Free-Boundary Estimation for American Put Options

Jorge N. Hernández & Enrique Villamor — Florida International University, Department of Mathematics and Statistics

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Overview

This repository contains the research code accompanying the working paper Approximating the Free Boundary of an American Option. The paper develops a multi-period binomial framework to estimate the early-exercise free boundary of American put options under two volatility regimes:

1. Constant volatility — standard recombining binomial tree with backward induction and linear interpolation to locate the free boundary at each time step.
2. Stochastic volatility (GARCH) — a non-recombining binomial tree where volatility is updated dynamically at each node using a locally recalibrated GARCH(1,1) model, producing path-dependent variance estimates that reflect volatility clustering and mean reversion.

The key contribution is a unified framework that simultaneously estimates the free boundary and prices the American option, bridging two problems typically treated separately in the literature.

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Mathematical Problem

The value $V(S,t)$ of an American put option satisfies the free-boundary problem:

$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV \leq 0 \qquad V(S,t) \geq \max(K-S, 0)$$

with complementary slackness: at each point either the Black-Scholes PDE holds (continuation region) or the option is exercised (exercise region). The free boundary $S_f(t)$ is the critical stock price at which:

$$V(S_f(t), t) = K - S_f(t) \qquad \frac{\partial V}{\partial S}(S_f(t), t) = -1$$

No closed-form solution exists due to the coupling of the PDE with these nonlinear free-boundary and smooth-pasting conditions. This work estimates $S_f(t)$ numerically via binomial trees.

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Methods

Constant Volatility

• Recombining binomial tree with up/down factors $u = e^{\sigma\sqrt{\Delta t}}$, $d = 1/u$
• Backward induction with early exercise: $V^*{i,j} = \max(V{i,j},, K - S_{i,j})$
• Free boundary located by bracketing the transition between continuation and exercise regions
• Linear interpolation to refine $S_f(t_j)$ between adjacent nodes
• Iterative adjustment of $S_0$ until convergence: $|V_{0,0} - \max(K-S_0,0)| < \varepsilon$

Stochastic Volatility (GARCH)

• Non-recombining binomial tree — each path is distinct
• GARCH(1,1) recalibrated locally at each node using only the price history of the unique path leading to it:

$$\sigma_{i,j}^2 = \omega_{i,j} + \beta_{i,j} \sigma_{\lfloor i/2 \rfloor, j-1}^2 + \alpha_{i,j}\left(\frac{S_{i,j} - S_{\lfloor i/2 \rfloor, j-1}}{S_{\lfloor i/2 \rfloor, j-1}}\right)^2$$

• Stock price evolution at each node:

$$S_{i,j} = S_{\lfloor i/2 \rfloor, j-1} \cdot e^{\mu \Delta t + (-1)^i \sigma_{\lfloor i/2 \rfloor, j-1} \sqrt{252 \Delta t}}$$

• After recalibration, the historical dataset is restored to prevent forward contamination across paths
• Risk-neutral probabilities are node-specific: $p_{i,j} = \frac{e^{r\Delta t} - d_{i,j}}{u_{i,j} - d_{i,j}}$

Continuous-Time Limit

The paper also derives the continuous-time SDE limit of the GARCH(1,1) process:

$$ d\sigma_t^2 = \bar{\gamma}\left[\Gamma_L - \sigma_t^2\right] \cdot 252\ dt + \bar{\alpha}\sqrt{2}\sqrt{252}\ \sigma_t^2\ dW_t $$

where $\Gamma_L$ is the long-run variance and $\bar{\gamma} = 1 - \bar{\alpha} - \bar{\beta}$. This confirms the discrete updates converge to a mean-reverting diffusion in continuous time.

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Results

Constant volatility (n = 100, K = 450, T = 1, MSFT 2019–2024):

• Free boundary converges to $S_0 = 313.73$ at $t = 0$
• Boundary exhibits the expected concave-up shape approaching maturity

Stochastic volatility (n = 10, K = 450, MSFT 2019–2024, $S_{0,0} = 391.46$, $\varepsilon = 0.01$):

• GARCH-driven boundary lies notably higher than the constant-vol boundary, reflecting increased early-exercise incentive under volatile conditions
• Non-recombining structure produces path-dependent boundaries that constant-vol models cannot capture
• Minor irregularities at small n diminish as n → ∞

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Repository Structure

Free-Boundary/
├── constant-volatility-free-boundary.ipynb   ← Constant vol binomial tree & boundary estimation
├── garch-volatility-free-boundary.ipynb      ← GARCH non-recombining tree & boundary estimation
├── MSFT.csv                                   ← Microsoft daily price data (Aug 2019 – Aug 2024)
└── requirements.txt                           ← Python dependencies

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Replication

git clone https://github.com/jorherna1/Free-Boundary.git
cd Free-Boundary
pip install -r requirements.txt
jupyter notebook

Open either notebook and run all cells. The GARCH notebook is computationally intensive for large n due to the non-recombining structure requiring GARCH recalibration at every node (2^n nodes at depth n).

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Citation

Hernández, J. N., & Villamor, E. (2024). Approximating the free boundary of an American option. Working paper, Florida International University.

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Authors

• Jorge N. Hernández — Florida International University, Institute of Environment & Mathematics and Statistics. jorherna@fiu.edu · ORCID: 0009-0000-0469-4327
• Enrique Villamor — Florida International University, Mathematics and Statistics. villamor@fiu.edu · ORCID: 0000-0003-0556-6460