that our model performs best when the distribution of jump-sizes is only specified through certain (co-) moment conditions. Monte Carlo experiments are employed to confirm this.     

Linear Accounting Valuation When Abnormal Earnings Are AR(2)

The Ohlson (1995) model assumes that abnormal earnings follow an AR(1) process primarily for reasons of mathematical tractability. However, the empirical literature on the Garman and Ohlson (1980) model finds that the data support an AR(2) lag structure for earnings, book values and dividends. Moreover, the AR(2) process encompasses a far richer variety of time series patterns than does the AR(1) process and includes the AR(1) process as a special case. This paper solves the Ohlson model directly for an AR(2) abnormal earnings dynamic. The model is estimated on a time series firm-level basis following the approach used by Myers (1999). It is found that, like the Ohlson AR(1) model, the Ohlson AR(2) model severely underestimates market prices even relative to book values. These results further bring into question the empirical validity of the Ohlson model.     

A Class of Gaussian Hybrid Processes for Modeling Financial Markets

This paper proposes a one-factor model of financial markets using a class of Gaussian process that can be decomposed into a Brownian motion and an Ornstein–Uhlenbeck process. It is shown that this “hybrid” process is obtained as a continuous-time scaling limit of the differenced first-order autoregressive integrated moving average (ARIMA(1,1,1)) process. Parameter estimations using an ARIMA(1,1,1) framework and its variance ratio test show the accuracy of the proposed model. Construction of the one-factor commodity futures price model is presented as an application. A multidimensional extension of the hybrid process is also presented in the Appendix.     

Jump Diffusion Option Valuation in Discrete Time

We develop a simple, discrete time model to value options when the underlying process follows a jump diffusion process. Multivariate jumps are superimposed on the binomial model of Cox, Ross, and Rubinstein (1979) to obtain a model with a limiting jump diffusion process. This model incorporates the early exercise feature of American options as well as arbitrary jump distributions. It yields an efficient computational procedure that can be implemented in practice. As an application of the model, we illustrate some characteristics of the early exercise boundary of American options with certain types of jump distributions.     

Knowledge-Based Simulation for Business Process Redesign

This research focuses on knowledge-based simulation modeling for process redesign. Though the proposed technique can be utilized for ‘starting with a clean slate’, it is particularly well suited for situations where an existing process is already documented, and an attempt is being made to improve or redesign this process. We present a methodology that utilizes the basic process structure (represented in a matrix form), and using a rule-based knowledge acquisition system, interacts with the analyst to construct the process knowledge base. Once all the knowledge has been acquired, the system automatically generates an executable simulation model. Major benefits of this algorithmic approach include (1) reduced model building time, (2) increased analyst productivity, and (3) the assurance that basic process characteristics are not accidentally omitted in the simulation model. To test the validity and applicability of the proposed technique a prototype system has been developed that generates simulation programs in SLAM.© 1997 John Wiley & Sons, Ltd.     

Junp-Diffusion Interest Rate Process: An Empirical Examination

We investigate a jump-diffusion process, which is a mixture of an O-U process used by Vasicek (1977) and a compound Poisson jump process, for the term structure of interest rates. We develop a methodology for estimating the jump-diffusion model and complete an empirical study in comparing the model with the Vasicek model, for the US money market interest rates. The results show that when the short-term interest rate is low, both models predict an upward sloping term structure, with the jump-diffusion model fitting the actual term structure quite well and the Vasicek model overestimating significantly. When the short-term interest rate is high, both models predict a downward sloping term structure, with the jump-diffusion model underestimating the actual term structure more significantly than the Vasicek model.     

An empirical investigation of the two-factor Brennan-Schwartz term structure model

This paper studies the pricing behaviors of default-free bonds based on the two-factor model by Brennan and Schwartz (1979), where a short-term spot rate and a long-term consol rate are the state variables. The logarithm of these two factors is assumed to follow a linear transformation of an Ornstein-Uhlenbeck process. An exact discrete time model is derived to estimate the parameters in the process. The model prices are then numerically solved. The sensitivity analysis indicates that the long-rate process, especially the long-rate volatility parameter, is important in characterizing the term structure of interest rates.     

