Nicolò Valenti: Regression under shape restriction and the option price model

Adviser: Sara van de Geer

August 2007

Abstract:

Many types of problems are concerned with identifying a meaningful structure in real world situations. A structure involving orderings and inequalities is often useful since it is easy to interpret, understand, and explain. In many settings, economic theory only restricts the direction of the relationship between variables, not the particular functional form of their relationship. Let c(X) denote the call price as function of the strike price X. By the no arbitrage principle, c is a convex, decreasing function of
X, i.e. it satisfies certain shape constraints. It can be argued that economic theory virtually places no other restrictions on c, and that the estimation of the state-price density should be carried out using only these shape restrictions (and some bounds on first and second derivative). Further
smoothness assumptions or parametric assumptions may not be justified and have the potential risk of misspecifying the state-price density. Our work consists of studying estimation under such shape restrictions. We first consider monotone regression function estimation, the so-called  sotonic regression problem. Second, we analyse the problem of convex regression estimation. Then we build a nonparametric estimator of the call pricing that is decreasing and convex for small samples.

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