Malliavin Calculus Techniques Research Articles

Malliavin calculus techniques for local asymptotic mixed normality and their application to hypoelliptic diffusions

Malliavin calculus techniques for local asymptotic mixed normality and their application to hypoelliptic diffusions

Convergence of densities of spatial averages of the parabolic Anderson model driven by colored noise

In this paper, we present a rate of convergence in the uniform norm for the densities of spatial averages of the solution to the d-dimensional parabolic Anderson model driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. The proof is based on the combination of Malliavin calculus techniques and the Stein's method for normal approximations.

On the Skew and Curvature of the Implied and Local Volatilities

ABSTRACT In this paper, we study the relationship between the short-end of the local and the implied volatility surfaces. Our results, based on Malliavin calculus techniques, recover the recent rule (where H denotes the Hurst parameter of the volatility process) for rough volatilities (see F. Bourgey, S. De Marco, P. Friz, and P. Pigato. 2022. “Local Volatility under Rough Volatility.” arXiv:2204.02376v1 [q-fin.MF] https://doi.org/10.48550/arXiv.2204.02376.), that states that the short-time skew slope of the at-the-money implied volatility is of the corresponding slope for local volatilities. Moreover, we see that the at-the-money short-end curvature of the implied volatility can be written in terms of the short-end skew and curvature of the local volatility and vice versa. Additionally, this relationship depends on H.

Pricing cumulative loss derivatives under additive models via Malliavin calculus

We show that integration by parts formulas based on Malliavin-Skorohod calculus techniques for additive processes help us to compute quantities like ${\E}(L_T h(L_T))$ for different suitable functions $h$ and different models for the cumulative loss process $L_T$. These quantities are important in Insurance and Finance. For example they appear in computing expected shortfall risk measures or stop-loss contracts. The formulas given in the present paper, obtained by simple proofs, generalize the formulas given in a recent paper by Hillairet, Jiao and Réveillac using Malliavin calculus techniques for the standard Poisson process, a particular case of additive process.

Anticipative information in a Brownian−Poisson market

The anticipative information refers to some information about future events that may be disclosed in advance. This information may regard, for example, financial assets and their future trends. In our paper, we assume the existence of some anticipative information in a market whose risky asset dynamics evolve according to a Brownian motion and a Poisson process. Using Malliavin calculus and filtration enlargement techniques, we derive the information drift of the mentioned processes and, both in the pure jump case and in the mixed one, we compute the additional expected logarithmic utility. Many examples are shown, where the anticipative information is related to some conditions that the constituent processes or their running maximum may verify, in particular, we show new examples considering Bernoulli random variables.

LAMN property for jump diffusion processes with discrete observations on a fixed time interval

We consider a one-dimensional stochastic differential equation with jumps driven by a Brownian motion and an independent Lévy process with finite Lévy measure, whose drift and diffusion coefficients depend on an unknown parameter. Under smoothness and non-degeneracy assumptions on the drift and diffusion coefficients and integrability assumption of jump size distribution, we prove the Local Asymptotic Mixed Normality property when the solution process is observed discretely at high frequency on a fixed time interval. The proof is essentially based on Malliavin calculus techniques and an analysis of the jump structure of the Lévy process.

Convergence of densities of spatial averages of stochastic heat equation

In this paper, we consider the one-dimensional stochastic heat equation driven by a space–time white noise. In two different scenarios, (i) initial condition u0=1 and general nonlinear coefficient σ, (ii) initial condition u0=δ0 and σ(x)=x (Parabolic Anderson Model), we establish rates of convergence with respect to the uniform distance between the density of spatial averages of (renormalized) solution and the density of the standard normal distribution. These results are based on a combination of Stein’s method for normal approximations and Malliavin calculus techniques. A key ingredient in Case (i) is a new estimate on the Lp-norm of the second Malliavin derivative.

LAMN property for multivariate inhomogeneous diffusions with discrete observations

We consider a class of multidimensional inhomogeneous diffusions whose drift coefficient depends on a multidimensional unknown parameter. Under some appropriate assumptions, we prove the local asymptotic mixed normality property for the drift parameter from high-frequency observations when the length of the observation window tends to infinity. To obtain the result, we use the Malliavin calculus techniques and the change of measures. Our approach is applicable for both ergodic and non-ergodic diffusions.

Smoothness of densities for path-dependent SDEs under Hörmander's condition

We establish the existence of smooth densities for solutions to a broad class of path-dependent SDEs under a Hörmander-type condition. The classical scheme based on the reduced Malliavin matrix turns out to be unavailable in the path-dependent context. We approach the problem by lifting the given n-dimensional path-dependent SDE into a suitable Lp-type Banach space in such a way that the lifted Banach-space-valued equation becomes a state-dependent reformulation of the original SDE. We then formulate Hörmander's bracket condition in Rn for non-anticipative SDE coefficients defining the Lie brackets in terms of vertical derivatives in the sense of the functional Itô calculus. Our pathway to the main result engages an interplay between the analysis of SDEs in Banach spaces, Malliavin calculus, and rough path techniques.

Lp-Theory for the fractional time stochastic heat equation with an infinite-dimensional fractional Brownian motion

In this paper, we study the [Formula: see text]-theory of the fractional time stochastic heat equation [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] denotes the Caputo derivative of order [Formula: see text], and [Formula: see text] is a sequence of i.i.d. fractional Brownian motions with a same Hurst index [Formula: see text]. The integral with respect to fractional Brownian motion is the Skorohod integral. By using the Malliavin calculus techniques and fractional calculus, we obtain a generalized Littlewood–Paley inequality, and prove the existence and uniqueness of [Formula: see text]-solution to such equation.

