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Hidden Risks and Optionalities in American Options

Noura El Hassan, Bacel Maddah, Nassim N. Taleb

TL;DR

The paper addresses hidden convexity and model risk in American options by injecting stochasticity into key determinants, notably the carry and funding rates, and by formalizing a fugit-based framework to quantify the resulting optionality. It develops fragility measures and practical heuristics, such as the fugit and $\Omega$ metrics, and demonstrates these through a binomial lattice implementation with Gauss-Hermite integration across normal, Vasicek/Hull-White, and lognormal rate dynamics. The main contributions are the explicit separation of rate stochasticity from exercise decisions, the introduction of the fugit-based pricing approach, and extensive numerical results showing substantial, moneyness- and dynamics-dependent hidden optionality that challenges traditional European-vs-American pricing. The practical impact is a robust risk-management toolkit that emphasizes scenario analysis over precise pricing, helping practitioners assess and hedge the latent convexity embedded in American options.

Abstract

We develop a practical framework for identifying and quantifying the hidden layers of risks and optionality embedded in American options by introducing stochasticity into one or more of their underlying determinants. The heuristic approach remedies the problems of conventional pricing systems, which treat some key inputs deterministically, hence systematically underestimate the flexibility and convexity inherent in early-exercise features.

Hidden Risks and Optionalities in American Options

TL;DR

The paper addresses hidden convexity and model risk in American options by injecting stochasticity into key determinants, notably the carry and funding rates, and by formalizing a fugit-based framework to quantify the resulting optionality. It develops fragility measures and practical heuristics, such as the fugit and metrics, and demonstrates these through a binomial lattice implementation with Gauss-Hermite integration across normal, Vasicek/Hull-White, and lognormal rate dynamics. The main contributions are the explicit separation of rate stochasticity from exercise decisions, the introduction of the fugit-based pricing approach, and extensive numerical results showing substantial, moneyness- and dynamics-dependent hidden optionality that challenges traditional European-vs-American pricing. The practical impact is a robust risk-management toolkit that emphasizes scenario analysis over precise pricing, helping practitioners assess and hedge the latent convexity embedded in American options.

Abstract

We develop a practical framework for identifying and quantifying the hidden layers of risks and optionality embedded in American options by introducing stochasticity into one or more of their underlying determinants. The heuristic approach remedies the problems of conventional pricing systems, which treat some key inputs deterministically, hence systematically underestimate the flexibility and convexity inherent in early-exercise features.
Paper Structure (25 sections, 27 equations, 15 figures, 6 tables)

This paper contains 25 sections, 27 equations, 15 figures, 6 tables.

Table of Contents

  1. Unaccounted Optionality
  2. A note on model nonlinearity as fragility
  3. Application to American Options
  4. Illustrative Practitioner Episodes
  5. Differential Valuation Cases
  6. Pricing Implementations
  7. Integrating an American Option across Stochastic Rates
  8. Various stopping times methodologies
  9. Single expected Stopping Time
  10. The Stochastic Fugit: Distribution of Exercise Times
  11. Fugit-weighted integration heuristic
  12. Extension to Two Stochastic Rates
  13. Various distribution of rates
  14. Main Numerical Implementation
  15. Further Numerical Results and Technical Details
  16. ...and 10 more sections

Figures (15)

  • Figure 1: Optionality versus standard deviation for a an equity put option under a normally distributed local interest rate with $OA(r^*)=14.3184$, $S/K=1.00$
  • Figure 2: Optionality versus standard deviation for a currency put option under a lognormally distributed local interest rate with $OA(r^*)=15.1700$ and $S/K=1.00$
  • Figure 3: Optionality versus standard deviation for a currency call option under a Hull-White distributed local interest rate with $OA(r^*)=12.7779$ and $S/K=1.00$
  • Figure 4: Effective sopping time $\tau^*$ versus moneyness for an equity put option.
  • Figure 5: Effective sopping time $\tau^*$ versus moneyness for an American currency put
  • ...and 10 more figures

Theorems & Definitions (1)

  • remark 1