Log-price Process Research Articles

Ruin in a continuous-time risk model with arbitrarily dependent insurance and financial risks triggered by systematic factors

This paper is devoted to asymptotic analysis for a continuous-time risk model with the insurance surplus process and the log-price process of the investment driven by two dependent jump-diffusion processes. We take into account arbitrary dependence between the insurance claims and their corresponding investment return jumps caused by a sequence of systematic factors, whose arrival times constitute a renewal counting process. Under the framework of regular variation, we obtain a simple and unified asymptotic formula for the finite-time ruin probability as the initial wealth becomes large. It turns out that, in the weakly dependent case, the tails of the claims determine the exact decay rate of the finite-time ruin probability while the investment return jumps only contribute to the coefficient of the asymptotic formula; however, in the strongly dependent case, they both produce essential impacts on the finite-time ruin probability which is under-estimated in the weakly dependent case.

Large deviations for fractional volatility models with non-Gaussian volatility driver

We study non-Gaussian fractional stochastic volatility models. The volatility in such a model is described by a positive function of a stochastic process that is a fractional transform of the solution to an SDE satisfying the Yamada–Watanabe condition. Such models are generalizations of a fractional version of the Heston model considered in Bäuerle and Desmettre (2020). We establish sample path and small-noise large deviation principles for the log-price process in a non-Gaussian model. We also illustrate how to compute the second order Taylor expansion of the rate function, in a simplified example.

Time-inhomogeneous Gaussian stochastic volatility models: Large deviations and super roughness

We introduce time-inhomogeneous stochastic volatility models, in which the volatility is described by a nonnegative function of a Volterra type continuous Gaussian process that may have very rough sample paths. The main results obtained in the paper are sample path and small-noise large deviation principles for the log-price process in a time-inhomogeneous super rough Gaussian model under very mild restrictions. We use these results to study the asymptotic behavior of binary barrier options, exit time probability functions, and call options.

Multiple subordinated modeling of asset returns: Implications for option pricing

Motivated by behavioral finance, we introduce multiple embedded financial time clocks. Consistent with asset pricing theory in analyzing equity returns, the investors’ view is considered by introducing a behavioral subordinator. Subordinating to the Brownian motion process in the log-normal model results in a new log-price process whose parameter is as important as the mean and variance. We describe new distributions, demonstrating their use to model tail behavior. The models are applied to S&P 500 returns, treating the Chicago Board Options Exchange (CBOE) volatility index (VIX) as intrinsic-time change and CBOE Volatility-of-Volatility Index as the volatility subordinator. We find these volatility indexes fail as time-change subordinators. We employ a double subordinator model to explain the equity premium puzzle and the excess volatility puzzle. The results indicate the puzzles can be explained by fitting a double subordinator model to the historical data.

Edgeworth corrections for spot volatility estimator

We develop Edgeworth expansion theory for spot volatility estimator under general assumptions on the log-price process that allow for drift and leverage effect. The result is based on further estimation of skewness and kurtosis, when compared with existing second order asymptotic normality result. Thus our theory can provide with a refinement result for the finite sample distribution of spot volatility. We also construct feasible confidence intervals (one-sided and two-sided) for spot volatility by using Edgeworth expansion. The Monte Carlo simulation study we conduct shows that the intervals based on Edgeworth expansion perform better than the conventional intervals based on normal approximation, which justifies the correctness of our theoretical conclusion.

Option Pricing with Mixed Lévy Subordinated Price Process and Implied Probability Weighting Function

It is essential to incorporate the impact of investor behavior when modeling the dynamics of asset returns. In this paper, we reconcile behavioral finance and rational finance by incorporating investor behavior within the framework of dynamic asset pricing theory. To include the views of investors, we employ the method of subordination which has been proposed in the literature by including business (intrinsic, market) time. We define a mixed Levy subordinated model by adding a single subordinated Levy process to the well-known log-normal model, resulting in a new log-price process. We apply the proposed models to study the behavioral finance notion of "greed and fear" disposition from the perspective of rational dynamic asset pricing theory. The greedy or fearful disposition of option traders is studied using the shape of the probability weighting function. We then derive the implied probability weighting function for the fear and greed deposition of option traders in comparison to spot traders. Our result shows the diminishing sensitivity of option traders. Diminishing sensitivity results in option traders overweighting the probability of big losses in comparison to spot traders.

