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Volatility Smile by Multilevel Least Square Yves Achdou

Summary: Volatility Smile by Multilevel Least Square
Yves Achdou 
Olivier Pironneau y
October 1, 2001
The aim of this paper is to propose several algorithms for nding the local volatility from
partial observations of the price of an European vanilla option. Dupire's equation is used.
The local volatility and the price of the option are discretized by nite elements with highly
non uniform meshes and with a coarser mesh for the local volatility. The inverse problem
is formulated as a least square problem and the minimization is done by an interior point
method. The gradient of the cost function is computed exactly by solving an adjoint problem.
A multilevel approach is proposed for accelerating the computations. Also, a suboptimal
time-stepping algorithm is considered. For all the proposed algorithms, numerical tests are
1 Introduction
The volatility, , is the diĘcult parameter of the Black-Scholes model of nance (see Wilmott
[23] for an introduction). It is convenient but unrealistic to take it to be constant but on the
other hand little is known to relate it explicitly to any observable nancial data. Therefore much
research is being conducted to solve eĘciently the inverse problem whereby one adjusts  so
that the solution of the Black-Scholes model ts nancial observations. Such nancial parameter


Source: Achdou, Yves - Laboratoire Jacques-Louis Lions, Université Pierre-et-Marie-Curie, Paris 6


Collections: Mathematics