CVA and vulnerable options pricing by correlation expansions

We consider the problem of computing the Credit Value Adjustment ({CVA}) of a European option in presence of the Wrong Way Risk ({WWR}) in a default intensity setting. Namely we model the asset price evolution as solution to a linear equation that might depend on different stochastic factors and we provide an approximate evaluation of the option's price, by exploiting a correlation expansion approach, introduced in \cite{AS}. We compare the numerical performance of such a method with that recently proposed by Brigo et al. (\cite{BR18}, \cite{BRH18}) in the case of a call option driven by a GBM correlated with the CIR default intensity. We additionally report some numerical evaluations obtained by other methods.Comment: 21 page

A primal-dual algorithm for BSDEs

We generalize the primal-dual methodology, which is popular in the pricing of early-exercise options, to a backward dynamic programming equation associated with time discretization schemes of (reflected) backward stochastic differential equations (BSDEs). Taking as an input some approximate solution of the backward dynamic program, which was pre-computed, e.g., by least-squares Monte Carlo, our methodology allows to construct a confidence interval for the unknown true solution of the time discretized (reflected) BSDE at time 0. We numerically demonstrate the practical applicability of our method in two five-dimensional nonlinear pricing problems where tight price bounds were previously unavailable

Pricing synthetic CDO tranche on ABS

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Random Time Forward Starting Options

We introduce a natural generalization of the forward-starting options, first discussed by M. Rubinstein. The main feature of the contract presented here is that the strike-determination time is not fixed ex-ante, but allowed to be random, usually related to the occurrence of some event, either of financial nature or not. We will call these options {\bf Random Time Forward Starting (RTFS)}. We show that, under an appropriate "martingale preserving" hypothesis, we can exhibit arbitrage free prices, which can be explicitly computed in many classical market models, at least under independence between the random time and the assets' prices. Practical implementations of the pricing methodologies are also provided. Finally a credit value adjustment formula for these OTC options is computed for the unilateral counterparty credit risk.Comment: 19 pages, 1 figur

Approximate Pricing of Swaptions in Affine and Quadratic Models

This paper proposes new bounds on the prices of European-style swaptions for affine and quadratic interest rate models. These bounds are computable whenever the joint characteristic function of the state variables is known. In particular, our lower bound involves the computation of a one-dimensional Fourier transform independently of the swap length. In addition, we control the error of our method by providing a new upper bound on swaption price that is applicable to all considered models. We test our bounds on different affine models and on a quadratic Gaussian model. We also apply our procedure to the multiple curve framework. The bounds are found to be accurate and computationally efficient

A generative adversarial network approach to calibration of local stochastic volatility models

We propose a fully data-driven approach to calibrate local stochastic volatility (LSV) models, circumventing in particular the ad hoc interpolation of the volatility surface. To achieve this, we parametrize the leverage function by a family of feed-forward neural networks and learn their parameters directly from the available market option prices. This should be seen in the context of neural SDEs and (causal) generative adversarial networks: we generate volatility surfaces by specific neural SDEs, whose quality is assessed by quantifying, possibly in an adversarial manner, distances to market prices. The minimization of the calibration functional relies strongly on a variance reduction technique based on hedging and deep hedging, which is interesting in its own right: it allows the calculation of model prices and model implied volatilities in an accurate way using only small sets of sample paths. For numerical illustration we implement a SABR-type LSV model and conduct a thorough statistical performance analysis on many samples of implied volatility smiles, showing the accuracy and stability of the method.Comment: Replacement for previous version: Major update of previous version to match the content of the published versio

Optimization in Quasi-Monte Carlo Methods for Derivative Valuation

Computational complexity in financial theory and practice has seen an immense rise recently. Monte Carlo simulation has proved to be a robust and adaptable approach, well suited for supplying numerical solutions to a large class of complex problems. Although Monte Carlo simulation has been widely applied in the pricing of financial derivatives, it has been argued that the need to sample the relevant region as uniformly as possible is very important. This led to the development of quasi-Monte Carlo methods that use deterministic points to minimize the integration error. A major disadvantage of low-discrepancy number generators is that they tend to lose their ability of homogeneous coverage as the dimensionality increases. This thesis develops a novel approach to quasi-Monte Carlo methods to evaluate complex financial derivatives more accurately by optimizing the sample coordinates in such a way so as to minimize the discrepancies that appear when using lowdiscrepancy sequences. The main focus is to develop new methods to, optimize the sample coordinate vector, and to test their performance against existing quasi-Monte Carlo methods in pricing complicated multidimensional derivatives. Three new methods are developed, the Gear, the Simulated Annealing and the Stochastic Tunneling methods. These methods are used to evaluate complex multi-asset financial derivatives (geometric average and rainbow options) for dimensions up to 2000. It is shown that the two stochastic methods, Simulated Annealing and Stochastic Tunneling, perform better than existing quasi-Monte Carlo methods, Faure' and Sobol'. This difference in performance is more evident in higher dimensions, particularly when a low number of points is used in the Monte Carlo simulations. Overall, the Stochastic Tunneling method yields the smallest percentage root mean square relative error and requires less computational time to converge to a global solution, proving to be the most promising method in pricing complex derivativesImperial Users onl