Importance sampling and statistical Romberg method for Lévy processes

We develop a computational method for expected functionals of the drawdown and its duration in exponential Levy models. It is based on a novel simulation algorithm for the joint law of the state, supremum and time the supremum is attained of the Gaussian approximation of a general Levy process. We bound the bias for various locally Lipschitz and discontinuous payoffs arising in applications and analyse the computational complexities of the corresponding Monte Carlo and multilevel Monte Carlo estimators. Monte Carlo methods for Levy processes (using Gaussian approximation) have been analysed for Lipschitz payoffs, in which case the computational complexity of our algorithm is up to two orders of magnitude smaller when the jump activity is high. At the core of our approach are bounds on certain Wasserstein distances, obtained via the novel SBG coupling between a Levy process and its Gaussian approximation. Numerical performance, based on the implementation in the dedicated GitHub repository, exhibits a good agreement with our theoretical bounds.

Simulation of the drawdown and its duration in L\'{e}vy models via stick-breaking Gaussian approximation

Abstract We develop a computational method for expected functionals of the drawdown and its duration in exponential Lévy models. It is based on a novel simulation algorithm for the joint law of the state, supremum and time the supremum is attained of the Gaussian approximation for a general Lévy process. We bound the bias for various locally Lipschitz and discontinuous payoffs arising in applications and analyse the computational complexities of the corresponding Monte Carlo and multilevel Monte Carlo estimators. Monte Carlo methods for Lévy processes (using Gaussian approximation) have been analysed for Lipschitz payoffs, in which case the computational complexity of our algorithm is up to two orders of magnitude smaller when the jump activity is high. At the core of our approach are bounds on certain Wasserstein distances, obtained via the novel stick-breaking Gaussian (SBG) coupling between a Lévy process and its Gaussian approximation. Numerical performance, based on the implementation in Cázares and Mijatović (SBG approximation. GitHub repository. Available online at https://github.com/jorgeignaciogc/SBG.jl (2020)), exhibits a good agreement with our theoretical bounds. Numerical evidence suggests that our algorithm remains stable and accurate when estimating Greeks for barrier options and outperforms the “obvious” algorithm for finite-jump-activity Lévy processes. 

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Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation

We consider the problem of obtaining effective representations for the solutions of linear, vector-valued stochastic differential equations (SDEs) driven by non-Gaussian pure-jump L\'evy processes, and we show how such representations lead to efficient simulation methods. The processes considered constitute a broad class of models that find application across the physical and biological sciences, mathematics, finance and engineering. Motivated by important relevant problems in statistical inference, we derive new, generalised shot-noise simulation methods whenever a normal variance-mean (NVM) mixture representation exists for the driving L\'evy process, including the generalised hyperbolic, normal-Gamma, and normal tempered stable cases. Simple, explicit conditions are identified for the convergence of the residual of a truncated shot-noise representation to a Brownian motion in the case of the pure L\'evy process, and to a Brownian-driven SDE in the case of the L\'evy-driven SDE. These results provide Gaussian approximations to the small jumps of the process under the NVM representation. The resulting representations are of particular importance in state inference and parameter estimation for L\'evy-driven SDE models, since the resulting conditionally Gaussian structures can be readily incorporated into latent variable inference methods such as Markov chain Monte Carlo (MCMC), Expectation-Maximisation (EM), and sequential Monte Carlo.

Generalised shot noise representations of stochastic systems driven by non-Gaussian L\'evy processes

We construct a pathwise calculus for functionals of integer-valued measures and use it to derive an martingale representation formula with respect to a large class of integer-valued random measures. Using these results, we extend the Functional Ito Calculus to functionals of integer-valued random measures. We construct a 'stochastic derivative' operator with respect to an integer-valued random measure, and show it to be the inverse of the stochastic integral with respect to the compensated measure. This stochastic derivative yields an explicit martingale representation formula for square-integrable martingales. Our results extend beyond the class of Poisson random measures and allow for random and time-dependent compensators.

Functional Itô calculus and martingale representation formula for integer-valued measures

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 truncating or replacing the small jumps by a Brownian motion with the same variance. We will derive bounds for the errors generated by these two types of approximation.

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E. H. A. Dia

Papers

Error bounds for small jumps of Lvy processes