Contents -- Preface -- Foundations -- 1. Multilevel Monte Carlo methods for applications in finance Mike Giles and Lukasz Szpruch -- 1. Introduction -- 2. Multilevel Monte Carlo -- 2.1. Monte Carlo -- 2.2. Multilevel Monte Carlo Theorem -- 2.3. Improved MLMC -- 2.4. SDEs -- 2.5. Euler and Milstein discretizations -- 2.6. MLMC algorithm -- 3. Pricing with MLMC -- 3.1. Euler-Maruyama scheme -- 3.2. Milstein scheme -- 3.2.1. Lookback options -- 3.3. Conditional Monte Carlo -- 3.4. Barrier options -- 3.5. Digital options -- 4. Greeks with MLMC -- 4.1. Monte Carlo Greeks -- 4.2. Multilevel Monte Carlo Greeks -- 4.3. European call -- 4.4. Conditional Monte Carlo for Pathwise Sensitivity -- 4.5. Split pathwise sensitivities -- 4.6. Optimal number of samples -- 4.7. Vibrato Monte Carlo -- 5. MLMC for Jump-diffusion processes -- 5.1. A Jump-adapted Milstein discretization -- 5.1.1. Multilevel Monte Carlo for constant jump rate -- 5.1.2. MLMC for Path-dependent rates -- 5.2. Lévy processes -- 6. Multi-dimensional Milstein scheme -- 6.1. Antithetic MLMC estimator -- 6.2. Clark-Cameron Example -- 6.3. Milstein discretization - General theory -- 6.4. Piecewise linear interpolation analysis -- 6.5. Simulations for antithetic Monte Carlo -- 7. Other uses of multilevel method -- 7.1. SPDEs -- 7.2. Nested simulation -- 7.3. Truncated series expansions -- 7.4. Mixed precision arithmetic -- 8. Multilevel Quasi-Monte Carlo -- 9. Conclusions -- Bibliography -- 2. Convergence of numerical methods for SDEs in finance Peter Kloeden and Andreas Neuenkirch -- 1. Introduction -- 2. Pathwise Convergence Rates of the Euler Scheme and general Itô-Taylor Methods -- 3. The Explicit Euler Scheme: Criteria for Weak and Strong Convergence -- 4. Strong convergence of implicit and tamed Euler schemes.

5. Strong Convergence Rates for the approximation of the Cox- Ingersoll-Ross process and the Heston model -- 6. Summary and Outlook -- Bibliography -- 3. Inverse problems in finance J. Baumeister -- 1. Forward and inverse problems in finance -- 1.1. Classification of problems -- 1.2. Financial market and financial products -- Fiancial market -- Derivatives -- 1.3. The forward problem of option pricing -- The Black-Scholes-Merton model -- Pricing formula -- Sensitivity -- 1.4. Volatility -- Implied volatility -- Stochastic volatility -- Local volatility -- 1.5. Inverse problems of finance -- 2. An introduction into regularization of ill-posed problems -- 2.1. Ill-posedness/Well-posedness -- Definition -- Tutorial example: numerical differentiation -- Numerical methods: inverse crime -- 2.2. Regularization -- The linear equation -- Regularization scheme -- Filtering -- The classical method of Tikhonov -- Regularization by iteration -- Statistical regularization -- 2.3. Calibration -- Introduction -- Least squares formulation -- 2.4. Data assimilation -- 3. Elementary numerical approaches -- 3.1. Computation of the implied volatility -- The goal -- Newton's method -- Analysis of the computation scheme -- 3.2. Pricing surface -- Implied volatility surface -- Interpolation in option pricing -- Interpolation by shape-preserving splines in the presence of noise -- Computation of the option price function -- 3.3. Forward rates and discount factors -- Discount factor -- Forward rates -- Interpolation of market data -- 3.4. Parametric models for the forward rate -- The Nelson-Siegel model -- Numerical recipes for the calibration -- 3.5. Sensitivities -- Parameter dependent expectation values -- Sensitivities via difference quotients -- 4. Dupire's method: the dual equation -- 4.1. Dupire's equation -- 4.2. The uniqueness theorem for the inverse problem.

4.3. An integral equation for the unknown local volatility -- 4.4. Computation of the volatility via numerical differentiation -- The equation for the local volatility -- The computation of the local volatility via numerical differentiation -- 5. Calibration/Hilbert space methods -- 5.1. Calibration via gradient methods -- The least squares problem -- Regularization of the least squares problem -- Parametrization: a remark -- The gradient-type method of Lagnado and Osher -- 5.2. Black-Scholes equation: weak solutions -- Appropriate function spaces -- 5.3. Weak solutions of Dupire's equation -- Weak solutions -- Calibration via Dupires equation -- 5.4. Estimation of volatilities via Kaczmarz method -- Introduction -- The ART-algorithm -- The Landweber-Kaczmarz method -- The Kaczmarz-method for the computation of local volatilities -- Bibliography -- 4. Asymptotic and non asymptotic approximations for option valuation R. Bompis and E. Gobet -- 1. Introduction -- 2. An overview of approximation results -- 2.1. Large and small strikes, at fixed maturity -- 2.2. Long maturities, at fixed strike -- 2.3. Long maturities, with large/small strikes -- 2.4. Non large maturities and non extreme strikes -- 2.4.1. Small noise expansion -- 2.4.2. Short maturity -- 2.4.3. Fast volatility -- 2.4.4. Proxy expansion -- 2.5. Asymptotic expansion versus non-asymptotic expansion -- 3. Approximation based on proxy -- 3.1. Notations and definitions -- 3.2. Proxy approximation: a primer using the local volatility at spot -- 3.3. Towards Call option approximations with the local volatility at strike and at mid-point -- 4. Proofs: a comparative discussion between stochastic analysis and PDE techniques -- 4.1. A pure stochastic analysis approach -- 4.2. Mixing stochastic analysis and PDE -- 4.3. A pure PDE approach -- 5. Higher-order proxy approximation.

