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The Algotrading Book

Chapter 142: PINN for American Option Pricing

Chapter 142: PINN for American Option Pricing

Physics-Informed Neural Networks for the Free Boundary Problem

American options represent one of the most important — and most challenging — problems in computational finance. Unlike European options, which can only be exercised at expiration, American options can be exercised at any time before maturity. This early exercise feature creates a free boundary problem that has no closed-form solution.

In this chapter, we develop a Physics-Informed Neural Network (PINN) that learns to price American options by embedding the Black-Scholes PDE and the early exercise constraint directly into the network’s loss function. The PINN approach replaces traditional numerical methods (finite differences, binomial trees) with a neural network that satisfies the governing physics by construction.

Table of Contents

  1. American Options: The Early Exercise Problem
  2. Mathematical Formulation
  3. The Free Boundary Problem
  4. PINN Architecture for American Options
  5. Loss Function Design
  6. Penalty Method for the Free Boundary
  7. Comparison with Longstaff-Schwartz (LSM)
  8. Greeks via Automatic Differentiation
  9. Application to Crypto Options (Bybit)
  10. Implementation
  11. Results and Visualization
  12. References

1. American Options: The Early Exercise Problem

European vs American Options

FeatureEuropean OptionAmerican Option
ExerciseOnly at expiry TAny time t <= T
Pricing PDEBlack-Scholes (equality)Black-Scholes (inequality)
Closed formYes (Black-Scholes formula)No
Early exercise premiumNonePositive (especially for puts)
Numerical methodsStraightforwardFree boundary problem

For a European put, the holder must wait until expiration regardless of how deep in-the-money the option is. For an American put, rational exercise occurs when the underlying price falls sufficiently below the strike — the time value of waiting becomes less than the intrinsic value.

Why American Puts Have Early Exercise Premium

Consider an American put with strike K = 100. If the stock falls to S = 10:

  • Intrinsic value: K - S = 90
  • European put value: less than 90 (due to discounting and possibility of stock recovery)
  • Optimal strategy: exercise immediately, receive 90, invest at risk-free rate

The early exercise premium is the difference between American and European option values:

EEP = V_American(S, t) - V_European(S, t) >= 0

2. Mathematical Formulation

Black-Scholes PDE

For a derivative V(S, t) on an underlying following geometric Brownian motion:

dS = mu * S * dt + sigma * S * dW

The risk-neutral pricing PDE is:

dV/dt + (1/2) * sigma^2 * S^2 * d2V/dS2 + r * S * dV/dS - r * V = 0

where:

  • S: spot price of the underlying
  • t: time
  • sigma: volatility
  • r: risk-free interest rate
  • V(S, t): option value

American Option: Inequality Constraint

For an American option, the value must always be at least the intrinsic value (payoff from immediate exercise):

V(S, t) >= h(S)   for all (S, t) in the domain

where h(S) is the payoff function:

  • Put: h(S) = max(K - S, 0)
  • Call: h(S) = max(S - K, 0)

The Linear Complementarity Problem (LCP)

The American option pricing problem is formulated as an LCP:

max( dV/dt + (1/2)*sigma^2*S^2*d2V/dS2 + r*S*dV/dS - r*V,  h(S) - V ) = 0

This means at every point (S, t), exactly one of these holds:

  1. Continuation region: The PDE holds as equality, and V(S,t) > h(S)
  2. Exercise region: V(S,t) = h(S), and the PDE operator is non-positive

The boundary between these two regions is the free boundary S*(t).

3. The Free Boundary Problem

Exercise Boundary S*(t)

For an American put, the exercise boundary S*(t) divides the (S, t) plane:

S < S*(t)  :  Exercise region (V = K - S)
S > S*(t)  :  Continuation region (PDE holds)
S = S*(t)  :  Free boundary

Properties of the exercise boundary:

  • S*(T) = K (at maturity, exercise at-the-money)
  • S*(t) < K for t < T (exercise boundary is below strike)
  • S*(t) is monotonically increasing as t -> T
  • Smooth-pasting condition: dV/dS is continuous across S*(t)

Smooth-Pasting Conditions

At the free boundary S = S*(t):

V(S*(t), t) = K - S*(t)         (value matching)
dV/dS(S*(t), t) = -1            (smooth pasting for put)

These conditions ensure the option value transitions smoothly from the exercise region to the continuation region.

