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.
In this article, we consider a Markov process X, starting from x and solving a stochastic differential equation, which is driven by a Brownian motion and an independent pure jump component exhibiting state-dependent jump intensity and infinite jump activity. A second order expansion is derived for the tail probability …
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 …
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…
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).
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.
Volatility estimation based on high-frequency data is key to accurately measure and control the risk of financial assets. A Lévy process with infinite jump activity and microstructure noise is considered one of the simplest, yet accurate enough, models for financial data at high-frequency. Utilizing this model, we prop…
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 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.
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 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 show a concise extension of the monotone stability approach to backward stochastic differential equations (BSDEs) that are jointly driven by a Brownian motion and a random measure for jumps, which could be of infinite activity with a non-deterministic and time inhomogeneous compensator. The BSDE generator function c…
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 investigate the existence of affine realizations for Lévy driven interest rate term structure models under the real-world probability measure, which so far has only been studied under an assumed risk-neutral probability measure. For models driven by Wiener processes, all results obtained under the risk-neutral appro…
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…
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.
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 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.
We develop a general class of noise-robust estimators based on the existing estimators in the non-noisy high-frequency data literature. The microstructure noise is a parametric function of the limit order book. The noise-robust estimators are constructed as plug-in versions of their counterparts, where we replace the e…
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.
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.
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…
In this paper, we consider a framework adapting the notion of cointegration when two asset prices are generated by a driftless Itô-semimartingale featuring jumps with infinite activity, observed regularly and synchronously at high frequency. We develop a regression based estimation of the cointegrated relations method …
We present an agent behavior based microscopic model that induces jumps, spikes and high volatility phases in the price process of a traded asset. We transfer dynamics of thermally activated jumps of an unexcited/ excited two state system discussed in the context of quantum mechanics to agent socio-economic behavior an…
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.
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…
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.
User engagement in online social networking depends critically on the level of social activity in the corresponding platform--the number of online actions, such as posts, shares or replies, taken by their users. Can we design data-driven algorithms to increase social activity? At a user level, such algorithms may incre…
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.
We use the database leak of Mt. Gox exchange to analyze the dynamics of the price of bitcoin from June 2011 to November 2013. This gives us a rare opportunity to study an emerging retail-focused, highly speculative and unregulated market with trader identifiers at a tick transaction level. Jumps are frequent events and…
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.
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.
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.
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 …
New method estimates volatility for Lévy processes with unbounded jumps efficiently.
problem Efficient estimation of volatility for Lévy processes with unbounded jumps.
method Developed a new estimator based on high-order expansions of truncated moments.
result Method outperforms existing alternatives in estimating volatility.
We consider a univariate semimartingale model for (the logarithm of) an asset price, containing jumps having possibly infinite activity (IA). The nonparametric threshold estimator of the integrated variance IV proposed in Mancini 2009 is constructed using observations on a discrete time grid, and precisely it sums up t…
Study infinite-depth limits of neural networks with fixed width.
problem Understanding the behavior of neural networks as depth increases with fixed width.
method Analyzing finite-width residual networks with random Gaussian weights, focusing on the infinite-depth limit.
result The pre-activations converge to a zero-drift diffusion process, differing from the infinite-width limit.