5.1. Third order approximation with the local volatility at spot and at strike -- 5.2. Third order approximation with the local volatility at midpoint -- 5.3. Third order expansion of the implied volatility -- 6. Approximation of the Delta -- 7. Numerical experiments -- 7.1. The set of tests -- 7.2. Analysis of results -- 7.3. CEV Delta approximations -- 8. Appendix -- 8.1. Computations of derivatives of CallBS w.r.t the log spot, the log strike and the total variance -- 8.2. Derivatives of CallBA w.r.t the spot, the strike and the total variance -- 8.3. Derivatives of BS w.r.t the log spot, the log strike and the total variance -- 8.4. Proof of Lemma 1 -- 8.5. Applications of the expansions for time-independent CEV model -- Bibliography -- Algorithms -- 5. Discretization of backward stochastic Volterra integral equations Christian Bender and Stanislav Pokalyuk -- 1. Introduction -- 2. Preliminaries -- 3. Main result -- 4. Connection to parabolic Cauchy problems -- 4.1. Construction of the solutions of BSVIEs via PDEs -- 4.2. Regularity problem for parabolic Cauchy problems -- 4.3. Proof of Lemma 4.1 -- 5. Proof of Theorem 3.6 -- 5.1. Convergence of the finite dimensional distributions -- 5.2. Tightness -- 6. Numerical example -- 7. Proof of Lemma 3.3 -- Bibliography -- 6. Semi-Lagrangian schemes for parabolic equations Kristian Debrabant and Espen Robstad Jakobsen -- 1. Introduction -- 2. Some motivation for the schemes -- 3. Local PDEs of Bellman type -- 3.1. Examples of approximations L k -- 3.2. Linear interpolation SL scheme (LISL) -- 3.3. Error estimates -- 3.4. Boundary conditions -- 3.5. Convergence test for a super-replication problem -- 3.6. A super-replication problem -- 4. Nonlocal PDEs of Bellman type -- 4.1. Derivation of the scheme for linear problems -- 4.2. The scheme in the nonlinear case under (18).

5. Appendix: Well-posedness of the Bellman equation -- Bibliography -- 7. Derivative-free weak approximation methods for stochastic differential equations Kristian Debrabant and Andreas Rößler -- 1. Introduction and examples -- 2. Efficient second order stochastic Runge-Kutta methods -- 2.1. An efficient explicit second order SRK method -- 2.2. Implicit SRK methods -- 3. Third order SRK methods for SDEs with additive noise -- 4. Continuous weak approximation -- 5. Adaptive step size selection -- 5.1. Embedded stochastic Runge-Kutta methods -- 5.2. Step size control algorithm -- 5.3. Simulation of the conditional distributed random variables -- 6. Adaptive selection of Monte Carlo sample number and time stepping -- 6.1. Classical setting -- 6.2. Short outlook to multi-level Monte Carlo simulation -- Bibliography -- 8. Wavelet solution of degenerate Kolmogoro forward equations Oleg Reichmann and Christoph Schwab -- 1. Financial Modelling -- 2. Pricing of Derivative Contracts -- 2.1. Models -- 2.2. Contracts -- 2.2.1. European -- 2.2.2. Compound -- 2.2.3. American -- 2.2.4. Swing -- 3. Well-posedness of PDEs and PDIs -- 3.1. General results -- 3.2. Contracts of European type -- 3.2.1. Univariate models -- 3.2.2. Stochastic volatility models -- 3.3. Contracts of American type -- 3.3.1. Univariate models -- 3.3.2. Stochastic volatility models -- 3.4. Greeks -- 4. Discretization -- 4.1. Wavelet transformation -- 4.2. Norm equivalences -- 4.3. Weighted spaces -- 4.4. Space discretization -- 4.5. Discontinuous Galerkin time discretization -- 4.5.1. Derivation of the linear systems -- 5. Numerical examples -- Bibliography -- 9. Randomized multilevel quasi-Monte Carlo path simulation Thomas Gerstner and Marco Noll -- 1. Introduction -- 2. Multilevel Monte Carlo method -- 3. Randomized multilevel quasi-Monte Carlo method -- 4. Extended complexity theorem.

5. Numerical results.