Functional Expansions and Signatures

Bruno Dupire

Head of Quantitative Research

Bloomberg / NYU

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UCSB, September 29, 2022

CFMAR Workshop 2022

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Joint work with Valentin Tissot-Daguette (Bloomberg / Princeton)

OUTLINE

• Signatures
• Functional Itô Calculus
• Expansions of Functionals
• Wiener Chaos
• Intrinsic Value
• Functional Taylor
• Applications
• Claim Decomposition
• Hedging with Signature

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ONE DIMENSION + TIME

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(T,X) PATH VS (X1,X2) PATH

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SHORT WORDS EXAMPLES

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SIGNATURES: GEOMETRIC VISUALISATION

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PATH RECONSTRUCTION

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PATH RECONSTRUCTION

• We can compute signatures from a path.
• Can we reconstruct the path from the signatures?

Yes, and a subset of words is enough

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RECONSTRUCTION PROPERTY

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Further details in [V. Tissot-Daguette. “Short communication: Projection of functionals and fast pricing of exotic options”, SIFIN, 2022]

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FUNCTIONAL ITÔ CALCULUS

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�FUNCTIONAL ITÔ CALCULUS

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RESULTS AND APPLICATIONS

- Functional versions of Itô formula, Feynman-Kacs and BS PDE

- Super-replication (refinement of Kramkov decomposition)

- Lie Bracket of price and time functional derivatives

- Characterisation of attainable claims

- Decomposition of volatility risk

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�REVIEW OF ITÔ CALCULUS

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PATH SPACES

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FUNCTIONALS, T-FUNCTIONALS

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EXAMPLE OF A FUNCTIONAL

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FUNCTIONAL DERIVATIVES

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EXAMPLES

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PATHWISE ITÔ & STRATONOVICH FORMULA

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PATHWISE STRATONOVICH INTEGRATION

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FUNCTIONAL DERIVATIVES OF INTEGRALS

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FUNCTIONAL DERIVATIVES OF SIGNATURES

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SIGNATURE SPANNING

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SPACE SPANNED BY ORDER K SIGNATURES

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INCREMENTAL BASIS

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INCREMENTAL BASIS

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INCREMENTAL BASIS

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INDEPENDENT SIGNATURES

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FUNCTIONAL EXPANSIONS

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CLASSIFICATION OF EXPANSIONS

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Classical

Static (around a path)

Dynamic (after a path)

TAYLOR AS ITERATED INTEGRALS

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WIENER CHAOS EXPANSION

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INTRINSIC VALUE FUNCTIONAL

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INTRINSIC VALUE EXPANSION

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TAYLOR EXPANSION OF A FUNCTIONAL

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FUNCTIONAL TAYLOR (GENERAL CASE)

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2 USES

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SUMMARY

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FROM FUNCTIONAL EXPANSION TO T-FUNCTIONAL EXPANSION

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APPLICATIONS

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EUROPEAN AND EXOTIC OPTIONS

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EUROPEAN OPTION EXAMPLE

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FORWARD T1<T EXAMPLE

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FORWARD START EXAMPLE

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DEEP HEDGE MADE SIMPLE

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CONCLUSION

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REFERENCES

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• V. Bally, L. Caramellino, and R. Cont, “Stochastic integration by parts and functional Itô calculus”,�Advanced Courses in Mathematics - CRM Barcelona, 2016

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• B. Dupire, “Functional Itô Calculus", SSRN, republished in Quantitative Finance, 5, pp. 721-729, 2019

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• K.-T. Chen, “Integration of Paths, Geometric Invariants and a Generalized Baker-Hausdorff Formula”, Annals of Mathematics, 65, pp. 163–178, 1957

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• M. Fliess, "On the concept of derivatives and Taylor expansions for nonlinear input-output systems“, The 22nd IEEE Conference on Decision and Control, pp. 643-646, 1983

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• H. Föllmer, “Calcul d’Itô sans probabilités”. Séminaire de probabilités de Strasbourg, 15:143–150, 1981

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• C. Litterer and H. Oberhauser, “On a Chen–Fliess Approximation for Diffusion Functionals”, Monatshefte für Mathematik, 175, pp. 577-593, 2011

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• T. Lyons, S. Nejad, and I. Perez Arribas, ”Non-parametric Pricing and Hedging of Exotic Derivatives”, Applied Mathematical Finance, 27:6, pp. 457-494, 2020

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• V. Tissot-Daguette. “Short communication: Projection of functionals and fast pricing of exotic options”, SIAM Journal on Financial Mathematics, 13(2):SC74–SC86, 2022

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 arXiv:1809.09466

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Thank You

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