Presentations
http://hdl.handle.net/10754/575213
2020-09-19T21:39:20ZNumerical Smoothing with Multilevel Monte Carlo for Efficient Option Pricing and Density Estimation
http://hdl.handle.net/10754/664597
Numerical Smoothing with Multilevel Monte Carlo for Efficient Option Pricing and Density Estimation
Bayer, Christian; Ben Hammouda, Chiheb; Tempone, Raul
When approximating the expectation of a functional of a certain stochastic process, the robustness and performance of multilevel Monte Carlo (MLMC) method, may be highly deteriorated by the low regularity of the integrand with respect to the input parameters. To overcome this issue, a smoothing procedure is needed to uncover the available regularity and improve the performance of the MLMC estimator. In this work, we consider cases where we cannot perform an analytic smoothing. Thus, we introduce a novel numerical smoothing technique based on root-finding combined with a one dimensional integration with respect to a single well-chosen variable. Our study is motivated by option pricing problems and our main focus is on dynamics where a discretization of the asset price is needed. Through our analysis and numerical experiments, we demonstrate how numerical smoothing significantly reduces the kurtosis at the deep levels of MLMC, and also improves the strong convergence rate, when using Euler scheme. Due to the complexity theorem of MLMC, and given a pre-selected tolerance, $\text{TOL}$, this results in an improvement of the complexity from $\mathcal{O}\left(\text{TOL}^{-2.5}\right)$ in the standard case to $\mathcal{O}\left(\text{TOL}^{-2} \log(\text{TOL})^2\right)$. Moreover, we show how our numerical smoothing combined with MLMC enables us also to estimate density functions, which standard MLMC (without smoothing) fails to achieve.
2020-08-12T00:00:00ZImportance sampling for a robust and efficient multilevel Monte Carlo estimator
http://hdl.handle.net/10754/664595
Importance sampling for a robust and efficient multilevel Monte Carlo estimator
Ben Hammouda, Chiheb; Ben Rached, Nadhir; Tempone, Raul
The multilevel Monte Carlo (MLMC) method for continuous time Markov chains, first introduced by Anderson and Higham (SIAM Multiscal Model. Simul. 10(1), 2012), is a highly efficient simulation technique that can be used to estimate various statistical quantities for stochastic reaction networks (SRNs), and in particular for stochastic biological systems. Unfortunately, the robustness and performance of the multilevel method can be deteriorated due to the phenomenon of high kurtosis, observed at the deep levels of MLMC, which leads to inaccurate estimates for the sample variance. In this work, we address cases where the high kurtosis phenomenon is due to catastrophic coupling (characteristic of pure jump processes where coupled consecutive paths are identical in most of the simulations, while differences only appear in a very small proportion), and introduce a pathwise dependent importance sampling technique that improves the robustness and efficiency of the multilevel method. Our analysis, along with the conducted numerical experiments, demonstrates that our proposed method significantly reduces the kurtosis at the deep levels of MLMC, and also improves the strong convergence rate. Due to the complexity theorem of MLMC and given a pre-selected tolerance, TOL, this results in an improvement of the complexity from O(TOL^{-2} (log(TOL))^2) in the standard case to O(TOL^{-2}), which is the optimal complexity of the MLMC estimator. We achieve all these improvements with a negligible additional cost since our IS algorithm is only applied a few times across each simulated path.
2020-08-11T00:00:00ZHierarchical Approximation Methods for Option Pricing and Stochastic Reaction Networks
http://hdl.handle.net/10754/664851
Hierarchical Approximation Methods for Option Pricing and Stochastic Reaction Networks
Ben Hammouda, Chiheb
In biochemically reactive systems with small copy numbers of one or more reactant molecules, stochastic effects dominate the dynamics. In the first part of this thesis, we design novel efficient simulation techniques for a reliable and robust estimation of various statistical quantities for stochastic biological and chemical systems under the framework of Stochastic Reaction Networks (SRNs). In systems characterized by having simultaneously fast and slow timescales, existing discrete state-space stochastic path simulation methods can be very slow. In the first work in this part, we propose a novel hybrid multilevel Monte Carlo (MLMC), which uses a novel split-step implicit tau-leap scheme at the coarse levels, where the explicit scheme is not applicable due to numerical instability issues. Our analysis illustrates the advantage of our proposed method over MLMC combined with the explicit scheme. In a second work related to the first part, we solve another challenge present in this context called the high kurtosis phenomenon. We address cases where the high kurtosis, observed for the MLMC estimator, is due to catastrophic coupling. We propose a novel method that provides a more reliable and robust multilevel estimator. Our approach combines the MLMC method with a pathwise-dependent importance sampling technique for simulating the coupled paths. Through our theoretical estimates and numerical analysis, we show that our approach not only improves the robustness of the multilevel estimator by reducing the kurtosis significantly but also improves the strong convergence rate, and consequently, the complexity rate of the MLMC method. We achieve all these improvements with a negligible additional cost.
