Adjoint equation, Monte Carlo calibration, Multi-layer method, 65C05, 65K05, 90C30, 90C90, 91B28, C61, C63,
The pricing of derivatives has gained considerable importance in the finance industry and leads to challenging problems in numerical optimization. We focus on the numerical solution of a stochastic model for option prices. In particular, we are concerned with the calibration of these models to real data, which leads to large scale optimization problems. We consider the numerical solution of these optimization problems and give some indication how to reduce the complexity of these problems. Special emphasis is devoted to a multi-layer strategy which is embedded into the optimization iteration.In recent years optimization problems have gained increasing importance in the finance sector. Examples of applications are portfolio optimization, hedging strategies and model calibration. The last application is the basis for the two first ones, as the pricing of a portfolio including options requires an accurate model for the option prices; this applies likewise to hedging strategies for complex derivatives using standard options.For this reason we consider a well established model for plain vanilla options, the Heston model. In this model the stock price S t is driven by a Brownian motion and the variance v t is governed by a linear mean-reverting stochastic process in an interval [0, T ]. This leads to Heston's stochastic volatility model [2]where the two Brownian motions are correlated with correlation coefficient ρ. The constants κ, θ, σ, ρ, c v can be chosen in such a way that the model can be adapted to real market data. Suppose that several observed data C 1 , ..., C I are given for options on the same underlying with expiration times T i ≤ T and strike prices K i for i = 1, ..., I. Then one tries to mimimize the following least squares problemover a suitable set U , where S Ti (x) is the output of the stochastic process (1) which obviously depends on the vector x of parameters. It should be noted that the computation of the price of an option can be performed in various ways. For models like the Black-Scholes or Heston model there are explicit or semi-explicit solution formulas available, for an application to calibration see [1]. In many cases the price of an option can also be computed by solving a partial differential equation. However, closed form solutions do not exist in most cases and PDE solvers have to be tailored highly to the specific model structure and may not be applicable, if the number of space variables in the PDE is large.In contrast, a discretization of the stochastic differential equation within a Monte-Carlo framework allows to adopt fairly easily existing codes to new models even if the dimension of the SDE changes. On the other hand, the huge number of simulations is commonly considered as the major drawback for the Monte-Carlo method and has a considerable impact on the solution of the discretized calibration problem. In the sequel we propose various approaches how to alleviate this drawback within an optimization framework.In order to solve the calibration problem (2) numerically, we hence...
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