Modelling electricity prices: a time change approach

To capture mean reversion and sharp seasonal spikes observed in electricity prices, this paper develops a new stochastic model for electricity spot prices by time changing the Jump Cox-Ingersoll-Ross (JCIR) process with a random clock that is a composite of a Gamma subordinator and a deterministic clock with seasonal activity rate. The time-changed JCIR process is a time-inhomogeneous Markov semimartingale which can be either a jump-diffusion or a pure-jump process, and it has a mean-reverting jump component that leads to mean reversion in the prices in addition to the smooth mean-reversion force. Furthermore, the characteristics of the time-changed JCIR process are seasonal, allowing spikes to occur in a seasonal pattern. The Laplace transform of the time-changed JCIR process can be efficiently computed by Gauss–Laguerre quadrature. This allows us to recover its transition density through efficient Laplace inversion and to calibrate our model using maximum likelihood estimation. To price electricity derivatives, we introduce a class of measure changes that transforms one time-changed JCIR process into another time-changed JCIR process. We derive a closed-form formula for the futures price and obtain the Laplace transform of futures option price in terms of the Laplace transform of the time-changed JCIR process, which can then be efficiently inverted to yield the option price. By fitting our model to two major electricity markets in the US, we show that it is able to capture both the trajectorial and the statistical properties of electricity prices. Comparison with a popular jump-diffusion model is also provided.     

[Geometric Lévy Process & MEMM] Pricing Model and Related Estimation Problems

In this article the [Geometric Lévy Process & MEMM] pricingmodel is proposed. This model is an option pricing model for theincomplete markets, and this model is based on the assumptions that theprice processes are geometric Lévy processes and that the pricesof the options are determined by the minimal relative entropy methods.This model has many good points. For example, the theoretical part ofthe model is contained in the framework of the theory of Lévyprocess (additive process). In fact the price process is also aLévy process (with changed Lévy measure) under the minimalrelative entropy martingale measure (MEMM), and so the calculation ofthe prices of options are reduced to the computation of functionals ofLévy process. In previous papers, we have investigated thesemodels in the case of jump type geometric Lévy processes. In thispaper we extend the previous results for more general type of geometricLévy processes. In order to apply this model to real optionpricing problems, we have to estimate the price process of theunderlying asset. This problem is reduced to the estimation problem ofthe characteristic triplet of Lévy processes. We investigate thisproblem in the latter half of the paper.     

Estimation and pricing under long-memory stochastic volatility

We treat the problem of option pricing under a stochastic volatility model that exhibits long-range dependence. We model the price process as a Geometric Brownian Motion with volatility evolving as a fractional Ornstein–Uhlenbeck process. We assume that the model has long-memory, thus the memory parameter H in the volatility is greater than 0.5. Although the price process evolves in continuous time, the reality is that observations can only be collected in discrete time. Using historical stock price information we adapt an interacting particle stochastic filtering algorithm to estimate the stochastic volatility empirical distribution. In order to deal with the pricing problem we construct a multinomial recombining tree using sampled values of the volatility from the stochastic volatility empirical measure. Moreover, we describe how to estimate the parameters of our model, including the long-memory parameter of the fractional Brownian motion that drives the volatility process using an implied method. Finally, we compute option prices on the S&P 500 index and we compare our estimated prices with the market option prices.     

The weekend effect on the distribution of stock prices: Implications for option pricing

Evidence of weekend effects on the distribution of security returns suggests that returns are generated by a process operating closer to trading time rather than calendar time. In contrast, accumulation of interest over the weekend follows a calendar-time process. Since both the variance of returns and the interest rate are important parameters of the Black-Scholes option pricing model, this paper suggests that the model be stated to account for this by utilizing a trading-time variance and a calendar-time interest rate. Empirical evidence indicates that this allows the model to better explain market option prices.     

Why Has U.S. Inflation Become Harder to Forecast?

We examine whether the U.S. rate of price inflation has become harder to forecast and, to the extent that it has, what changes in the inflation process have made it so. The main finding is that the univariate inflation process is well described by an unobserved component trend-cycle model with stochastic volatility or, equivalently, an integrated moving average process with time-varying parameters. This model explains a variety of recent univariate inflation forecasting puzzles and begins to explain some multivariate inflation forecasting puzzles as well.     

A liquidity-based model for asset price bubbles

We provide a new liquidity-based model for financial asset price bubbles that explains bubble formation and bubble bursting. The martingale approach to modeling price bubbles assumes that the asset's market price process is exogenous and the fundamental price, the expected future cash flows under a martingale measure, is endogenous. In contrast, we define the asset's fundamental price process exogenously and asset price bubbles are endogenously determined by market trading activity. This enables us to generate a model that explains both bubble formation and bubble bursting. In our model, the quantity impact of trading activity on the fundamental price process—liquidity risk—is what generates price bubbles. We study the conditions under which asset price bubbles are consistent with no arbitrage opportunities and we relate our definition of the fundamental price process to the classical definition.     

Option pricing under stochastic volatility and tempered stable Lévy jumps

The purpose of this paper is to introduce a stochastic volatility model for option pricing that exhibits Lévy jump behavior. For this model, we derive the general formula for a European call option. A well known particular case of this class of models is the Bates model, for which the jumps are modeled by a compound Poisson process with normally distributed jumps. Alternatively, we turn our attention to infinite activity jumps produced by a tempered stable process. Then we empirically compare the estimated log-return probability density and the option prices produced from this model to both the Bates model and the Black–Scholes model. We find that the tempered stable jumps describe more precisely market prices than compound Poisson jumps assumed in the Bates model.     

