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# On Higher Derivatives of Expectations

## Author

Listed:
• Robert de Rozario

(University of NSW, Sydney, Australia)

## 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.

## Suggested Citation

• Robert de Rozario, 2003. "On Higher Derivatives of Expectations," Risk and Insurance 0308001, University Library of Munich, Germany.
• Handle: RePEc:wpa:wuwpri:0308001
Note: Type of Document - LaTex; prepared on IBM PC ; to print on PostScript; pages: 6 ; figures: included. In the process of being submitted
as

File URL: https://econwpa.ub.uni-muenchen.de/econ-wp/ri/papers/0308/0308001.pdf

## References listed on IDEAS

as
1. Eric Benhamou, 2003. "Optimal Malliavin Weighting Function for the Computation of the Greeks," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 37-53.
2. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
Full references (including those not matched with items on IDEAS)

### Keywords

price sensitivities; greeks; malliavin calculus;

### JEL classification:

• C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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