Spot estimation for fractional Ornstein–Uhlenbeck stochastic volatility model: consistency and central limit theorem

There has been an increasing interest for rough stochastic volatility models. However, little is known about the statistical inference for such models, especially for high frequency data. This paper investigates estimation of the fractional spot volatility from discrete observations of the price process on a grid with a time interval $$\Delta _n\rightarrow 0$$ as $$n\rightarrow \infty $$ . Namely, the model with fractional Ornstein–Uhlenbeck log-volatility and Ito-semimartingale log-price processes is considered. In this setup both consistency and central limit theorem are proven for truncated and non-truncated spot volatility estimators. Then, asymptotic confidence intervals are derived for a finite number of spot volatility estimators at different estimation times. Consequently, the highest possible rate of convergence achieved in the central limit theorem $$\Delta _n^{H/(2H+1)}$$ is a function of the Hurst parameter H of the fractional Brownian motion driving the volatility. This rate coincides with the already known highest convergence rate for the Brownian case when $$H=0.5$$ . Furthermore, simulations in this paper validate the consistency and central limit theorem numerically. Article class.

Multiple Subordinated Modeling of Asset Returns: Implications for Option Pricing

Subordination is an often used stochastic process in modeling asset prices. Subordinated Levy price processes and local volatility price processes are now the main tools in modern dynamic asset pricing theory. In this paper, we introduce the theory of multiple internally embedded financial time-clocks motivated by behavioral finance. To be consistent with dynamic asset pricing theory and option pricing, as suggested by behavioral finance, the investors' view is considered by introducing an intrinsic time process which we refer to as a behavioral subordinator. The process is subordinated to the Brownian motion process in the well-known log-normal model, resulting in a new log-price process. The number of embedded subordinations results in a new parameter that must be estimated and this parameter is as important as the mean and variance of asset returns. We describe new distributions, demonstrating how they can be applied to modeling the tail behavior of stock market returns. We apply the proposed models to modeling S&P 500 returns, treating the CBOE Volatility Index as intrinsic time change and the CBOE Volatility-of-Volatility Index as the volatility subordinator. We find that these volatility indexes are not proper time-change subordinators in modeling the returns of the S&P 500.

Time-Inhomogeneous Gaussian Stochastic Volatility Models: Large Deviations and Super Roughness

We introduce time-inhomogeneous stochastic volatility models, in which the volatility is described by a nonnegative function of a Volterra type continuous Gaussian process that may have extremely rough sample paths. The drift function and the volatility function are assumed to be time-dependent and locally ω-continuous for some modulus of continuity ω. The main results obtained in the paper are sample path and small-noise large deviation principles for the log-price process in a Gaussian model under very mild restrictions. We use these results to study the asymptotic behavior of binary up-and-in barrier options and binary call options.

Large Deviation Principles for Stochastic Volatility Models with Reflection and Three Faces of the Stein and Stein Model

We introduce stochastic volatility models, in which the volatility is described by a time-dependent nonnegative function of a reflecting diffusion. The idea to use reflecting diffusions as building blocks of the volatility came into being because of a certain volatility misspecification in the classical Stein and Stein model. A version of this model that uses the reflecting Ornstein-Uhlenbeck process as the volatility process is a special example of a stochastic volatility model with reflection. The main results obtained in the present paper are sample path and small-noise large deviation principles for the log-price process in a stochastic volatility model with reflection under rather mild restrictions. We use these results to study the asymptotic behavior of binary barrier options and call prices in the small-noise regime.

Ruin probabilities for a Lévy-driven generalised Ornstein–Uhlenbeck process

We study the asymptotics of the ruin probability for a process which is the solution of a linear SDE defined by a pair of independent Levy processes. Our main interest is a model describing the evolution of the capital reserve of an insurance company selling annuities and investing in a risky asset. Let $\beta >0$ be the root of the cumulant-generating function $H$ of the increment $V_{1}$ of the log-price process. We show that the ruin probability admits the exact asymptotic $Cu^{-\beta }$ as the initial capital $u\to \infty $, assuming only that the law of $V_{T}$ is non-arithmetic without any further assumptions on the price process.