In the second part of this thesis, we design novel numerical methods for pricing financial derivatives. Option pricing can be formulated as an integration problem, which is usually challenging due to a combination of two complications: 1) The high dimensionality of the input space, and 2) The low regularity of the integrand on the input parameters. We address these challenges by using different techniques for smoothing the integrand to uncover the available regularity and improve quadrature methods' convergence behavior. We develop different ways of smoothing that depend on the characteristics of the problem at hand. Then, we approximate the resulting integrals using hierarchical quadrature methods. In the first work in this part, we design a fast method for pricing European options under the rough Bergomi model. This model exhibits several numerical and theoretical challenges. As a consequence, classical numerical methods for pricing become either inapplicable or computationally expensive. In our approach, we first smoothen the integrand analytically and then use quadrature methods. These quadrature methods are coupled with Brownian bridge construction and Richardson extrapolation, to approximate the resulting integral. Numerical examples with different parameter constellations exhibit the performance of our novel methodology. Indeed, our hierarchical methods demonstrate substantial computational gains compared to the MC method, which is the prevalent method in this context. In the second work in this part, we consider cases where we cannot perform an analytic smoothing. Consequently, we perform a numerical smoothing based on root-finding techniques, with a particular focus on cases where discretization of the asset price dynamics is needed. We illustrate the advantage of our approach, which combines numerical smoothing with adaptive sparse grids' quadrature, over the MC approach. Furthermore, we demonstrate that our numerical smoothing procedure improves the robustness and the complexity rate of the MLMC estimator. Finally, our numerical smoothing, coupled with MLMC, enables us also to estimate density functions efficiently.
2020-07-02T00:00:00ZDiscrete changes in fault free-face roughness: constraining past earthquakes characteristics
http://hdl.handle.net/10754/662594
Discrete changes in fault free-face roughness: constraining past earthquakes characteristics
Zielke, Olaf; Benedetti, Lucilla; Mai, Paul Martin; Rizza, Magali; Fleury, Jules; Pousse Beltran, Lea; Puliti, Irene; Pace, Bruno
A driving motivator in many active tectonics studies is to learn more about the recurrence large and potentially destructive earthquakes, providing the means to assess the respective fault’s future seismic behavior. Doing so requires long records of earthquake recurrence. The lack of sufficiently long instrumental seismic records (that would be best suited for this task) has led to the development of other approaches that may constrain the recurrence of surface rupturing earthquakes along individual faults. These approaches take different forms, depending on the specific tectonic and geographic conditions of an investigated region.
For example, around the Mediterranean Sea, we frequently find bedrock scarps along normal faults. Assuming that bedrock (i.e., fault free-face) exposure is caused by the occurrence of sub-sequent large earthquakes, we may measure certain rock properties to constrain the time and size of past earthquakes as well as the fault’s geologic slip-rate. A now-classic example in this regard is the measurement of $^{36}$Cl concentrations along exposed fault scarps in limestones.
For the presented study, we looked at another property of the exposed fault free-face, namely its morphologic roughness. We aim to identify whether fault free-face roughness contains information to constrain earthquake occurrence and fault slip-rates following the assumption that  sub-sequent exposure to the elements and sub-areal erosional conditions may leave a signal in how rough (or smooth) the fault free-face is (assuming a somewhat uniform pre-exposure roughness). Here, we present observations of fault free-face surface roughness for the Mt. Vettore fault (last ruptured in 2016) and the Rocca Preturo fault (The underlying models of fault free-face morphology were generated using the Structure-from-Motion approach and a large suite of unregistered optical images.). Employing different metrics to quantify morphologic roughness, we were indeed able to observe a) an increase in surface roughness with fault scarp height (i.e., longer exposure to sub-areal erosion causes higher roughness), and b) distinct (rather than gradual) changes in surface roughness, suggesting a correlation to individual exposure events such as earthquakes. Hence, fault free-face morphology of bedrock faults may serve as an additional metric to reconstruct earthquake recurrence patterns.
2020-03-09T00:00:00Z