Functional Itô calculus^†

† The Editors of Quantitative Finance are delighted that Bruno Dupire has accepted their invitation to publish his seminal paper on functional Itô calculus in honour of his 60th birthday. Well-known to both academics and practitioners, Bruno's 2009 paper on SSRN continues to be highly-cited, with new applications in mathematical finance and the theory of stochastic optimal control appearing regularly ever since.

Abstract

We extend some results of the Itô calculus to functionals of the current path of a process to reflect the fact that often the impact of randomness is cumulative and depends on the history of the process, not merely on its current value. We express the differential of the functional in terms of adequately defined partial derivatives to obtain an Itô formula. We develop an extension of the Feynman-Kac formula to the functional case and an explicit expression of the integrand in the Martingale Representation Theorem. We establish that under certain conditions, even path dependent options prices satisfy a partial differential equation in a local sense. We exploit this fact to find an expression of the price difference between two models and compute variational derivatives with respect to the volatility surface.

Keywords:

• Path dependent functionals and options
• Functional Itô and Feynman-Kac formulae
• Martingale representation
• Functional PDE
• Delta hedging
• Model impact
• Submartingale bounds

Acknowledgments

It is my pleasure to thank Nicole El Karoui, Rama Cont, Philip Protter, Raphael Douady, the participants of the Princeton conference on Implied Volatility Models at Huntington Beach in October 2008 where this work was first presented, and my colleagues at Bloomberg, for enriching conversations. Thank you to the Editors for their invitation and copy edit of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

† The Editors of Quantitative Finance are delighted that Bruno Dupire has accepted their invitation to publish his seminal paper on functional Itô calculus in honour of his 60th birthday. Well-known to both academics and practitioners, Bruno's 2009 paper on SSRN continues to be highly-cited, with new applications in mathematical finance and the theory of stochastic optimal control appearing regularly ever since.