Gaussian stochastic volatility models: Scaling regimes, large deviations, and moment explosions

In this paper, we establish sample path large and moderate deviation principles for log-price processes in Gaussian stochastic volatility models, and study the asymptotic behavior of exit probabilities, call pricing functions, and the implied volatility. In addition, we prove that if the volatility function in an uncorrelated Gaussian model grows faster than linearly, then, for the asset price process, all the moments of order greater than one are infinite. Similar moment explosion results are obtained for correlated models.

Gaussian Stochastic Volatility Models: Scaling Regimes, Large Deviations, and Moment Explosions

In this paper, we establish sample path large and moderate deviation principles for log-price processes in Gaussian stochastic volatility models, and study the asymptotic behavior of exit probabilities, call pricing functions, and the implied volatility. In addition, we prove that if the volatility function in an uncorrelated Gaussian model grows faster than linearly, then, for the asset price process, all the moments of order greater than one are infinite. Similar moment explosion results are obtained for correlated models.

Variational Formulation of American Option Prices in the Heston Model

We give an analytical characterization of the price function of an American option in Heston-type models. Our approach is based on variational inequalities and extends recent results of Daskalopoulos and Feehan (2011). We study the existence and uniqueness of a weak solution of the associated degenerate parabolic obstacle problem. Then, we use suitable estimates on the joint distribution of the log-price process and the volatility process in order to characterize the analytical weak solution as the solution to the optimal stopping problem. We also rely on semi-group techniques and on the affine property of the model.

Pricing European options under uncertainty with application of Levy processes and the minimal Lq equivalent martingale measure

The study involves pricing of European options with usage of stochastic and fuzzy methods. We use a geometric Levy process as the model of the underlying asset, assuming that the log-price of a primary financial instrument is a jump–diffusion with jump part described by a linear combination of time-homogeneous Poisson processes. Analytical option pricing formulas in crisp case, using the minimal L q equivalent martingale measure, are derived. The pricing expressions are achieved employing probability and stochastic analysis. The fuzzy counterparts of some model parameters are explored due to the fact that they are imprecisely evaluated. Applying fuzzy arithmetic, we derive the analytical option pricing expressions with fuzzy parameters. Moreover, we conceptualize a method of decision-making, taking into consideration the obtained fuzzy formulas. At last, we go through numerical examples to illustrate our theoretical results. Our main achievement in the paper is overcoming difficulties related to derivation of the option pricing formulas by advanced analysis of Jacod–Grigelionis characteristics of the log-price process used in our approach for the minimal L q equivalent martingale measure as well as the skilled use of fuzzy arithmetic methods.

A nonparametric eigenvalue-regularized integrated covariance matrix estimator for asset return data

In high-frequency data analysis, the extreme eigenvalues of a realized covariance matrix are biased when its dimension p is large relative to the sample size n. Furthermore, with non-synchronous trading and contamination of microstructure noise, we propose a nonparametrically eigenvalue-regularized integrated covariance matrix estimator (NERIVE) which does not assume specific structures for the underlying integrated covariance matrix. We show that NERIVE is positive definite in probability, with extreme eigenvalues shrunk nonlinearly under the high dimensional framework p∕n→c>0. We also prove that in portfolio allocation, the minimum variance optimal weight vector constructed using NERIVE has maximum exposure and actual risk upper bounds of order p−1∕2. Incidentally, the same maximum exposure bound is also satisfied by the theoretical minimum variance portfolio weights. All these results hold true also under a jump–diffusion model for the log-price processes with jumps removed using the wavelet method proposed in Fan and Wang (2007). They are further extended to accommodate the existence of pervasive factors such as a market factor under the setting p3∕2∕n→c>0. The practical performance of NERIVE is illustrated by comparing to the usual two-scale realized covariance matrix as well as some other nonparametric alternatives using different simulation settings and a real data set.