Analysis of Markov chain approximation for Asian options and occupation-time derivatives: Greeks and convergence rates

The continuous-time Markov chain (CTMC) approximation method is a powerful tool that has recently been utilized in the valuation of derivative securities, and it has the advantage of yielding closed-form matrix expressions suitable for efficient implementation. For two types of popular path-dependent derivatives, the arithmetic Asian option and the occupation-time derivative, this paper obtains explicit closed-form matrix expressions for the Laplace transforms of their prices and the Greeks of Asian options, through the novel use of pathwise method and Malliavin calculus techniques. We for the first time establish the exact second-order convergence rates of the CTMC methods when applied to the prices and Greeks of Asian options. We propose a new set of error analysis methods for the CTMC methods applied to these path-dependent derivatives, whose payoffs depend on the average of asset prices. A detailed error and convergence analysis of the algorithms and numerical experiments substantiate the theoretical findings.

Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise

The numerical approximation of the mild solution to a semilinear stochastic wave equation driven by additive noise is considered. A standard finite element method is employed for the spatial approximation and a a rational approximation of the exponential function for the temporal approximation. First, strong convergence of this approximation in both positive and negative order norms is proven. With the help of Malliavin calculus techniques this result is then used to deduce weak convergence rates for the class of twice continuously differentiable test functions with polynomially bounded derivatives. Under appropriate assumptions on the parameters of the equation, the weak rate is found to be essentially twice the strong rate. This extends earlier work by one of the authors to the semilinear setting. Numerical simulations illustrate the theoretical results.

Local asymptotic properties for Cox‐Ingersoll‐Ross process with discrete observations

AbstractIn this paper, we consider a one‐dimensional Cox‐Ingersoll‐Ross (CIR) process whose drift coefficient depends on unknown parameters. Considering the process discretely observed at high frequency, we prove the local asymptotic normality property in the subcritical case, the local asymptotic quadraticity in the critical case, and the local asymptotic mixed normality property in the supercritical case. To obtain these results, we use the Malliavin calculus techniques developed recently for CIR process together with the estimation for positive and negative polynomial moments of the CIR process. In this study, we require the same conditions of high frequency and infinite horizon as in the case of ergodic diffusions with globally Lipschitz coefficients studied earlier in the literature. However, in the non‐ergodic cases, additional assumptions on the decreasing rate are required due to the fact that the square root diffusion coefficient of the CIR process is not regular enough.

SPDEs with linear multiplicative fractional noise: Continuity in law with respect to the Hurst index

In this article, we consider the one-dimensional stochastic wave and heat equations driven by a linear multiplicative Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index H∈(14,1). We prove that the solution of each of the above equations is continuous in terms of the index H, with respect to the convergence in law in the space of continuous functions. The proof is based on a tightness criterion on the plane and Malliavin calculus techniques in order to identify the limit law.

APPROXIMATING EXPECTED VALUE OF AN OPTION WITH NON-LIPSCHITZ PAYOFF IN FRACTIONAL HESTON-TYPE MODEL

In this paper, we consider option pricing in a framework of the fractional Heston-type model with [Formula: see text]. As it is impossible to obtain an explicit formula for the expectation [Formula: see text] in this case, where [Formula: see text] is the asset price at maturity time and [Formula: see text] is a payoff function, we provide a discretization schemes [Formula: see text] and [Formula: see text] for volatility and price processes correspondingly and study convergence [Formula: see text] as the mesh of the partition tends to zero. The rate of convergence is calculated. As we allow [Formula: see text] to be non-Lipschitz and/or to have discontinuities of the first kind which can cause errors if [Formula: see text] is replaced by [Formula: see text] under the expectation straightforwardly, we use Malliavin calculus techniques to provide an alternative formula for [Formula: see text] with smooth functional under the expectation.

Interest rate option hedging portfolios without bank account

Purpose This paper aims to study the conditions for the hedging portfolio of any contingent claim on bonds to have no bank account part. Design/methodology/approach Hedging and Malliavin calculus techniques recently developed under a stochastic string framework are applied. Findings A necessary and sufficient condition for the hedging portfolio to have no bank account part is found. This condition is applied to a barrier option, and an example of a contingent claim whose hedging portfolio has a bank account part different from zero is provided. Originality/value To the best of the authors’ knowledge, this is the first time that this issue has been addressed in the literature.

A note on the implied volatility of floating strike Asian options

In this paper, we study the short-time behavior of the implied volatility for short-time floating strike Asian options. Our method is based on Malliavin calculus techniques and allows us to construct an approximation formula for the corresponding option prices. Numerical examples are given.

SMALL-TIME ASYMPTOTICS IN GEOMETRIC ASIAN OPTIONS FOR A STOCHASTIC VOLATILITY JUMP-DIFFUSION MODEL

The aim of this paper is to study the small time to maturity of the behavior of the geometric Asian option price and implied volatility under a general stochastic volatility model with Lévy process. The volatility process does not need to be a diffusion or a Markov process, but the future average volatility in the model is a nonadapted process. An anticipating Itô formula for Lévy process and the decomposition of the price (Hull–White formula) are obtained using the Malliavin calculus techniques. The decomposition formula is applied to find the small-time limit of the geometric Asian option price and the implied volatility for the model in at-the-money and out-of-the-money cases.

The implied volatility of Forward-Start options: ATM short-time level, skew and curvature

ABSTRACTUsing Malliavin Calculus techniques, we derive closed-form expressions for the at-the-money behaviour of the forward implied volatility, its skew and its curvature, in general Markovian stochastic volatility models with continuous paths.

Computation of the Delta of European options under stochastic volatility models

We employ Malliavin calculus techniques to compute the Delta of European type options in the presence of stochastic volatility. We obtain a general formula for the Malliavin weight and apply the derived formula to the well known models of Stein-Stein and Heston in order to show the numerical accuracy and efficiency of our approach.