Continuous-time Diffusion Processes Research Articles

Realized volatility or price range: Evidence from a discrete simulation of the continuous time diffusion process

The study provides evidence in favor of the price range as a proxy estimator of volatility in financial time series, in the cases that either intra-day datasets are unavailable or they are available at a low sampling frequency.A stochastic differential equation with time varying volatility of the instantaneous log-returns process is simulated, in order to mimic the continuous time diffusion analogue of the discrete time volatility process. The simulations provide evidence that the price range measures are superior to the realized volatility constructed at low sampling frequency. The high-low price range volatility estimator is more accurate than the realized volatility estimator based on five, or less, equidistance points in time. The open-high-low-close price range is more accurate than the realized volatility estimator based on eight, or less, intra-period log-returns.

Estimating the diffusion coefficient function for a diversified world stock index

This paper deals with the estimation of continuous-time diffusion processes which model the dynamics of a well diversified world stock index (WSI). We use the nonparametric kernel-based estimation to empirically identify a square root type diffusion coefficient function in the dynamics of the discounted WSI. A square root process turns out to be an excellent building block for a parsimonious model for the WSI. Its dynamics allow capturing various empirical stylized facts and long term properties of the index, as well as, the explicit computation of various financial quantities.

Estimating the Diffusion Coefficient Function for a Diversified World Stock Index

This paper deals with the estimation of continuous-time diffusion processes which model the dynamics of a well diversified world stock index (WSI). We use nonparametric kernel-based estimation to empirically identify a square root type diffusion coefficient function in the dynamics of the discounted WSI. A square root process turns out to be an excellent building block for a parsimonious model for the WSI. Its dynamics allow capturing various empirical stylized facts and long term properties of the index, as well as, the explicit computation of various financial quantities.

On Generating Monte Carlo Samples of Continuous Diffusion Bridges

Diffusion processes are widely used in engineering, finance, physics, and other fields. Usually continuous-time diffusion processes can be observed only at discrete time points. For many applications, it is often useful to impute continuous-time bridge samples that follow the diffusion dynamics and connect each pair of the consecutive observations. The sequential Monte Carlo (SMC) method is a useful tool for generating the intermediate paths of the bridge. The paths often are generated forward from the starting observation and forced in some ways to connect with the end observation. In this article we propose a constrained SMC algorithm with an effective resampling scheme guided by backward pilots carrying the information of the end observation. This resampling scheme can be easily combined with any forward SMC sampler. Two synthetic examples are used to demonstrate the effectiveness of the resampling scheme.

A damped diffusion framework for financial modeling and closed-form maximum likelihood estimation

Asset price bubbles can arise unintentionally when one uses continuous-time diffusion processes to model financial quantities. We propose a flexible damped diffusion framework that is able to break many types of bubbles and preserve the martingale pricing approach. Damping can be done on either the diffusion or drift function. Oftentimes, certain solutions to the valuation PDE can be ruled out by requiring the solution to be a limit of martingale prices for damped diffusion models. Monte Carlo study shows that with finite time-series length, maximum likelihood estimation often fails to detect the damped diffusion function while fabricates nonlinear drift function. An alternative method based on Aït-Sahalia's specification test on parametric models is proposed.

A Damped Diffusion Framework for Financial Modeling and Closed-Form Maximum Likelihood Estimation

Asset price bubbles can arise unintentionally when one uses continuous-time diffusion processes to model financial quantities. We propose a flexible damped diffusion framework that is able to break many types of bubbles and preserve the martingale pricing approach. Damping can be done on either the diffusion or drift function. Oftentimes, certain solutions to the valuation PDE can be ruled out by requiring the solution to be a limit of martingale prices for damped diffusion models. Monte Carlo study shows that with finite time-series length, maximum likelihood estimation often fails to detect the damped diffusion function while fabricates nonlinear drift function. An alternative method based on A\{i}t-Sahalia's specification test on parametric models is proposed.

Parameter estimation and bias correction for diffusion processes

This paper considers parameter estimation for continuous-time diffusion processes which are commonly used to model dynamics of financial securities including interest rates. To understand why the drift parameters are more difficult to estimate than the diffusion parameter, as observed in previous studies, we first develop expansions for the bias and variance of parameter estimators for two of the most employed interest rate processes, Vasicek and CIR processes. Then, we study the first order approximate maximum likelihood estimator for linear drift processes. A parametric bootstrap procedure is proposed to correct bias for general diffusion processes with a theoretical justification. Simulation studies confirm the theoretical findings and show that the bootstrap proposal can effectively reduce both the bias and the mean square error of parameter estimates, for both univariate and multivariate processes. The advantages of using more accurate parameter estimators when calculating various option prices in finance are demonstrated by an empirical study.

