246 research outputs found
Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation
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
Pricing exotic options using improved strong convergence
Today, better numerical approximations are required for multi-dimensional SDEs to improve on the poor performance of the standard Monte Carlo integration. With this aim in mind, the material in the thesis is divided into two main categories, stochastic calculus and mathematical finance. In the former, we introduce a new scheme or discrete time approximation based on an idea of Paul Malliavin where, for some conditions, a better strong convergence order is obtained than the standard Milstein scheme without the expensive simulation of the Lévy Area. We demonstrate when the conditions of the 2−Dimensional problem permit this and give an exact solution for the orthogonal transformation (θ Scheme or OrthogonalMilstein Scheme). Our applications are focused on continuous time diffusion models for the volatility and variance with their discrete time approximations (ARV). Two theorems that measure with confidence the order of strong and weak convergence of schemes without an exact solution or expectation of the system are formally proved and tested with numerical examples. In addition, some methods for simulating the double integrals or Lévy Area in the Milstein approximation are introduced.
For mathematical finance, we review evidence of non-constant volatility and consider the implications for option pricing using stochastic volatility models. A general stochastic volatility model that represents most of the stochastic volatility models that are outlined in the literature is proposed. This was necessary in order to both study and understand the option price properties. The analytic closed-form solution for a European/Digital option for both the Square Root Model and the 3/2 Model are given. We present the Multilevel Monte Carlo path simulation method which is a powerful tool for pricing exotic options. An improved/updated version of the ML-MC algorithm using multi-schemes and a non-zero starting level is introduced. To link the contents of the thesis, we present a wide variety of pricing exotic option examples where considerable computational savings are demonstrated using the new θ Scheme and the improved Multischeme Multilevel Monte Carlo method (MSL-MC). The computational cost to achieve an an accuracy of O() is reduced from O() to O() for some applications
Multilevel Monte Carlo method for parabolic stochastic partial differential equations
We analyze the convergence and complexity of multilevel Monte Carlo discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show under low regularity assumptions on the solution that the judicious combination of low order Galerkin discretizations in space and an Euler-Maruyama discretization in time yields mean square convergence of order one in space and of order1/2 in time to the expected value of the mild solution. The complexity of the multilevel estimator is shown to scale log-linearly with respect to the corresponding work to generate a single path of the solution on the finest mesh, resp. of the corresponding deterministic parabolic problem on the finest mes
Numerical methods for Lévy processes
We survey the use and limitations of some numerical methods for pricing derivative contracts in multidimensional geometric Lévy model
Generative Modelling of L\'{e}vy Area for High Order SDE Simulation
It is well known that, when numerically simulating solutions to SDEs,
achieving a strong convergence rate better than O(\sqrt{h}) (where h is the
step size) requires the use of certain iterated integrals of Brownian motion,
commonly referred to as its "L\'{e}vy areas". However, these stochastic
integrals are difficult to simulate due to their non-Gaussian nature and for a
d-dimensional Brownian motion with d > 2, no fast almost-exact sampling
algorithm is known.
In this paper, we propose L\'{e}vyGAN, a deep-learning-based model for
generating approximate samples of L\'{e}vy area conditional on a Brownian
increment. Due to our "Bridge-flipping" operation, the output samples match all
joint and conditional odd moments exactly. Our generator employs a tailored
GNN-inspired architecture, which enforces the correct dependency structure
between the output distribution and the conditioning variable. Furthermore, we
incorporate a mathematically principled characteristic-function based
discriminator. Lastly, we introduce a novel training mechanism termed
"Chen-training", which circumvents the need for expensive-to-generate training
data-sets. This new training procedure is underpinned by our two main
theoretical results.
For 4-dimensional Brownian motion, we show that L\'{e}vyGAN exhibits
state-of-the-art performance across several metrics which measure both the
joint and marginal distributions. We conclude with a numerical experiment on
the log-Heston model, a popular SDE in mathematical finance, demonstrating that
high-quality synthetic L\'{e}vy area can lead to high order weak convergence
and variance reduction when using multilevel Monte Carlo (MLMC)
09391 Abstracts Collection -- Algorithms and Complexity for Continuous Problems
From 20.09.09 to 25.09.09, the Dagstuhl Seminar 09391
Algorithms and Complexity for Continuous Problems was held in the
International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, participants presented their current research, and
ongoing work and open problems were discussed. Abstracts of the
presentations given during the seminar are put together in this paper. The
first section describes the seminar topics and goals in general. Links to
extended abstracts or full papers are provided, if available
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