Comparing Predictive Accuracy under Long Memory, With an Application to Volatility Forecasting*

This paper extends the popular Diebold-Mariano test to situations when the forecast error loss differential exhibits long memory. It is shown that this situation can arise frequently, since long memory can be transmitted from forecasts and the forecast objective to forecast error loss differentials. The nature of this transmission mainly depends on the (un)biasedness of the forecasts and whether the involved series share common long memory. Further results show that the conventional Diebold-Mariano test is invalidated under these circumstances. Robust statistics based on a memory and autocorrelation consistent estimator and an extended fixed-bandwidth approach are considered. The subsequent Monte Carlo study provides a novel comparison of these robust statistics. As empirical applications, we conduct forecast comparison tests for the realized volatility of the Standard and Poors 500 index among recent extensions of the heterogeneous autoregressive model. While we find that forecasts improve significantly if jumps in the log-price process are considered separately from continuous components, improvements achieved by the inclusion of implied volatility turn out to be insignificant in most situations.

A LOCAL GAUSSIAN BOOTSTRAP METHOD FOR REALIZED VOLATILITY AND REALIZED BETA

This article introduces a local Gaussian bootstrap method useful for the estimation of the asymptotic distribution of high-frequency data-based statistics such as functions of realized multivariate volatility measures as well as their asymptotic variances. The new approach consists of dividing the original data into nonoverlapping blocks ofMconsecutive returns sampled at frequencyh(whereh−1denotes the sample size) and then generating the bootstrap observations at each frequency within a block by drawing them randomly from a mean zero Gaussian distribution with a variance given by the realized variance computed over the corresponding block.Our main contributions are as follows. First, we show that the local Gaussian bootstrap is first-order consistent when used to estimate the distributions of realized volatility and realized betas under assumptions on the log-price process which follows a continuous Brownian semimartingale process. Second, we show that the local Gaussian bootstrap matches accurately the first four cumulants of realized volatility up too(h), implying that this method provides third-order refinements. This is in contrast with the wild bootstrap of Gonçalves and Meddahi (2009,Econometrica77(1), 283–306), which is only second-order correct. Third, we show that the local Gaussian bootstrap is able to provide second-order refinements for the realized beta, which is also an improvement of the existing bootstrap results in Dovonon, Gonçalves, and Meddahi (2013,Journal of Econometrics172, 49–65) (where the pairs bootstrap was shown not to be second-order correct under general stochastic volatility). In addition, we highlight the connection between the local Gaussian bootstrap and the local Gaussianity approximation of continuous semimartingales established by Mykland and Zhang (2009,Econometrica77, 1403–1455) and show the suitability of this bootstrap method to deal with the new class of estimators introduced in that article. Lastly, we provide Monte Carlo simulations and use empirical data to compare the finite sample accuracy of our new bootstrap confidence intervals for integrated volatility with the existing results.

Estimation of Quadratic Variation via a Generalized Ito Isometry

Recently, two types of innovative estimators of the quadratic variation (QV), namely RVB and RVQ estimators, are established in Yu (2020) for an asset's log-price process permitting time-varying spot volatility. Both RVB and RVQ utilize either higher-order- or cross-terms of log-return and hence they have been demonstrated having better convergence property than existing ones (e.g. the widely-known realized volatility). However, the proof of the convergence property of RVB/RVQ is rather cumbersome and algebraically complicated, and the restriction on spot volatility (being a deterministic function of time) in the log-price process is a little bit strong. This article mainly addresses two issues: (1)relax the restriction for asset's price process; (2)substantially shorten the original proofs by alternatively employing two useful lemmas (called high-order It^o isometry and adjoint It^o identity) developed in this study.

Large Deviation Principle for Volterra Type Fractional Stochastic Volatility Models

We study fractional stochastic volatility models in which the volatility process is a positive continuous function $\sigma$ of a continuous Gaussian process $\widehat{B}$. Forde and Zhang established a large deviation principle for the log-price process in such a model under the assumptions that the function $\sigma$ is globally Hölder continuous and the process $\widehat{B}$ is fractional Brownian motion. In the present paper, we prove a similar small-noise large deviation principle under weaker restrictions on $\sigma$ and $\widehat{B}$. We assume that $\sigma$ satisfies a mild local regularity condition, while the process $\widehat{B}$ is a Volterra type Gaussian process. Under an additional assumption of the self-similarity of the process $\widehat{B}$, we derive a large deviation principle in the small-time regime. As an application, we obtain asymptotic formulas for binary options, call and put pricing functions, and the implied volatility in certain mixed regimes.