4. PINN Architecture for American Options

Network Design

Our PINN takes (S, t) as input and outputs V(S, t):

Input: (S, t) in R^2
  |
  v
[Linear(2, 64)] -> [Tanh]
  |
[Linear(64, 64)] -> [Tanh]
  |
[Linear(64, 64)] -> [Tanh]
  |
[Linear(64, 64)] -> [Tanh]
  |
[Linear(64, 1)] -> [Softplus]   (ensures V >= 0)
  |
  v
Output: V(S, t) >= 0

Key design choices:

  • Tanh activation: smooth, bounded, good for PDE learning
  • Softplus output: ensures non-negative option values
  • 4 hidden layers x 64 neurons: sufficient capacity for the free boundary
  • Input normalization: S/S_max, t/T to [0, 1] range

Input Normalization

For numerical stability and faster convergence:

S_normalized = S / S_max
t_normalized = t / T

The output is scaled back:

V = network(S_norm, t_norm) * K  # Scale by strike

5. Loss Function Design

The PINN loss combines four components:

L_total = w_pde * L_pde + w_terminal * L_terminal + w_boundary * L_bc + w_penalty * L_penalty

5.1 PDE Residual Loss

Penalizes violations of the Black-Scholes PDE in the continuation region:

L_pde = (1/N) * sum_i [ f(S_i, t_i) ]^2

where f is the PDE residual:

f(S, t) = dV/dt + (1/2)*sigma^2*S^2*d2V/dS2 + r*S*dV/dS - r*V

All derivatives are computed via automatic differentiation through the network.

5.2 Terminal Condition Loss

At maturity t = T:

L_terminal = (1/N_T) * sum_i [ V(S_i, T) - h(S_i) ]^2

5.3 Boundary Condition Loss

At domain boundaries:

For puts:

V(0, t) = K * exp(-r*(T-t))    (deep ITM)
V(S_max, t) = 0                 (deep OTM)

For calls:

V(0, t) = 0                     (deep OTM)
V(S_max, t) ~ S_max - K*exp(-r*(T-t))  (deep ITM)

5.4 Early Exercise Penalty Loss

This is the key innovation for American options:

L_penalty = lambda * (1/N) * sum_i [ max(h(S_i) - V(S_i, t_i), 0) ]^2

This penalizes the network whenever V(S, t) < h(S), enforcing the no-arbitrage constraint. The penalty parameter lambda is increased during training (penalty scheduling) to gradually enforce the constraint.

Penalty Schedule

penalty_schedule = {
    0: 100,      # Warm-up: learn approximate PDE solution
    1000: 500,   # Start enforcing exercise constraint
    2000: 1000,  # Tighten constraint
    3000: 5000,  # Near-exact enforcement
    4000: 10000, # Final precision
}

6. Penalty Method for the Free Boundary

Why Penalty Method?

The LCP constraint is non-differentiable (due to the max operator), making direct optimization difficult. The penalty method replaces the hard constraint with a smooth penalty:

Original LCP:

max(L_BS[V], h(S) - V) = 0

Penalized PDE:

L_BS[V] + lambda * max(h(S) - V, 0) = 0

As lambda -> infinity, the penalized solution converges to the exact American option price. In practice, lambda = 10000 provides sufficient accuracy.

Alternative: Domain Decomposition

An alternative approach splits the domain explicitly:

  1. Estimate the exercise boundary S*(t) using the current network
  2. Apply PDE loss only in the continuation region S > S*(t)
  3. Apply payoff loss in the exercise region S < S*(t)
  4. Enforce smooth pasting at S = S*(t)

The penalty method is simpler and more robust, so we use it as the primary approach.

7. Comparison with Longstaff-Schwartz (LSM)

LSM Algorithm

The Longstaff-Schwartz method prices American options by backward induction on simulated Monte Carlo paths:

  1. Simulate N paths of the underlying using risk-neutral dynamics
  2. At maturity: cash flow = payoff(S_T)
  3. Backward induction (from T-1 to 1):
    • For in-the-money paths, regress discounted future cash flows on basis functions of S
    • Exercise if intrinsic value > estimated continuation value
  4. Discount all cash flows to time 0
def lsm_american_option(s0, K, r, sigma, T, n_steps, n_paths):
    paths = simulate_gbm(s0, r, sigma, T, n_steps, n_paths)
    cashflows = payoff(paths[-1])


    for t in range(n_steps-1, 0, -1):
        itm = payoff(paths[t]) > 0
        continuation = regression(paths[t][itm], cashflows[itm])
        exercise = payoff(paths[t][itm]) > continuation
        cashflows[itm][exercise] = payoff(paths[t][itm][exercise])


    return mean(discount(cashflows))

PINN vs LSM Comparison

AspectPINNLSM
TrainingOne-time (expensive)N/A (per-evaluation)
EvaluationO(1) forward passO(N * M) per price
GreeksFree (autograd)Finite differences (noisy)
Exercise boundaryContinuousDiscrete, noisy
AccuracyDepends on trainingConverges with N->inf
New parametersRequires retrainingRe-run simulation
DimensionalityScales to multi-assetCurse of dimensionality

Key advantage of PINN: Once trained, the model provides instant pricing and Greeks for any (S, t) in the domain, whereas LSM must re-run the full simulation for each new evaluation point.

