Binomial Trees for Option Pricing

Binomial trees decompose option pricing into a sequence of discrete decisions—like climbing a decision ladder where each rung offers two possible outcomes. This intuitive framework handles early exercise, extracts Greeks without calculus, and provides a foundation for understanding more sophisticated models.

Framework: Building the Tree

Step 1: Define Parameters

Given inputs:

• S = Spot price = $100
• K = Strike price = $100
• σ = Volatility = 30%
• r = Risk-free rate = 5%
• T = Time to expiration = 0.25 years (3 months)
• n = Number of steps = 3

Step 2: Calculate u, d, and p

The up and down factors determine price movements:

• Δt = T/n = 0.25/3 = 0.0833 (each step is ~1 month)
• u = e^(σ√Δt) = e^(0.30 × √0.0833) = e^0.0866 = 1.0905
• d = 1/u = 0.9170
• p = (e^(rΔt) - d) / (u - d) = (1.0042 - 0.9170) / (1.0905 - 0.9170) = 0.503

The risk-neutral probability p ≈ 0.503 means up and down moves are approximately equally likely under the risk-neutral measure.

Step 3: Build Stock Price Tree

The tree recombines: an up-down path reaches the same node as a down-up path ($100 × u × d = $100).

Step 4: Calculate Terminal Payoffs

For a call option (max(S - K, 0)):

Step 5: Backward Induction

Working backward, each node's value is the discounted expected value:

At Step 2:

Node uu ($118.91): Value = e^(-rΔt) × [p × $29.66 + (1-p) × $9.05] Value = 0.9958 × [0.503 × $29.66 + 0.497 × $9.05] = $19.33

Node ud ($100.00): Value = e^(-rΔt) × [p × $9.05 + (1-p) × $0] Value = 0.9958 × [0.503 × $9.05] = $4.53

Node dd ($84.10): Value = e^(-rΔt) × [p × $0 + (1-p) × $0] = $0

At Step 1:

Node u ($109.05): Value = e^(-rΔt) × [p × $19.33 + (1-p) × $4.53] = $11.92

Node d ($91.70): Value = e^(-rΔt) × [p × $4.53 + (1-p) × $0] = $2.27

At Step 0:

Root node ($100): Value = e^(-rΔt) × [p × $11.92 + (1-p) × $2.27] = $7.10

The 3-step binomial call price is $7.10.

Early Exercise and American Options

For American options, at each node check whether immediate exercise exceeds continuation value:

American Put Example: At node d (Step 1), stock = $91.70:

• Continuation value: $2.27 (calculated above)
• Immediate exercise: max($100 - $91.70, 0) = $8.30

Since $8.30 > $2.27, exercise immediately. The American put uses $8.30 at this node.

Repeat this check at every node during backward induction. The American option value will exceed the European value due to early exercise optionality.

Extracting Delta and Gamma

Delta from Adjacent Nodes

At Step 0, delta is calculated from Step 1 values:

Delta = (V_u - V_d) / (S_u - S_d) Delta = ($11.92 - $2.27) / ($109.05 - $91.70) = $9.65 / $17.35 = 0.556

The option gains $0.556 for each $1 increase in the underlying (at current levels).

Gamma from Three Nodes

Gamma requires values at Step 2:

Delta_up = (V_uu - V_ud) / (S_uu - S_ud) = ($19.33 - $4.53) / ($118.91 - $100) = 0.783 Delta_down = (V_ud - V_dd) / (S_ud - S_dd) = ($4.53 - $0) / ($100 - $84.10) = 0.285

Gamma = (Delta_up - Delta_down) / ((S_uu - S_dd) / 2) Gamma = (0.783 - 0.285) / (($118.91 - $84.10) / 2) = 0.498 / 17.41 = 0.0286

When Trees Beat PDE or Monte Carlo

Binomial trees excel when:

• Early exercise matters: Trees naturally handle American options via backward induction
• Discrete dividends: Trees incorporate dividend drops at specific nodes
• Transparency required: Each node is traceable for audit or explanation
• Greeks needed: Simple formulas extract sensitivities without adjoint methods

Trees become impractical when:

• Path-dependent payoffs require non-recombining trees (exponential complexity)
• Many underlying assets (curse of dimensionality)
• Extreme precision needed (requires many steps)

For a 100-step tree on a single asset, computation is nearly instantaneous. For 3 assets, a 100-step tree has 100³ = 1 million nodes—still manageable. Beyond 4-5 assets, Monte Carlo becomes preferable.

Comparison to Black-Scholes

Using the same inputs:

• Black-Scholes call price: $7.14
• 3-step binomial: $7.10
• 50-step binomial: $7.13
• 200-step binomial: $7.14

The binomial converges to Black-Scholes as steps increase. The 3-step tree provides intuition; 50+ steps provide pricing accuracy.

Next Steps

For the continuous-time limit of binomial pricing, see Black-Scholes Model Inputs and Outputs.

To compare tree methods with other American option approaches, review American Option Pricing Approaches.