DE MARCO Stefano

In this thesis, we examine several stochastic approximation methods for both the computation of financial risk measures and the pricing of derivatives.Since explicit formulas are rarely available for such quantities, the need for fast, efficient and reliable analytical approximations is of paramount importance to financial institutions.In the first part, we study several multilevel Monte Carlo approximation methods and apply them to two practical problems: the estimation of quantities involving nested expectations (such as initial margin) and the discretization of integrals appearing in rough models for the forward variance for VIX option pricing.In both cases, we analyze the asymptotic optimality properties of the corresponding multilevel estimators and numerically demonstrate their superiority over a classical Monte Carlo method.In the second part, motivated by the numerous examples from credit risk modeling, we propose a general metamodeling framework for large sums of weighted Bernoulli random variables, which are conditionally independent with respect to a common factor X. Our generic approach is based on the polynomial decomposition of the chaos of the common factor and on a Gaussian approximation. L2 error estimates are given when the factor X is associated with classical orthogonal polynomials.Finally, in the last part of this thesis, we focus on the short-time asymptotics of U.S. implied volatility and U.S. option prices in local volatility models. We also propose a law approximation of the VIX index in rough models for forward variance, expressed in terms of lognormal proxies, and derive expansion results for VIX options with explicit coefficients.

The Multilevel Monte-Carlo (MLMC) method developed by Giles [Gil08] has a natural application to the evaluation of nested expectation of the form E [g(E [f (X, Y)|X])], where f, g are functions and (X, Y) a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of Initial Margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotical optimality. at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal/dual algorithms for stochastic control problems.

We introduce a new class of anticipative backward stochastic differential equations with a dependence of McKean type on the law of the solution, that we name MKABSDE. We provide existence and uniqueness results in a general framework with relatively general regularity assumptions on the coefficients. We show how such stochastic equations arise within the modern paradigm of derivative pricing where a central counterparty (CCP) requires the members to deposit variation and initial margins to cover their exposure. In the case when the initial margin is proportional to the Conditional Value-at-Risk (CVaR) of the contract price, we apply our general result to define the price as a solution of a MKABSDE. We provide several linear and non-linear simpler approximations, which we solve using different numerical (deterministic and Monte-Carlo) methods.

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In this paper, we develop the reversible shaking transformation methods on path space of Gobet and Liu [GL15] to estimate the rare event statistics arising in different financial risk settings which are embedded within a unified framework of isonormal Gaussian process. Namely, we combine splitting methods with both Interacting Particle System (IPS) technique and ergodic transformations using Parallel-One-Path (POP) estimators. We also propose an adaptive version for the POP method and prove its convergence. We demonstrate the application of our methods in various examples which cover usual semi-martingale stochastic models (not necessarily Markovian) driven by Brownian motion and, also, models driven by fractional Brownian motion (non semi-martingale) to address various financial risks. Interestingly, owing to the Gaussian process framework, our methods are also able to efficiently handle the important problem of sensitivities of rare event statistics with respect to the model parameters.

This is a companion paper based on our previous work [ADGL15] on rare event simulation methods. In this paper, we provide an alternative proof for the ergodicity of shaking transformation in the Gaussian case and propose two variants of the existing methods with comparisons of numerical performance. In numerical tests, we also illustrate the idea of extreme scenario generation based on the convergence of marginal distributions of the underlying Markov chains and show the impact of the discretization of continuous time models on rare event probability estimation.

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The one-dimensional SDE with non Lipschitz diffusion coefficient $dX_{t} = b(X_{t})dt + \sigma X_{t}^{\gamma} dB_{t}, \ X_{0}=x, \ \gamma.

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Some recent works in the field of mathematical finance have brought new light on the importance of studying the regularity and the tail asymptotics of distributions for certain classes of diffusions with non-globally smooth coefficients. In this Ph.D. dissertation we deal with some issues in this framework. In a first part, we study the existence, smoothness and space asymptotics of densities for the solutions of stochastic differential equations assuming only local conditions on the coefficients of the equation. Our analysis is based on Malliavin calculus tools and on « tube estimates » for Ito processes, namely estimates for the probability that the trajectory of an Ito process remains close to a deterministic curve. We obtain significant estimates of densities and distribution functions in general classes of option pricing models, including generalisations of CIR and CEV processes and Local-Stochastic Volatility models. In the latter case, the estimates we derive have an impact on the moment explosion of the underlying price and, consequently, on the large-strike behaviour of the implied volatility. Parametric implied volatility modeling, in its turn, makes the object of the second part. In particular, we focus on J. Gatheral's SVI model, first proposing an effective quasi-explicit calibration procedure and displaying its performances on market data. Then, we analyse the capability of SVI to generate efficient approximations of symmetric smiles, building an explicit time-dependent parameterization.

Recent work in the field of financial mathematics has highlighted the importance of studying the regularity and fine-grained behavior of distribution tails for certain classes of diffusions with non-globally regular coefficients. In this thesis, we deal with problems arising from this context. We first study the existence, regularity and asymptotics in density space for solutions of stochastic differential equations by imposing only local conditions on the coefficients of the equation. Our analysis is based on the tools of Malliavin calculus and on estimates for Ito processes confined in a tube around a deterministic curve. We obtain significant estimates of the distribution function and density in classes of models including generalizations of the CIR and CEV and models with local-stochastic volatility: in the latter case, the estimates lead to the explosion of the moments of the underlying and thus have an impact on the asymptotic strike behavior of the implied volatility. The parametric modeling of the volatility surface, in turn, is the subject of the second part. We consider the SVI model of J. Gatheral, proposing a new quasi-explicit calibration strategy, whose performance on market data is illustrated. Then, we analyze the ability of SVI to generate approximations for symmetric smiles, by generalizing it to a time-dependent model. We test its application to a Heston model (without and with displacement), generating semi-closed approximations for the volatility smile.