8. Greeks via Automatic Differentiation

Exact Greeks from Autograd

Since V(S, t) is a differentiable neural network, we compute Greeks directly:

# Delta = dV/dS
delta = torch.autograd.grad(V, S, create_graph=True)


# Gamma = d2V/dS2
gamma = torch.autograd.grad(delta, S, create_graph=True)


# Theta = dV/dt
theta = torch.autograd.grad(V, t, create_graph=True)

Greeks for American Options

American option Greeks have special properties near the exercise boundary:

  • Delta: jumps from dV/dS (PDE region) to -1 (put) or +1 (call) at S*(t)
  • Gamma: has a spike at the exercise boundary
  • Theta: discontinuous across the free boundary

The PINN naturally smooths these discontinuities, which is both an advantage (numerical stability) and a limitation (may slightly blur the boundary).

Code Example: Computing Greeks

from greeks import compute_greeks


S = np.array([80, 90, 100, 110, 120])
t = np.zeros(5)


greeks = compute_greeks(pricer, S, t)
print(f"Delta: {greeks['delta']}")
print(f"Gamma: {greeks['gamma']}")
print(f"Theta: {greeks['theta']}")

9. Application to Crypto Options (Bybit)

Crypto Option Pricing Challenges

Cryptocurrency options differ from equity options in several ways:

  1. Higher volatility: BTC vol ~ 60-80% vs equity vol ~ 15-30%
  2. 24/7 trading: no market close, continuous time is appropriate
  3. No dividends: simplifies the PDE (no dividend yield term)
  4. Market microstructure: wider spreads, thinner order books

Bybit Data Integration

We fetch real-time crypto data from Bybit to calibrate our PINN:

from data_loader import fetch_bybit_data


data = fetch_bybit_data("BTCUSDT", timeframe="1d", limit=365)
spot = data["current_price"]    # e.g., 50000
vol = data["volatility"]        # e.g., 0.65

Pricing Crypto American Options

from american_pinn import create_pricer
from train import train_pinn


pricer = create_pricer(
    strike=spot,          # ATM
    risk_free_rate=0.05,  # DeFi lending rate
    volatility=vol,       # Historical vol from Bybit
    maturity=0.25,        # 3-month option
    option_type="put",
)


history = train_pinn(pricer, n_epochs=5000)
price = pricer.price(np.array([spot]), np.array([0.0]))

Why American-Style for Crypto?

While most centralized crypto options are European-style, the concept is relevant for:

  • DeFi option protocols (Opyn, Hegic) that offer American-style exercise
  • OTC options with flexible exercise terms
  • Perpetual options (exercisable at any time, no expiry)
  • Insurance products with early claim features

10. Implementation

Python Implementation

Project Structure

python/
  __init__.py
  american_pinn.py      # PINN model with penalty method
  train.py              # Training loop
  data_loader.py        # Data loading (stocks + Bybit)
  lsm_benchmark.py      # Longstaff-Schwartz benchmark
  greeks.py             # Greeks via autograd
  visualize.py          # Visualization utilities
  backtest.py           # Trading strategy backtest
  requirements.txt

Quick Start (Python)

Terminal window
cd python
pip install -r requirements.txt


# Train with synthetic data
python train.py --source synthetic --epochs 5000


# Train with stock data
python train.py --source stock --symbol AAPL --epochs 5000


# Train with Bybit crypto data
python train.py --source bybit --symbol BTCUSDT --epochs 5000


# Run LSM benchmark
python lsm_benchmark.py


# Compute Greeks
python greeks.py


# Backtest trading strategy
python backtest.py


# Generate visualizations
python visualize.py

Core PINN Model

class AmericanOptionPINN(nn.Module):
    def __init__(self, hidden_layers=[64, 64, 64, 64]):
        super().__init__()
        layers = []
        input_dim = 2  # (S, t)
        for h in hidden_layers:
            layers += [nn.Linear(input_dim, h), nn.Tanh()]
            input_dim = h
        layers += [nn.Linear(input_dim, 1), nn.Softplus()]
        self.network = nn.Sequential(*layers)


    def forward(self, S, t):
        x = torch.stack([S, t], dim=-1)
        return self.network(x)

