# On Higher Derivatives of Expectations

*Robert de Rozario*

Risk and Insurance from EconWPA

**Abstract:**
It is understood that derivatives of an expectation $E [\phi(S(T)) | S(0) = x]$ with respect to $x$ can be expressed as $E [\phi(S(T)) \pi | S(0) = x]$, where $S(T)$ is a stochastic variable at time $T$ and $\pi$ is a stochastic weighting function (weight) independent of the form of $\phi$. Derivatives of expectations of this form are encountered in various fields of knowledge. We establish two results for weights of higher order derivatives under the dynamics given by (\ref{dynamics}). Specifically, we derive and solve a recursive relationship for generating weights. This results in a tractable formula for weights of any order.

**Keywords:** price sensitivities; greeks; malliavin calculus (search for similar items in EconPapers)

**JEL-codes:** C63 (search for similar items in EconPapers)

**Date:** 2003-08-19

**Note:** Type of Document - LaTex; prepared on IBM PC ; to print on PostScript; pages: 6 ; figures: included. In the process of being submitted

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**Persistent link:** http://EconPapers.repec.org/RePEc:wpa:wuwpri:0308001

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