This paper uses Malliavin calculus to price and compute delta of financial derivatives in jump-diffusion models.
problem Pricing and delta computation of financial derivatives in jump-diffusion models with stochastic intensity.
method Utilizes Malliavin calculus to price and compute delta, applying the Euler scheme for convergence analysis.
result Established the convergence of approximated solution, financial derivative, and its delta Greeks.
We extend the Bismut-Elworthy-Li formula to non-degenerate jump diffusions and "payoff" functions depending on the process at multiple future times. In the spirit of Fournie et al [13] and Davis and Johansson [9] this can improve Monte Carlo numerics for stochastic volatility models with jumps. To this end one needs so…
A new machine learning method solves high-dimensional Kolmogorov PDEs efficiently.
problem Solving high-dimensional Kolmogorov PDEs and SDEs.
method Stochastic weighted minimization and stochastic gradient descent with Malliavin weights.
result Accurate approximation of high-dimensional Kolmogorov PDEs and SDEs without curse of dimensionality.
Researchers develop Malliavin calculus for signatures, simplifying option Greeks computation.
problem Lack of tractability and explicit representations in Malliavin calculus.
method Focus on finite linear combinations of time-extended Brownian motion signatures, derive explicit formulas for Malliavin derivative, and compute Greeks for path-dependent options.
result Closed-form expressions for classical operators of Malliavin calculus, providing algebraic formulations.
The paper provides an efficient method to price path-dependent derivatives using multiscale stochastic volatility models.
problem Pricing path-dependent derivatives under multiscale stochastic volatility models.
method Derives a Malliavin representation for the first-order approximation of the price of path-dependent derivatives.
result An efficient Monte Carlo approximation for pricing path-dependent derivatives is derived.
In mathematical Finance calculating the Greeks by Malliavin weights has proved to be a numerically satisfactory procedure for finite-dimensional Itô-diffusions. The existence of Malliavin weights relies on absolute continuity of laws of the projected diffusion process and a sufficiently regular density. In this article…
The extremely useful method of Malliavin calculus has not yet gained adequate popularity because of the complicated analytic apparatus of this method. The author attempts here to propose a simplified algebraic formalism similar to Malliavin calculus, but based on the notion of creation-annihilation operators instead of…
Develops a method to approximate convexity adjustments for interest rate products.
problem Finding accurate convexity adjustments for interest rate products.
method Uses Malliavin calculus to develop an approximation method.
result Excellent numerical accuracy of the formulas for various interest rate products.
Paper proves SVV model reproduces power-law skew in implied volatilities.
problem Reproducing power-law behavior in implied volatility skew.
method Analytical proof using Malliavin calculus and Volterra kernel selection.
result SVV model reproduces power-law skew under correct kernel choice.
This study introduces computation of option sensitivities (Greeks) using the Malliavin calculus under the assumption that the underlying asset and interest rate both evolve from a stochastic volatility model and a stochastic interest rate model, respectively. Therefore, it integrates the recent developments in the Mall…
Enhancing the Black-Scholes model with Lévy processes and Malliavin calculus
problem Improving option valuation by incorporating stochastic volatility and jumps
method Deriving a pricing formula and exact implied volatility using multidimensional Itô calculus and Malliavin calculus
result Better capture of empirical features like volatility smiles
An explicit martingale representation for random variables described as a functional of a Levy process will be given. The Clark-Ocone theorem shows that integrands appeared in a martingale representation are given by conditional expectations of Malliavin derivatives. Our goal is to extend it to random variables which a…
This paper extends Heston model to fractional Brownian motion for option pricing.
problem Developing a new financial model for option pricing with fractional Brownian motion.
method Extending Malliavin differentiability to fractional Heston-type model.
result Proves fractional Heston-type model is Malliavin differentiable and derives option pricing expressions.
We introduce an asymptotic small noise expansion, a so called vol-of-vol expansion, for potentially infinite dimensional and rough stochastic volatility models. Thereby we extend the scope of existing results for finite dimensional models and validate claims for infinite dimensional models. Furthermore we provide new, …
New approach to score function in diffusion models using Malliavin calculus.
problem Estimating score function for complex data distributions.
method Combines Malliavin calculus with Bismut-type formula to derive exact score function expression.
result Derives exact, closed-form expression for score function in diffusion models.
The abstract theorem extends a Lie group result to Lie groupoids.
problem Expressing functions on Lie groupoids as convolutions of two functions.
method Using a lemma from Dixmier-Malliavin, Lie algebroids, and exponential map.
result Every smooth, compactly-supported function on a Lie groupoid can be expressed as a finite sum of convolutions of two such functions.
