A linearization-based solution to an inverse problem in financial markets

Valuation of options and other financial derivatives critically depends on the specification of the stochastic process for the underlying assets. Values of financial derivatives are directly observable. These may be used to recover unobservable volatility surfaces (functions). This kind of inverse problems, where one looks for causes of observed effects, are usually “ill-posed”. This paper is concerned with the estimation of a local volatility surface by linearizing this inverse problem. The MAPLE programs that numerically implement the linearization algorithm show that the method is reasonably accurate for reconstruction of a volatility which is useful in the accurate pricing of exotic options.