Quantitative Derivatives Lab

Bjerksund-Stensland (1993) American Option Lab

Evaluate American call and put options, optimal early exercise triggers, and early exercise premiums with Petter Bjerksund and Gunnar Stensland's closed-form flat boundary model.

Market Archetypes:

Contract & Market Inputs

Cost of carry b = r - q = 1.00%.

Valuation Summary & Early Exercise Metrics

American Option Price
$0.00
Bjerksund-Stensland
European Price (BS)
$0.00
Black-Scholes
Early Exercise Premium
$0.00
0.0% of Value
Critical Trigger Boundary
$0.00
Boundary (I)
Model Architecture American Value European Value Early Exercise Premium Computational Method
Bjerksund-Stensland (1993) $0.00 $0.00 $0.00 Closed-form flat boundary
Barone-Adesi & Whaley (1987) $0.00 $0.00 $0.00 Semi-analytical Newton root
CRR Binomial Lattice (100 steps) $0.00 $0.00 $0.00 Discrete backward induction

Option Sensitivity Greeks

Delta (Δ) 0.0000
Gamma (Γ) 0.0000
Vega (ν) 0.0000
Theta (Θ / day) 0.0000
Rho (ρ) 0.0000
Exercise Status: S < I. Continuation is optimal. Do not exercise early.

Visual Analytics & Boundary Profile

Blue: Bjerksund-Stensland American Value | Red Dash: European Black-Scholes | Green: Intrinsic Payoff | Gold Dot: Critical Boundary Trigger (I)

Matrix 1: Spot Price (S) vs. Volatility (σ) on American Price

Shows American option pricing across spot price shifts (±20%) and volatility levels (±10%).

Matrix 2: Dividend Yield (q) vs. Expiry (T) on Critical Boundary (I)

Examines the early exercise critical price boundary I across dividend yields and expiration horizons.

Mathematical Foundation: Bjerksund & Stensland (1993)

1. Flat Boundary Formulation

Bjerksund and Stensland (1993) introduced an analytical approximation for American options by approximating the continuous, time-varying early-exercise boundary with an optimal flat trigger level I. The cost of carry is defined as b = r - q.

β = (1/2 - b/σ²) + √((b/σ² - 1/2)² + 2r/σ²)
B_∞ = β / (β - 1) · K
B_0 = max(K, (r / (r - b)) · K)
h(T) = -(b·T + 2σ√T) · B_0 / (B_∞ - B_0)
I = B_0 + (B_∞ - B_0) · (1 - e^{h(T)})

When b ≥ r (non-dividend paying assets), early exercise of a call is never optimal (I → ∞), and the American call exactly equals the European Black-Scholes formula.

2. Closed-Form Valuation & Put-Call Symmetry

When b < r and spot S < I, the American call value is computed using auxiliary normal cumulative distribution functions φ(S, T, γ, H, I):

C = α·S^β - α·φ(S, T, β, I, I) + φ(S, T, 1, I, I) - φ(S, T, 1, K, I) - K·φ(S, T, 0, I, I) + K·φ(S, T, 0, K, I)
where α = (I - K) · I^{-β}

American puts are evaluated through exact put-call symmetry: P(S, K, T, r, b, σ) = C(K, S, T, r - b, -b, σ). This eliminates the need for separate put boundary equations and delivers instantaneous sub-millisecond execution.

Bjerksund-Stensland Mastery Quiz

Frequently Asked Questions

The Bjerksund-Stensland (1993, 2002) model is an analytical closed-form approximation for pricing American options. Unlike numerical binomial lattices or iterative Newton-Raphson solvers like Barone-Adesi Whaley, Bjerksund-Stensland posits an explicit flat or two-state boundary for the early exercise trigger and evaluates option prices directly using cumulative standard normal distributions.

While both models evaluate American options semi-analytically, BAW requires a 1D numerical root-finding algorithm (Newton-Raphson) to locate the critical price S* at every evaluation. In contrast, Bjerksund-Stensland provides an explicit closed-form formula for the critical exercise boundary I, making it computationally faster and often more accurate for long-dated options (up to 30 years) and high dividend yields.

American puts are priced using a mathematical put-call symmetry transformation: P(S, K, T, r, b, sigma) = C(K, S, T, r - b, -b, sigma). By exchanging the strike and spot prices and flipping the cost-of-carry sign, the closed-form call formula directly values the American put without requiring a separate numerical boundary solver.

When the underlying pays dividends or has a negative cost of carry (b < r), an American call should be exercised early whenever the spot price S exceeds the critical boundary I (S >= I). If b >= r (e.g. non-dividend paying stock), early exercise is never optimal, and the American call price exactly equals the European Black-Scholes call price.

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