Adjoint-Based Calibration of Local Volatility Models

Auszug

Financial derivatives such as like options and futures have gained significant importance over the last 40 years. Not only the total number of contracts but also the variety did grow in a remarkable way. Nowadays, in addition to standard European-style plain vanilla call and put options, exotic derivatives like digital or barrier options of American or Bermudan-type, Asian-style options like lookbacks, as well as chooser options, cliquets or any reasonable combination are frequently traded derivatives. In order to price and hedge exotic options adequately, practitioners, such as traders and risk managers, need to extract accurate market dynamic information from liquidly traded standard instruments. This is done by calibrating financial market models to the current market situation. Pricing options and model calibration are key issues in the financial market literature. They pose interesting mathematical challenges in several areas like mathematical modeling, stochastic differential equations (SDEs), partial differential equations (PDEs), optimization and numerical analysis.
This work concerns the efficient calibration of financial market models and focuses mainly on local volatility models. While mostly PDE-constrained calibration problems are considered, several notes are also given on the efficient calibration using stochastic differential equations referred to Monte Carlo calibration. Recently adjoint techniques (henceforth called adjoints for reasons of simplicity) have gained considerable importance in financial literature, because they allow to quickly compute option sensitivities with respect to a large number of model parameters.
In this work, adjoints are considered from various perspectives; for instance, as a ‘dual’ version of Black–Scholes PDE or as an efficient way to compute gradient and Hessian information used for optimization. More precisely, a fairly basic but mathematically rigorous procedure is presented to derive discrete adjoint equations applicable to PDE discretization schemes like the Crank–Nicolson method and to SDE discretization schemes like the Euler–Maruyama method. Furthermore, important numerical issues are addressed to increase the applicability of the proposed methods in practice. For instance, proper parameterizations for the model parameters are discussed and the impact on the numerical effort for the computation of gradients and Jacobians using adjoint equations is investigated theoretically and numerically. To remedy numerical oscillations imposed by non-smooth initial data, the use of the so-called Rannacher time-marching scheme is discussed, which replaces the classical Crank–Nicolson one.
The last part of this work focuses on the reconstruction of non-parametric local volatility surfaces. The resulting large-scale optimization problem is solved using an inexact Gauss–Newton method embedded in a trust-region framework to obtain global convergence. The Gauss–Newton subproblems are solved by way of a preconditioned conjugate-gradient (PCG) method, where the preconditioner is not meant to accelerate convergence, but it is rather applied to overcome the ill-posedness of the calibration problem. Necessary matrix-vector products are computed without saving the Jacobian through adjoint and sensitivity techniques. Furthermore, the special choice of the trust-region norm imposes a smoothing effect on the calibrated local volatility surface.
To illustrate the performance of the proposed methods and to demonstrate the applicability in practice, detailed numerical examples on real market data are given.