Study short-term behavior of up-and-in barrier options using Malliavin calculus.
problem Analyzing the decay rate of up-and-in barrier option prices as maturity decreases.
method Use Malliavin calculus to analyze the law of the supremum of the log-price process.
result Derive upper bound on asymptotic decay rate of up-and-in barrier option prices.
Optimizes reinsurance and investment strategies to minimize ruin probability.
problem Optimizing reinsurance and investment strategies to minimize ruin probability.
method Stochastic projected gradient method based on Malliavin calculus.
result Effectiveness of the proposed method demonstrated through numerical experiments.
This paper is devoted to pricing American options using Monte Carlo and the Malliavin calculus. Unlike the majority of articles related to this topic, in this work we will not use localization fonctions to reduce the variance. Our method is based on expressing the conditional expectation E[f(St)/Ss] using the Malliavin…
We investigate the use of Malliavin calculus in order to calculate the Greeks of multidimensional complex path-dependent options by simulation. For this purpose, we extend the formulas employed by Montero and Kohatsu-Higa to the multidimensional case. The multidimensional setting shows the convenience of the Malliavin …
Abstract: Generalizes SGMs to infinite-dimensional Hilbertian setting.
problem Difficulties in extending SGMs to infinite-dimensional settings.
method Uses Gamma and Malliavin Calculus, Dirichlet forms, Wiener chaoses, and time-reversal formula.
result Generalized SGMs to Hilbertian setting with finite-dimensional entropic convergence bounds.
Paper develops methods for solving complex stochastic equations using Malliavin calculus.
problem Existence, uniqueness, and regularity of solutions to BSVIEs.
method Malliavin calculus for tackling diagonal processes and nonlinear dependence.
result Developed well-posedness results for BSVIEs, including probabilistic interpretation of PDEs and portfolio optimization.
Dupire's functional Itô calculus provides an alternative approach to the classical Malliavin calculus for the computation of sensitivities, also called Greeks, of path-dependent derivatives prices. In this paper, we introduce a measure of path-dependence of functionals within the functional Itô calculus framework. Name…
New financial model with sandwiched volatility for option pricing.
problem Developing a new financial model for option pricing.
method Introducing a new model with stochastic volatility driven by a Gaussian Volterra process, ensuring the solution is sandwiched between two arbitrary Hölder continuous functions.
result Developed an algorithm for pricing options with discontinuous payoffs using Malliavin calculus.
Study examines how risk tolerance impacts long-term investment returns.
problem Understanding the impact of risk tolerance on investment returns over time.
method Used Malliavin calculus and Hansen--Scheinkman decomposition.
result Risk aversion affects long-term investment utility through eigenvalues and eigenfunctions.
The paper derives inequalities and formulas for generalized Ricci flow.
problem Understanding and characterizing generalized Ricci flow.
method Using Bochner formula and adapted Malliavin gradient, the paper derives inequalities and characterizes generalized Ricci flow.
result Characterizations of generalized Ricci flow via inequalities for the associated Malliavin gradient.
In a 2006 article (\cite{A1}), Allouba gave his quadratic covariation differentiation theory for Itô's integral calculus. He defined the derivative of a semimartingale with respect to a Brownian motion as the time derivative of their quadratic covariation and a generalization thereof. He then obtained a systematic diff…
Myopic optimization outperforms reinforcement learning in portfolio management, leading to lower returns and higher risks.
problem Reinforcement learning strategies in portfolio management yield lower or negative returns and higher risks compared to myopic optimization.
method Modeling execution/liquidation frictions with mark-to-market accounting, using Malliavin calculus to derive policy gradients and risk shadow price, and quantifying phantom profit.
result Myopic optimization outperforms reinforcement learning in portfolio management, leading to better returns and lower risks.
Volatility roughness studied using fractional noise-driven models.
problem Volatility roughness interpretation.
method Data-reconstructed fractional volatility model with fractional noise.
result Option pricing equation and solution derived using Malliavin calculus.
Characterizes nodal volumes of Gaussian fields on manifolds, extending previous work.
problem Understanding the law and regularity of nodal volumes for Gaussian fields on manifolds.
method Gaussian measures, Morse theory, Malliavin-Sobolev spaces, ray absolute continuity.
result Extension and generalization of previous work on stationary fields to arbitrary dimensions.
In this article, we give a brief informal introduction to Malliavin Calculus for newcomers. We apply these ideas to the simulation of Greeks in Finance. First to European-type options where formulas can be computed explicitly and therefore can serve as testing ground. Later we study the case of Asian options where clos…
Researchers compute Greeks for rough Volterra SV models using Malliavin calculus.
problem Computing Greeks under rough Volterra stochastic volatility models.
method Malliavin calculus techniques, extending integration by parts to non-square integrable functionals.
result Formulas for computing Greeks (Delta, Gamma, Rho, Vega) under various rough Volterra SV models.
