The article derives small-time expansions for jump-diffusion models with infinite jump activity.
problem Analyzing state-dependent jump-diffusion models with infinite jump activity.
method Derives a second-order expansion for tail probabilities and option prices.
result Obtains a second-order expansion for out-of-the-money European call option prices.
Develops a numerical method for LRM strategies in BNS models with infinite active jumps.
problem Calculating locally risk-minimizing strategies for non-martingale BNS models with infinite active jumps.
method Modified Malliavin calculus expression and Monte Carlo method for non-martingale BNS models.
result Proposes a numerical method for LRM strategies in non-martingale BNS models with infinite active jumps.
We analyse the behaviour of the implied volatility smile for options close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalisation of the strike variable with the property that the implied volatility converges to a non-constant limiting shape, which is a function of …
Bayesian method corrects misspecified volatility estimation in high-frequency financial data.
problem Volatility estimation in financial data with infinite jump activity and microstructure noise.
method Proposes a misspecified posterior corrected by a simple estimate of the location shift and re-scaling of the log likelihood.
result Establishes a Bernstein-von Mises theorem for the adjusted posterior, showing asymptotic Gaussianity and consistent estimation.
Study near-maturity convergence rates of American put prices in Lévy models.
problem Analyzing convergence rates of optimal exercise prices in Lévy models.
method Examined two settings: jumps of unbounded and bounded variation, deriving near-maturity expansions.
result Near-maturity convergence rate of optimal exercise price is of order √(T-t).
Observing prices of European put and call options, we calibrate exponential Lévy models nonparametrically. We discuss the efficient implementation of the spectral estimation procedures for Lévy models of finite jump activity as well as for self-decomposable Lévy models. Based on finite sample variances, confidence inte…
Deep learning method improves numerical approximation of FBSDEs with jumps.
problem Improving numerical solutions for FBSDEs with jumps.
method Deep learning-based approach for decoupled FBSDEs with jumps.
result A priori and a posteriori error estimates for finite and infinite activity cases.
Develops methods to simulate option prices for a specific stochastic volatility model.
problem No method exists to compute option prices numerically for a non-martingale jump-type model.
method Develops two Monte Carlo simulation methods under change of measure.
result Conducts numerical experiments to validate the developed methods.
Study the hedging of cryptocurrency options in a volatile market.
problem Hedging options in a volatile, non-stationary cryptocurrency market.
method Calibrated to SVI-implied volatility surfaces, Monte Carlo price paths generated using SVCJ, GARCH, and historical data. Delta, Delta-Gamma, Delta-Vega, and Minimum Variance strategies applied. Wide range of market models tested.
result Calibration results indicate stochastic volatility, low jump frequency, and infinite activity. Short-dated options less sensitive to volatility or Gamma hedges; longer-dated options benefit from multiple-instrument hedges.
Deep learning improves option pricing for a non-martingale asset model.
problem Computing call option prices for the Barndorff-Nielsen and Shephard model with infinite jumps.
method Developed a supervised deep-learning scheme using Monte Carlo teaching data and a Black-Scholes-derived variable.
result Significant improvement in accuracy of option pricing.
Extends stability approach to BSDEs with jumps, providing criteria for existence and uniqueness.
problem Existence and uniqueness of solutions to BSDEs with jumps.
method Monotone stability approach, non-convex generator, non-global Lipschitz conditions.
result Concrete criteria for existence and uniqueness of solutions, comparison, and bounds.
High-frequency data cointegration framework developed with rigorous theory and tests.
problem Cointegration in high-frequency data with jumps and infinite activity.
method Regression-based estimation method and Dickey-Fuller type residual tests.
result Consistent and asymptotic limit theory for cointegration tests.
Extends deep solver to FBSDEs with jumps for option pricing.
problem Solving FBSDEs with jumps for financial applications.
method Discretization, ANN parametrization, reinforcement learning, loss function minimization.
result Successfully applied to option pricing in low and high dimensions.