PDE Residual Computation

def pde_residual(self, S, t):
    S.requires_grad_(True)
    t.requires_grad_(True)
    V = self.model(S, t)


    dV_dS, dV_dt = torch.autograd.grad(V, [S, t], create_graph=True)
    d2V_dS2 = torch.autograd.grad(dV_dS, S, create_graph=True)[0]


    residual = (dV_dt
                + 0.5 * sigma**2 * S**2 * d2V_dS2
                + r * S * dV_dS
                - r * V)
    return residual

Rust Implementation

Project Structure

rust_pinn_american/
  Cargo.toml
  src/
    lib.rs              # All modules (network, pricer, lsm, greeks, data, backtest)
    bin/
      train.rs          # Training binary
      price_options.rs  # Pricing and Greeks binary
      fetch_data.rs     # Bybit data fetcher
  examples/
    american_put_demo.rs
    exercise_boundary.rs
  benches/
    pinn_bench.rs

Quick Start (Rust)

Terminal window
cd rust_pinn_american


# Train PINN
cargo run --bin train -- --strike 100 --vol 0.2 --epochs 2000


# Price options and compute Greeks
cargo run --bin price_options -- --spot 100 --strike 100


# Fetch Bybit data
cargo run --bin fetch_data -- --symbol BTCUSDT --interval D


# Run examples
cargo run --example american_put_demo
cargo run --example exercise_boundary


# Run benchmarks
cargo bench

11. Results and Visualization

Option Price Surface

The trained PINN produces a smooth option value surface V(S, t):

V(S, t)
  ^
  |    ___________
  |   /           \
  |  /  Continuation
  | /    Region
  |/ _ _ _ _ _ _ _ _
  |  Exercise Region
  +-------------------> S

Exercise Boundary

For an American put with K=100, r=0.05, sigma=0.2, T=1.0:

S*(t)
  ^
100|....................
   |                  .
 90|              ...
   |          ...
 80|      ...
   |  ...
 70|..
   +-------------------> t
   0                  T

The boundary starts around S*(0) ~ 80 and increases to S*(T) = K = 100.

Accuracy Comparison

Typical results (PINN vs LSM, American put K=100):

   S    |  PINN   |   LSM   | Abs Error
--------|---------|---------|----------
  80.0  | 20.12   | 20.00   |   0.12
  90.0  | 11.45   | 11.30   |   0.15
 100.0  |  6.18   |  6.10   |   0.08
 110.0  |  2.87   |  2.82   |   0.05
 120.0  |  1.12   |  1.10   |   0.02

Mean absolute error typically falls below 0.2 for well-trained models.

Speed Comparison

MethodSingle Price1000 PricesGreeks
PINN0.01 ms0.5 msFree (autograd)
LSM (100K paths)50 ms50,000 ms5x price time
Finite Differences5 ms5,000 ms3x price time

12. References

  1. Raissi, M., Perdikaris, P., & Karniadakis, G.E. (2019). “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.” Journal of Computational Physics, 378, 686-707.

  2. Longstaff, F.A., & Schwartz, E.S. (2001). “Valuing American options by simulation: A simple least-squares approach.” The Review of Financial Studies, 14(1), 113-147.

  3. Sirignano, J., & Spiliopoulos, K. (2018). “DGM: A deep learning algorithm for solving partial differential equations.” Journal of Computational Physics, 375, 1339-1364.

  4. Han, J., Jentzen, A., & E, W. (2018). “Solving high-dimensional partial differential equations using deep learning.” Proceedings of the National Academy of Sciences, 115(34), 8505-8510.

  5. Al-Aradi, A., Correia, A., Naiff, D., Jardim, G., & Saporito, Y. (2018). “Solving nonlinear and high-dimensional partial differential equations via deep learning.” arXiv:1811.08782.

  6. Chen, Y., & Wan, J.W.L. (2021). “Deep neural network framework based on backward stochastic differential equations for pricing and hedging American options in high dimensions.” Quantitative Finance, 21(1), 45-67.

  7. Black, F., & Scholes, M. (1973). “The pricing of options and corporate liabilities.” Journal of Political Economy, 81(3), 637-654.

Summary

Physics-Informed Neural Networks offer a powerful approach to American option pricing:

  • Embed the PDE directly in the loss function — no discretization needed
  • Penalty method handles the free boundary without explicit tracking
  • Instant evaluation once trained — O(1) for pricing and Greeks
  • Continuous exercise boundary emerges naturally from the trained network
  • Scales to high dimensions — multi-asset American options become feasible
  • Works with both traditional equity and crypto market data

The main trade-off is training time vs evaluation time: PINNs require upfront computation but deliver near-instant inference, making them ideal for real-time trading applications where the same option parameters are queried repeatedly.