Extends Clark-Ocone theorem to non-Malliavin differentiable random variables using Ito's formula.
problem Extending Clark-Ocone theorem to non-Malliavin differentiable random variables.
method Uses Ito's formula instead of Malliavin calculus.
result Explicit representation of locally risk-minimizing strategies for digital options in Levy models.
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.
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…
Investigates quadratic-exponential growth BSDEs with jumps and proves existence and differentiability.
problem Existence and differentiability of solutions to quadratic-exponential growth BSDEs with jumps.
method Proves existence and differentiability of solutions under general quadratic-exponential structure using local Lipschitz continuity and A_gamma-condition.
result Proves existence and differentiability of solutions under general quadratic-exponential structure.
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.
Study computes option sensitivities using Malliavin calculus for hybrid stochastic models.
problem Computing option sensitivities (Greeks) under hybrid stochastic volatility and interest rate models.
method Integrates Malliavin calculus for Delta, Vega, and Rho computation; extends to non-differentiable payoffs.
result Malliavin calculus enables effective numerical implementations for various option types.
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.
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.
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.
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.
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.
We provide sufficient conditions for the existence and uniqueness of solutions to a stochastic differential equation which arises in a price impact model. These conditions are stated as smoothness and boundedness requirements on utility functions or Malliavin differentiability of payoffs and endowments.
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.
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.
Study optimal portfolio strategy under uncertain market conditions.
problem Optimizing portfolio under partial information and drift uncertainty.
method Developed optimal strategy using BSDE and particle representation.
result An efficient numerical scheme approximates the optimal portfolio.
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.
The paper studies BSDEs and their densities in biology and finance.
problem Existence of densities for solutions of BSDEs.
method Conditions for Malliavin differentiability and application to gene expression and finance.
result Results on existence of densities for BSDEs solutions.
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…
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.
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 analyzes hypoelliptic heat kernels near a manifold's cut locus.
problem Analyzing hypoelliptic heat kernels near a manifold's cut locus.
method Probabilistic approach using S. Watanabe's distributional Malliavin calculus and T. Lyons' rough path theory.
result Obtained a short time asymptotic expansion of hypoelliptic heat kernels up to any order.
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.
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.
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.
Study volatility of forward-start options using Malliavin Calculus.
problem Implied volatility of Forward-Start options, focusing on ATM behavior.
method Closed-form expressions derived using Malliavin Calculus in Markovian models.
result Derives expressions for at-the-money, skew, and curvature of forward implied volatility.
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
Study uses Malliavin calculus to find exact option pricing under stochastic volatility.
problem Exact pricing of European options in markets with stochastic volatility.
method Applying Malliavin calculus to models defined by Ornstein-Uhlenbeck or Cox-Ingersoll-Ross processes.
result Established the density function of the volatility average over time to maturity.
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.
The paper provides valuation formulas for insurance contracts using Malliavin calculus.
problem Valuation of insurance contracts with dependent claims.
method Using Malliavin calculus to express expected cash flows in terms of a building block.
result Formulas for expected cash flows in actuarial and financial contracts.
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 …
Study quantos in energy markets using HJM framework and Malliavin calculus.
problem Analyzing sensitivity of energy quanto options.
method Using HJM framework and Malliavin calculus, derive delta and cross-gamma formulas.
result Extension of existing work on quanto options in energy markets.
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.
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.
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 …
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…
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.
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.
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…
Study small-time fluctuations for sub-Riemannian diffusion loops.
problem Analyzing fluctuations of sub-Riemannian diffusion processes.
method Analyzes small-time fluctuations of diffusion processes with sub-Riemannian structure, identifying degenerate and non-degenerate covariance matrices.
result Rescaled fluctuations converge to a non-degenerate limiting diffusion loop.
Derives a new closed-form strategy for mean-variance hedging of additive processes.
problem Developing efficient numerical methods for mean-variance hedging strategies.
method Uses Malliavin calculus to derive a closed-form representation.
result Derives an explicit closed-form representation for mean-variance hedging.
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.
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.
Optimizes model points for life insurance portfolios using stochastic integration and Malliavin calculus.
problem Finding an optimal set of model points for life insurance policies to minimize risk.
method Representation theorem, stochastic integration in Banach space, Malliavin calculus.
result Representation theorem provides two formulations of risk functional.