Study large deviations for hypoelliptic diffusion on sub-Riemannian manifolds.
problem Large deviations for hypoelliptic diffusion measures on sub-Riemannian manifolds.
method Rough path theory and manifold-valued Malliavin calculus.
result Proved a large deviation principle for pinned hypoelliptic diffusion measures.
A new algorithm solves high-dimensional nonlinear BSDEs using deep learning.
problem Solving high-dimensional nonlinear backward stochastic differential equations (BSDEs).
method Backward differential deep learning, reformulating BSDEs as differential deep learning problems, using Malliavin calculus, discretizing integrals with Euler-Maruyama method, approximating processes with DNNs, backwardly optimizing DNN parameters.
result The proposed algorithm efficiently approximates solutions and their derivatives for high-dimensional BSDEs.
Study on skew and curvature of implied and local volatilities using Malliavin calculus.
problem Relationship between short-end of local and implied volatility surfaces.
method Malliavin calculus techniques
result Recover the $ rac{1}{H+3/2}$ rule for rough volatilities and relationships between skew and curvature.
A new algorithm solves high-dimensional nonlinear BSDEs efficiently.
problem Solving high-dimensional nonlinear backward stochastic differential equations (BSDEs).
method Transformed BSDE into a differential deep learning problem using Malliavin calculus. Discretized integrals using Euler-Maruyama method. Approximated solution with three deep neural networks. Optimized parameters using a differential learning loss function.
result Our algorithm is more accurate and faster than other methods.
Enhances Fourier estimator performance for asynchronous event-data.
problem Improving correlation and covariance estimation on event-data.
method Implement and test NUFFT methods with different averaging kernels.
result Demonstrates improved performance and relationship between averaging scales.
We apply results of Malliavin-Thalmaier-Watanabe for strong and weak Taylor expansions of solutions of perturbed stochastic differential equations (SDEs). In particular, we work out weight expressions for the Taylor coefficients of the expansion. The results are applied to LIBOR market models in order to deal with the …
Researchers develop explicit approximations for European put options in stochastic volatility models.
problem Developing accurate approximations for European put option prices in stochastic volatility models.
method Exploits expansions of the mixing representation of the put option price using Malliavin calculus.
result Explicit formulas for option prices and error bounds are derived, with closed-form solutions under piecewise-constant parameters.
We study the small-time fluctuations for diffusion processes which are conditioned by their initial and final positions, under the assumptions that the diffusivity has a sub-Riemannian structure and that the drift vector field lies in the span of the sub-Riemannian structure. In the case where the endpoints agree and t…
We investigate a class of quadratic-exponential growth BSDEs with jumps. The quadratic structure introduced by Barrieu & El Karoui (2013) yields the universal bounds on the possible solutions. With local Lipschitz continuity and the so-called A_gamma-condition for the comparison principle to hold, we prove the existenc…
New method modifies diffusions for singular rewards.
problem Handling singular rewards in diffusions.
method Malliavin calculus for non-differentiable rewards.
result Stable and reliable training of diffusions.
The article is devoted to models of financial markets with stochastic volatility, which is defined by a functional of Ornstein-Uhlenbeck process or Cox-Ingersoll-Ross process. We study the question of exact price of European option. The form of the density function of the random variable, which expresses the average of…
We obtain explicit representations of locally risk-minimizing strategies of call and put options for the Barndorff-Nielsen and Shephard models, which are Ornstein--Uhlenbeck-type stochastic volatility models. Using Malliavin calculus for Levy processes, Arai and Suzuki (2015) obtained a formula for locally risk-minimiz…
We focus on mean-variance hedging problem for models whose asset price follows an exponential additive process. Some representations of mean-variance hedging strategies for jump type models have already been suggested, but none is suited to develop numerical methods of the values of strategies for any given time up to …
Researchers tackle insider trading in incomplete markets using a discrete-time jump process approach.
problem Tackles insider trading in incomplete markets under the trinomial model.
method Uses a marked binomial process and stochastic analysis with Malliavin calculus.
result Identifies insider expected additional utility with Shannon entropy of extra information.
The study proves a quantitative functional CLT for neural networks with smooth activation functions.
problem Understanding the convergence rates of neural networks with different activation functions.
method Functional versions of the Stein-Malliavin approach and a quantitative functional central limit theorem.
result Rates of convergence depend on the smoothness of the activation function, ranging from logarithmic to sqrt(n).
The paper provides formulas for volatility in various models, including rough volatility.
problem Calibrating SPX and VIX options with rough volatility models.
method Developed explicit formulae using Malliavin calculus for Gaussian processes.
result New insights on joint calibration of SPX and VIX options.