In this paper we consider two semimartingales driven by diffusions and jumps. We allow both for finite activity and for infinite activity jump components. Given discrete observations we disentangle the {\it integrated covariation} (the covariation between the two diffusion parts, indicated by IC) from the co-jumps. Thi…
We consider a Markov process X, which is the solution of a stochastic differential equation driven by a Lévy process Z and an independent Wiener process W. Under some regularity conditions, including non-degeneracy of the diffusive and jump components of the process as well as smoothness of the Lévy density of $Z…
In quantitative finance, we often model asset prices as semimartingales, with drift, diffusion and jump components. The jump activity index measures the strength of the jumps at high frequencies, and is of interest both in model selection and fitting, and in volatility estimation. In this paper, we give a novel estimat…
It is well documented that a model for the underlying asset price process that seeks to capture the behaviour of the market prices of vanilla options needs to exhibit both diffusion and jump features. In this paper we assume that the asset price process S is Markov with cadlag paths and propose a scheme for computing…
Develops active learning for Jump Gaussian Process models.
problem Optimizing experimental designs and steering data acquisition in complex systems.
method Active learning of piecewise Jump Gaussian Process (Jump GP) models, accounting for model bias.
result Demonstrates the importance of accounting for model bias in Jump GP models.
We introduce an algorithm for the pricing of finite expiry American options driven by Lévy processes. The idea is to tweak Carr's `Canadisation' method, cf. Carr [9] (see also Bouchard et al [5]), in such a way that the adjusted algorithm is viable for any Lévy process whose law at an independent, exponentially distrib…
New method estimates volatility for processes with jumps of unbounded variation.
problem Estimating volatility of processes with jumps of unbounded variation.
method Developed a new volatility estimator using debiasing of truncated realized quadratic variation.
result Method outperforms existing alternatives in simulations.
The paper introduces walks with jumps for modeling neuron activity in hyperbolic space.
problem Encoding neuron activity sequences in hyperbolic space.
method Introducing walks with jumps in hyperbolic geometry to model neuron activity.
result Endpoints of walks with jumps do not fully encode the sequence of jump times.
Investigates real-world interest rate dynamics using affine models.
problem Existence of affine realizations for Lévy-driven interest rate models.
method Transfers results from risk-neutral to real-world probability measure.
result Severe restrictions on market price of risk in infinite activity jump models.
The paper examines conditions for stochastic invariance of cones in SPDEs with jumps.
problem Stochastic invariance of cones in SPDEs with jumps.
method Sufficient conditions for stochastic invariance of closed convex cones in abstract L2-spaces. result Conditions for stochastic invariance of cones are provided and analyzed.
We consider a general class of high order weak approximation schemes for stochastic differential equations driven by Lévy processes with infinite activity. These schemes combine a compound Poisson approximation for the jump part of the Lévy process with a high order scheme for the Brownian driven component, applied bet…
We investigate the extension of the multilevel Monte Carlo path simulation method to jump-diffusion SDEs. We consider models with finite rate activity, using a jump-adapted discretisation in which the jump times are computed and added to the standard uniform dis- cretisation times. The key component in multilevel analy…
Develops robust methods for infinite-dimensional stochastic processes.
problem Measuring covariations in stochastic evolution equations in infinite dimensions.
method Asymptotic theory for jump robust measurement of covariations.
result Identifies scaling limits for realized covariations.
New algorithm for Bayesian inference in population Markov Jump processes.
problem Challenges in Bayesian inference for continuous time, discrete state systems with infinite state-space.
method Pseudo-marginal sampling algorithms based on random truncation method.
result Significant savings in computational time compared to state-of-the-art methods.
Extends Alòs' formula to Barndorff-Nielsen and Shephard model.
problem Modeling call option prices in a stochastic volatility model.
method Uses Alòs' decomposition formula and Ito's formula for an Ornstein-Uhlenbeck model with infinite jumps.
result First Alòs type decomposition formula for Barndorff-Nielsen and Shephard model.
Formula for European option pricing under jump diffusion model.
problem Option pricing under complex stochastic processes.
method Infinite series of Black-Scholes terms for Levy-driven processes.
result Series solution converges with a radius of convergence.
We derive asymptotic expansions for option data to detect infinite variation volatility.
problem Detecting infinite variation volatility in high-frequency option data.
method Nonparametric higher-order asymptotic expansions for small-time changes of characteristic functions of Itô semimartingales.
result Evidence of infinite variation volatility in high-frequency option data.
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…
Study reveals frequent price jumps in Bitcoin market, influenced by trader behavior and market structure.
problem Understanding price dynamics and trader behavior in the Bitcoin market.
method Analysis of Mt. Gox exchange database to study Bitcoin price movements at tick level.
result Jumps in Bitcoin prices are frequent and predictable, influenced by order flow imbalance and trader aggressiveness.