The analysis and valuation of interest rate options

This paper provides a simple, alternative model for the valuation of European-style interest rate options. The assumption that drives the hedging argument in the model is that the forward prices of bonds follow an arbitrary two-state process. Later, this assumption is made more specific by postulating that the discount on a zero-coupon bond follows a multiplicative binomial process. In contrast to the Black-Scholes assumption applied to zero-coupon bonds, the limiting distribution of this process has the attractive features that the zero-bond price has a natural barrier at unity (thus precluding negative interest rates), and that the bond price is negatively skewed. The model is used to price interest rate options in general, and interest rate caps and floors in particular. The model is then generalized and applied to European-style options on bonds. A relationship is established between options on swaps and options on coupon bonds. The generalized model then provides a computationally simple formula, closely related to the Black-Scholes formula, for the valuation of European-style options on swaps.     

Purchasing power parity: Modeling and testing mean reversion

A model of mean reversion of exchange rates to purchasing power parity is developed and tested where exchange rates are assumed to follow a mean reverting elastic random walk toward a stochastic PPP rate. The model recognizes the possibility that mean reversion towards PPP may be nonlinear which allows greater flexibility in the adjustment process. Regression equations consistent with the theoretical model are derived. The model is tested using long- and short-term data for six countries. While the results are generally consistent with the findings of previous studies, evidence is presented which demonstrates that the mean reversion process is not linear for some countries.     

The Variance Gamma Process and Option Pricing

A three parameter stochastic process, termed the variance gamma process, that generalizes Brownian motion is developed as a model for the dynamics of log stock prices. The process is obtained by evaluating Brownian motion with drift at a random time given by a gamma process. The two additional parameters are the drift of the Brownian motion and the volatility of the time change. These additional parameters provide control over the skewness and kurtosis of the return distribution. Closed forms are obtained for the return density and the prices of European options. The statistical and risk neutral densities are estimated for data on the S & P500 Index and the prices of options on this Index. It is observed that the statistical density is symmetric with some kurtosis, while the risk neutral density is negatively skewed with a larger kurtosis. The additional parameters also correct for pricing biases of the Black Scholes model that is a parametric special case of the option pricing model developed here.     

An analysis of simultaneous company defaults using a shot noise process

During the subprime mortgage crisis, it became apparent that practical models, such as the one-factor Gaussian copula, had underestimated company default correlations. Complex models that attempt to incorporate default dependency are difficult to implement in practice. In this study, we develop a model for a company asset process, based on which we calculate simultaneous default probabilities using an option-theoretic approach. In our model, a shot noise process serves as the key element for controlling correlations among companies’ assets. The risk factor driving the shot noise process is common to all companies in an industry but the shot noise parameters are assumed company-specific; therefore, every company responds differently to this common risk factor. Our model gives earlier warning of financial distress and predicts higher simultaneous default probabilities than commonly used geometric Brownian motion asset model. It is also computationally simple and can be extended to analyze any finite number of companies.     

THE FAMA‐FRENCH MODEL,LEVERAGE, AND THE MODIGLIANI‐MILLER PROPOSITIONS

For a cost‐of‐equity model to conform to the Modigliani‐Miller cost‐of‐capital propositions, any sensitivity coefficients in the model must be related to the firm's leverage. In this paper I apply these principles to the Fama‐French model for the cost of equity and develop the relation between its sensitivity coefficients and firm leverage. I then examine an empirical process developed by Fama and French (1997) to model the evolution through time of their sensitivity coefficients and show that this empirical process is inconsistent with the Modigliani‐Miller propositions. Separable functions are proposed for these sensitivity coefficients that are consistent with the Modigliani‐Miller propositions.     

Rational expectations and corporate dividend policy

The problem of expectations formation has been either ignored or treated with very restrictive assumptions in traditional dividend adjustment models. Since these models are typically used to explain the dividend decisions of individual firms, a more satisfactory treatment of the process of expectations formation is needed. In order to analyze the dynamic dividend adjustment process, this article proposes a model, more general than previous ones, that is consistent with the rational expectations hypothesis. A nonlinear regression method is used to estimate the parameters of the model and to test the validity of the rational expectations hypothesis in dividend decisions making. The partial adjustment model with rational expectations explains dividend adjustments reasonably well. The overall results suggest that firms make use of available earnings information to form optimal future earnings forecasts; specifically, a firm's dividend adjustment process is completed in about two and a half quarters. This article also finds that the fourth-order serial correlation problem disappears after a generalized Tobit model is used for the parameter estimation.