An empirical comparison of continuous-time models of implied volatility indices

We explore the ability of alternative popular continuous-time diffusion and jump-diffusion processes to capture the dynamics of implied volatility indices over time. The performance of the various models is assessed under both econometric and financial metrics. To this end, data are employed from major European and American implied volatility indices and the rapidly growing CBOE volatility futures market. We find that the addition of jumps is necessary to capture the evolution of implied volatility indices under both metrics. Mean reversion is of second-order importance though. The results are consistent across the various metrics, markets, and construction methodologies.

Applying Game Options in a Loan Exit Model

This study explores market exit decisions of financial institutions into a duopolistic loan in the evaluation concept. A game options approach is used to analyze how uncertainty influences loan decisions. Suppose that the loan market only contains two financial institutions (the leader and follower), and these two roles can be chosen freely. The financial effect for outside shock follows the Poisson downward jump process, and the leader and follower exit the thresholds and are compared to the loan market in the continuous time diffusion process. This study obtains the following results: In the exiting model, the exit costs for the leader are influenced by those of the follower. If the ratio of the leader to follower exit costs is stable, the first-in-first-out or last-in-first-out optimal evaluation models are derived during two financial institutions exiting the loan market.

Diffusion Estimation and Structural Stability Tests of Short Term Interest Rate Models

This paper proposes a simple method for estimating the diffusion function of continuous-time diffusion processes without specifying the drift function structure. This estimation method is suitable for practitioners choosing an interest rate derivative pricing model among the short-term rate type. We estimate the diffusion function parameters by using one-month Euro rates; US Dollar, German Mark, and Japanese Yen. We conduct structural stability tests of the diffusion function parameters and find a statistically significant structural change in the German Mark process. We, however, do not detect structural changes in the US Dollar and the Japanese Yen processes during the same period.

Estimators of diffusions with randomly spaced discrete observations: A general theory

We provide a general method to analyze the asymptotic properties of a variety of estimators of continuous time diffusion processes when the data are not only discretely sampled in time but the time separating successive observations may possibly be random. We introduce a new operator, the generalized infinitesimal generator, to obtain Taylor expansions of the asymptotic moments of the estimators. As a special case, our results apply to the situation where the data are discretely sampled at a fixed nonrandom time interval. We include as specific examples estimators based on maximum-likelihood and discrete approximations such as the Euler scheme.

Diffusion processes in electromagnetic scattering generating K ‐distributed noise

Wepresent a classification of continuous-time diffusion processes for the local scattering cross-section and scattered intensity for an electromagnetic field scattering from a random medium, such that the asymptotic marginal distribution for the intensity is the K ‐distribution. These processes are represented as Ito stochastic differential equations, which enables identification of their stochastic volatilities with certain free functions that serve to calibrate the model. A study is made of the volatility behaviour of the electromagnetic intensity scattered from a random phase screen, which establishes the form of the volatility function to a high accuracy. This model is used to derive a stochastic differential equation for the complex‐valued amplitude, which is then applied to radar scattering from the sea surface for the purpose of anomaly detection.

Risk‐sensitive control of stochastic hybrid systems on infinitetime horizon

A risk‐sensitive optimal control problem is considered for a hybrid system that consists of continuous time diffusion process that depends on a discrete valued mode variable that is modeled as a Markov chain. Optimality conditions are presented and conditions for the existence of optimal controls are derived. It is shown that the optimal risk‐sensitive control problem is equivalent to the upper value of an associated stochastic differential game, and insight into the contributions of the noise input and mode variable to the risk sensitivity of the cost functional is given. Furthermore, it is shown that due to the mode variable risk sensitivity, the equivalence relationship that has been observed between risk‐sensitive and H∞ control in the nonhybrid case does not hold for stochastic hybrid systems.