Neural Jump ODEs extend to infinite-dimensional function spaces for optimal prediction.
problem Handling continuous-time stochastic processes in infinite-dimensional function spaces.
method Developing a new approximation strategy for infinite-dimensional function-valued processes.
result Proved convergence of the NJ-ODE to the optimal prediction process.
Develops robust estimators for high-frequency data with market microstructure noise.
problem Estimating prices in the presence of market microstructure noise.
method Plug-in versions of existing estimators, using raw price and limit order book data.
result Noise-robust estimators can be applied to various high-frequency data problems.
New method estimates active subspaces for jump-discontinuous functions.
problem Estimating active subspaces for discontinuous functions like ABMs.
method Extending active subspaces to discontinuous functions, using Gaussian process.
result Identifies important parameters in ABM simulations of refugee movement.
Generalizes NTK for surrogate gradient learning in neural networks.
problem Lack of theoretical foundation for surrogate gradient learning.
method Generalizes neural tangent kernel (NTK) for surrogate gradient learning (SGL).
result Surrogate gradient NTK provides a good characterization of SGL.
The value function of an optimal stopping problem for jump diffusions is known to be a generalized solution of a variational inequality. Assuming that the diffusion component of the process is nondegenerate and a mild assumption on the singularity of the Lévy measure, this paper shows that the value function of this op…
The paper prices and replicates various financial contracts on a risky asset with stochastic volatility and jumps.
problem Pricing and replicating financial contracts on assets with stochastic volatility and jumps.
method Develops pricing and hedging formulas for various financial contracts, independent of the volatility process dynamics.
result Pricing and hedging formulas for financial contracts are derived without dependence on the volatility process dynamics.
We consider a stochastic volatility model with jumps where the underlying asset price is driven by the process sum of a 2-dimensional Brownian motion and a 2-dimensional compensated Poisson process. The market is incomplete, resulting in infinitely many equivalent martingale measures. We find the set equivalent marting…
The paper develops algorithms to increase social activity online.
problem Increasing user engagement in social networks.
method Modeling social activity as marked temporal point processes and deriving SDEs with jumps to develop online algorithms.
result The developed algorithms consistently steer social activity more effectively than existing methods.
Study provides short-time expansions for LETF options using Lévy models.
problem Analyzing small-time behavior of LETF option prices with local volatility and jumps.
method Closed-form expressions for leading order terms of LETF option prices near expiration.
result Price of out-of-the-money LETF options is asymptotically equivalent to underlying ETF options with modified prices.
Develops a PIDE framework for option pricing with stochastic volatility and jumps.
problem Option pricing under stochastic volatility and jumps.
method PIDE framework derived from Lévy-type process, implemented via finite-difference discretization with FFT for nonlocal jump operator, calibrated using GMM.
result Stochastic volatility accounts for most pricing improvement, reducing implied-volatility RMSE by 39% compared to Black-Scholes.
The implied volatility skew has received relatively little attention in the literature on short-term asymptotics for financial models with jumps, despite its importance in model selection and calibration. We rectify this by providing high-order asymptotic expansions for the at-the-money implied volatility skew, under a…
Generative Bayesian Computation improves surrogates for expensive simulations.
problem Limitations of Gaussian process surrogates in handling complex, non-stationary data.
method Generative Bayesian Computation via Implicit Quantile Networks (IQNs).
result Generative Bayesian Computation outperforms traditional Gaussian process methods across various benchmarks.
Extended CIR process with jumps at fixed dates for modeling overnight rates.
problem Modeling overnight rates with jumps at predetermined dates.
method Formal definition and existence proof of a CIR process with stochastic discontinuities.
result Extended CIR process inherits affine property and non-negativity.
Extends active subspace analysis to infinite dimensions.
problem Dimension reduction in infinite dimensional functionals.
method Defines an operator for Hilbert space, extends Euclidean properties, proposes Monte Carlo procedure.
result Desirable properties extend to infinite dimensional setting.
Modeling financial volatility using quantum mechanics principles.
problem Capturing high volatility and spikes in financial asset prices.
method Agent-based model linking quantum mechanical jumps to socio-economic behavior.
result Model dynamics converge to Itô-diffusion price processes in large market limits.
The pricing of options in exponential Levy models amounts to the computation of expectations of functionals of Levy processes. In many situations, Monte-Carlo methods are used. However, the simulation of a Levy process with infinite Levy measure generally requires either to truncate small jumps or to replace them by a …