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Sensitivity analysis of path-dependent options in an incomplete market with pathwise functional Ito calculus

Siboniso Confrence Nkosi, Farai Julius Mhlanga

TL;DR

This work develops a pathwise, non-anticipative framework—functional Itô calculus—for pricing and hedging path-dependent derivatives in incomplete markets. It derives a functional Itô formula and a functional Feynman–Kac representation, enabling PDIE-based pricing and Monte Carlo computation of Greeks (Delta, Gamma, Vega) via explicit weight functions built from horizontal, vertical, and jump derivatives and their Lie bracket. The authors provide closed-form representations of the Greeks as expectations involving the payoff and tangent/weight processes, and illustrate the approach with digital options under a jump-diffusion model with zero interest rate. The results offer a robust, pathwise alternative to Malliavin methods, with potential extensions to American options, stochastic volatility, and multi-asset derivatives, enhancing both theoretical insight and numerical efficiency in path-dependent option pricing.

Abstract

Functional It^o calculus is based on an extension of the classical It^o calculus to functionals depending on the entire past evolution of the underlying paths and not only on its current value. The calculus builds on Follmer's deterministic proof of the It^o formula, see [3], and a notion of pathwise functional derivatives introduced by [5]. There are no smoothness assumptions required on the functionals, however, they are required to possess certain directional derivatives which may be computed pathwise, see [6, 9, 8]. Using functional It^o calculus and the notion of quadratic variation, we derive the functional It^o formula along with the Feynman-Kac formula for functional processes. Furthermore, we express the Greeks for path-dependent options as expectations, which can be efficiently computed numerically using Monte Carlo simulations. We illustrate these results by applying the formulae to digital options within the Black-Scholes model framework.

Sensitivity analysis of path-dependent options in an incomplete market with pathwise functional Ito calculus

TL;DR

This work develops a pathwise, non-anticipative framework—functional Itô calculus—for pricing and hedging path-dependent derivatives in incomplete markets. It derives a functional Itô formula and a functional Feynman–Kac representation, enabling PDIE-based pricing and Monte Carlo computation of Greeks (Delta, Gamma, Vega) via explicit weight functions built from horizontal, vertical, and jump derivatives and their Lie bracket. The authors provide closed-form representations of the Greeks as expectations involving the payoff and tangent/weight processes, and illustrate the approach with digital options under a jump-diffusion model with zero interest rate. The results offer a robust, pathwise alternative to Malliavin methods, with potential extensions to American options, stochastic volatility, and multi-asset derivatives, enhancing both theoretical insight and numerical efficiency in path-dependent option pricing.

Abstract

Functional It^o calculus is based on an extension of the classical It^o calculus to functionals depending on the entire past evolution of the underlying paths and not only on its current value. The calculus builds on Follmer's deterministic proof of the It^o formula, see [3], and a notion of pathwise functional derivatives introduced by [5]. There are no smoothness assumptions required on the functionals, however, they are required to possess certain directional derivatives which may be computed pathwise, see [6, 9, 8]. Using functional It^o calculus and the notion of quadratic variation, we derive the functional It^o formula along with the Feynman-Kac formula for functional processes. Furthermore, we express the Greeks for path-dependent options as expectations, which can be efficiently computed numerically using Monte Carlo simulations. We illustrate these results by applying the formulae to digital options within the Black-Scholes model framework.

Paper Structure

This paper contains 10 sections, 8 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. A Primer on pathwise functional Itô calculus
  3. Mathematical setup
  4. Functional It$\hat{\text{o}}$ calculus
  5. Greeks for path-dependent functionals
  6. Delta
  7. Gamma
  8. Vega
  9. Applications
  10. conclusion

Key Result

Theorem 4.1

Let $x_t$ be a semimartingale. Let $f(x)$ be a function o the $\mathcal{C}^{2}$-class. Then, for any $t\in[0,T]$, $f(X_t)$ is again a semimartingale and the following holds: where we have suppressed the arguments of $\gamma$.

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9: Boundedness-preserving functionals
  • Definition 2.10
  • ...and 17 more