Implied Binomial Trees

Despite its success, the Black-Scholes formula has become increasingly unreliable over time in the very markets where one would expect it to be most accurate. In addition, attempts by financial economists to extract probabilistic information from option prices have been puny in comparison to what is clearly possible. This paper develops a new method for inferring risk-neutral probabilities (or state- contingent prices) from the simultaneously observed prices of European options. These probabilities are then used to infer a unique fully specified recombining binomial tree that is consistent with these probabilities (and hence consistent with all the observed option prices). If specified exogenously, the model can also accommodate local interest rates and underlying asset payout rates which are general functions of the concurrent underlying asset price and time. In a 200 step lattice, for example, there are a total of 60,301 unknowns: 40,200 potentially different move sizes, 20,100 potentially different move probabilities, and 1 interest rate to be determined from 60,301 independent equations, many of which are non-linear in the unknowns. Despite this, a backwards recursive solution procedure exists which is only slightly more time-consuming than for a standard binomial tree with given constant move sizes and move probabilities. Moreover, closed-form expressions exist for the values and hedging parameters of European options maturing with or before the end of the tree. The tree can also be used to value and hedge American and several types of exotic options. Interpreted in terms of continuous-time diffusion processes, the model here assumes that the drift and local volatility are at most functions of the underlying asset price and time. But instead of beginning with a parameterization of these functions (as in previous research), the model derives these functions endogenously to fit current option prices. As a result, it can be thought of as an attempt to exhaust the potential for single state-variable path-independent diffusion processes to rectify problems with the Black- Scholes formula that arise in practice.

The Preisach model with stochastic input as a model for aftereffect

In previous papers, we have suggested to use the Preisach model driven by stochastic inputs as a model for aftereffect. However, in these papers the stochastic inputs have been modeled by discrete time I.I.D. (independent identically distributed) random processes. Here, we further extend the aforementioned approach by modeling the stochastic inputs by continuous time diffusion processes. It is shown that the mathematical machinery of the ‘‘exit problem’’ is instrumental for calculations of time evolutions of the expected value of the output of the Preisach model.

Optimal Control of Favorable Games with a Time Limit

This paper shows how to optimally control a stochastic process if one seeks to reach a certain value before a fixed time without first hitting zero. The process has a drift in the favorable direction. Discrete time random walks, as well as continuous time diffusion processes, are considered. The controls available are such that zero variance means zero drift, and more variance means more drift. To be more precise, in the continuous time case it holds that $dX_t = \sigma (X_t ,t)(\mu _t dt + dB_t )$, where $\sigma $ is the control variable. As corollaries of the results, some interesting inequalities for stochastic processes are obtained.

Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application

In this paper, we examine the pricing of European call options on stocks that have vari? ance rates that change randomly. We study continuous time diffusion processes for the stock return and the standard deviation parameter, and we find that one must use the stock and two options to form a riskless hedge. The riskless hedge does not lead to a unique option pricing function because the random standard deviation is not a traded security. One must appeal to an equilibrium asset pricing model to derive a unique option pricing function. In general, the option price depends on the risk premium associated with the random standard deviation. We find that the problem can be simplified by assuming that volatility risk can be diversified away and that changes in volatility are uncorrelated with the stock return. The resulting solution is an integral ofthe Black-Scholes formula and the distribution function for the variance of the stock price. We show that accurate option prices can be computed via Monte Carlo simulations and we apply the model to a set of actual prices.

A Generalization of the CAPM Based on a Property of the Covariance Operator

A key assumption behind the traditional capital asset pricing model (CAPM) is the joint normality of security returns. Recently, however, this assumption has been relaxed in at least two directions. First, the emergence of continuous-time models has shifted emphasis from discrete-time random variables to continuous-time diffusion processes, with log-normality (as opposed to normality) for security prices in the stationary case. Second, the recognition that the CAPM is difficult to test empirically has led to the development of an asset pricing theory based on an arbitrage argument in large markets and free of any distributional assumption.

On the steady state gametic frequency distribution for two linked loci in a finite population with recurrent reversible mutation

A theoretical study is made of some random-mating models of a finite population of monoecious, diploid individuals. Two linked loci are studied at each of which there are two possible alleles, and mutation from each allele to the other can take place at every generation. The equilibrium distribution of gene frequencies and linkage disequilibrium under these conditions is investigated, and differently structured discrete generation models involving zygotic and gametic sampling, respectively, are compared. These models are also compared with the continuous time diffusion process which approximates the discrete time process. To aid the comparisons, some numerical results are obtained for small and moderate sized populations in the three cases in which there is no recombination, close linkage, and free recombination between the loci. Moments of the distribution connected with the variance of linkage disequilibrium show that the values calculated using the diffusion approximation are sometimes closer to the more realistic zygotic sampling model than are those given by the gametic sampling model, the latter model having been used elsewhere to test the accuracy of the diffusion approximation.