American Option Pricing Approaches

Pricing American options requires solving a free-boundary problem—at each point in time, the holder decides whether to exercise or continue. Like chess clock decisions, the optimal strategy depends on future possibilities. Binomial trees, finite-difference methods, and Longstaff-Schwartz simulation each approach this problem differently.

Early Exercise Logic and Boundary

The optimal exercise boundary divides the (S, t) space into exercise and continuation regions:

For American puts:

• If S < S*(t), exercise immediately
• If S ≥ S*(t), continue holding

The boundary S*(t) rises as expiration approaches, reaching K at t = T.

Early exercise is optimal when:

• Puts: Interest earned on strike exceeds time value (deep ITM puts on non-dividend stocks)
• Calls on dividend stocks: Dividend capture exceeds remaining time value
• Calls on non-dividend stocks: Never optimal to exercise early

Threshold example: For an American put with K = $100, dividend yield = 0%, carry = 1.8%:

• Exercising captures $100 cash, earning 1.8% interest
• If time value of continuing < 1.8% × K × remaining_time, exercise

For a call on a dividend stock with dividend yield = 2.1%, carry = 1.8%:

• Dividend yield (2.1%) > carry rate (1.8%)
• Early exercise may be optimal to capture dividends

Method Comparison

Binomial Tree Implementation

At each node during backward induction:

continuation_value = discount × (p × V_up + (1-p) × V_down)
intrinsic_value = max(K - S, 0)  // for put
V_node = max(continuation_value, intrinsic_value)

If intrinsic_value > continuation_value, exercise is optimal at that node.

Advantages:

• Intuitive and transparent
• Delta and gamma directly from node differences
• Handles discrete dividends naturally

Limitations:

• Accuracy requires many steps (100+)
• Not practical beyond 3-4 underlying assets

Finite Difference Implementation

At each grid point during time stepping:

V_j^n = max(V_j^n_implicit_solve, intrinsic_j)

The implicit scheme solves for continuation values, then compares to intrinsic.

Advantages:

• Very accurate for single-asset problems
• Handles complex boundaries (barriers)
• Efficient for repeated pricing

Limitations:

• Grid construction requires care
• Dimensional curse for multiple assets
• Boundary conditions must be specified

Longstaff-Schwartz (Least Squares Monte Carlo)

Algorithm:

1. Simulate paths: Generate N paths of underlying from t=0 to T
2. Calculate terminal payoffs: At T, payoff = max(K - S_T, 0) for put
3. Backward recursion: At each exercise date before T:
   • Identify ITM paths
   • Regress continuation value on basis functions of S
   • Compare fitted continuation to intrinsic
   • Exercise if intrinsic > continuation
4. Average and discount: Compute mean of discounted payoffs

Basis functions: Common choices are 1, S, S² (Laguerre polynomials also used)

Advantages:

• Works for any number of underlying assets
• Handles path-dependent American features
• Flexible payoff specifications

Limitations:

• Regression noise affects exercise boundary
• Requires careful basis function selection
• More paths needed than European pricing

Model Validation Checklist

• Convergence test: Verify price stabilizes as steps/paths increase
• European comparison: American ≥ European for same terms (must hold)
• Put-call parity bounds: American call - American put bounded by European parity
• Early exercise boundary: Verify boundary is monotonic and reaches strike at expiration
• Extreme parameter tests: Test at σ = 200%, r = 0%, and with large dividends

Stress Scenarios

High volatility stress (σ = 200%):

At extreme vol, trees and PDE methods need more steps for stability. LS Monte Carlo handles high vol naturally but has higher variance.

Zero rate stress (r = 0%):

For puts: Early exercise incentive disappears (no interest on strike) American put → European put at r = 0

This is a useful validation check—American and European put prices should converge.

Method runtime comparison:

Runtime is relatively constant across moneyness for all methods.

Practical Guidance

Single asset American options: Use binomial trees (100-200 steps) for speed and transparency, or finite difference for production accuracy.

Multi-asset American options: Longstaff-Schwartz is the only practical choice. Expect 100,000+ paths for acceptable accuracy.

Validation before production: Compare new implementation to established library (QuantLib) on benchmark cases. Verify within 1% for normal parameters, 5% for stress parameters.

Next Steps

For the tree approach in detail, see Binomial Trees for Option Pricing.

For PDE methods, review Finite Difference Methods Overview.