Binomial Option Pricing

CRR model, multi-step trees, American options, early exercise boundary, calibration, and convergence to BSM.

When to Activate

• User pricing American options where closed-form solutions do not exist
• Teaching or explaining option pricing intuition through discrete models
• Pricing options with discrete dividends or complex exercise features
• Calibrating tree parameters to match market prices
• Understanding early exercise boundaries for American puts and calls on dividend-paying stocks
• Validating other pricing methods (Monte Carlo, finite differences) against tree results

Core Concepts

Binomial Model Intuition

At each time step, the stock price can move to one of two states:

• Up state: S_u = S * u (up factor)
• Down state: S_d = S * d (down factor)
• Risk-neutral probability of up move: p = (exp(r*dt) - d) / (u - d)
• Option value at each node: discounted expected value under risk-neutral probabilities
• At expiry: option value = intrinsic value (max(S-K, 0) for call, max(K-S, 0) for put)
• Work backward from expiry to today, computing option value at each node

Cox-Ross-Rubinstein (CRR) Parameterization

The standard parameterization ensuring the tree converges to GBM:

• u = exp(sigma * sqrt(dt))
• d = 1/u = exp(-sigma * sqrt(dt))
• p = (exp(r*dt) - d) / (u - d)
• dt = T/N where N = number of time steps

Properties:

• u * d = 1 (recombining tree — the tree reconnects, keeping the number of nodes manageable)
• After N steps: 2^N paths but only N+1 distinct terminal prices (due to recombination)
• Terminal stock prices: S * u^j * d^(N-j) for j = 0, 1, ..., N
• Risk-neutral probability of reaching node (j ups out of N steps): C(N,j) * p^j * (1-p)^(N-j)

Alternative Parameterizations

Jarrow-Rudd (Equal Probability)

• p = 0.5
• u = exp((r - sigma^2/2)dt + sigmasqrt(dt))
• d = exp((r - sigma^2/2)dt - sigmasqrt(dt))
• Better for some convergence properties but less commonly used

Tian (Moment-Matching)

• Matches the first three moments of the log-normal distribution
• u = 0.5 * exp(rdt) * M2 * (M2 + 1 + sqrt(M2^2 + 2M2 - 3))
• d = 0.5 * exp(rdt) * M2 * (M2 + 1 - sqrt(M2^2 + 2M2 - 3))
• M2 = exp(sigma^2 * dt)
• Faster convergence than CRR for some option types

Leisen-Reimer

• Designed to minimize oscillation in convergence
• Uses the Peizer-Pratt inversion for d1 and d2 from BSM
• Very smooth convergence even with moderate N (50-100 steps)

Methodology

European Option Pricing on a Binomial Tree

1. Build the stock tree: compute stock price at each node (i, j) where i = time step, j = up moves
   • S(i, j) = S_0 * u^j * d^(i-j) for j = 0, 1, ..., i
2. Compute terminal payoffs: at step N, payoff(N, j) = max(S(N,j) - K, 0) for calls
3. Backward induction: for i = N-1, N-2, ..., 0:
   • V(i, j) = exp(-r*dt) * [p * V(i+1, j+1) + (1-p) * V(i+1, j)]
4. Option price: V(0, 0) at the root node

American Option Pricing

The key addition: at each node, compare holding value to exercise value.

1. Build stock tree (same as European)
2. Terminal payoffs (same as European)
3. Backward induction with early exercise check:
   • Continuation value: C(i, j) = exp(-r*dt) * [p * V(i+1, j+1) + (1-p) * V(i+1, j)]
   • Exercise value: E(i, j) = max(K - S(i,j), 0) for put; max(S(i,j) - K, 0) for call
   • V(i, j) = max(C(i, j), E(i, j))
4. If E(i, j) > C(i, j), early exercise is optimal at that node
5. The set of nodes where early exercise is optimal defines the early exercise boundary

Early Exercise Boundary

American Put

• Always optimal to exercise sufficiently deep ITM (where time value is negative)
• Critical stock price S*(t): exercise if S < S*(t)
• S*(t) decreases as t -> T (boundary converges to K at expiry)
• For r > 0: S*(T) = K (exercise ATM puts at expiry)
• The boundary is smooth and monotonically increasing toward K

American Call on Dividend-Paying Stock

• Early exercise may be optimal just before ex-dividend dates
• Exercise if the dividend exceeds the time value of the option
• Rule of thumb: exercise is optimal if dividend > K * (1 - exp(-r*dt_remaining))
• For continuous dividend yield q: early exercise boundary exists for all t if q > 0

American Call on Non-Dividend Stock

• Never optimal to exercise early (no boundary exists)
• American call = European call (no early exercise premium)
• Proof: call value >= S - K*exp(-rT) > S - K (intrinsic value) when r > 0

Discrete Dividends on Trees

Method 1: Reduce stock price at ex-dividend node

• At the ex-dividend step, subtract the known dividend D from all stock prices
• Problem: tree no longer recombines after the dividend (branches split)
• Fix: use the "Black approximation" — reduce S_0 by PV of dividends, build tree on adjusted S

Method 2: Dividend yield approach

• Model dividend as a proportional reduction: S -> S * (1 - delta) at ex-date
• Tree remains recombining
• Less accurate for stocks with fixed dollar dividends

Method 3: Interpolation method

• Build recombining tree, at dividend date interpolate between nodes after dividend adjustment
• More accurate but computationally more complex

Convergence to BSM

• As N -> infinity, CRR binomial price converges to BSM price for European options
• Rate of convergence: O(1/N) — error decreases proportionally with number of steps
• Oscillation: CRR oscillates around the true value (even/odd N give systematically different prices)
• Practical: N = 200-500 steps gives accuracy within 1 cent for most options
• Richardson extrapolation: average prices from N and N+1 steps to cancel oscillation
• Leisen-Reimer: smooth convergence with N = 50-100

Examples

3-Step CRR Tree for European Call

Parameters: S=100, K=100, r=5%, sigma=30%, T=0.75 (3 quarters)
dt = 0.25, u = exp(0.30*sqrt(0.25)) = 1.1618, d = 1/u = 0.8607
p = (exp(0.05*0.25) - 0.8607) / (1.1618 - 0.8607) = 0.5073

Stock tree:
  t=0:     100.00
  t=0.25:  116.18 / 86.07
  t=0.50:  135.00 / 100.00 / 74.08
  t=0.75:  156.83 / 116.18 / 86.07 / 63.76

Call payoffs at T:
  156.83-100=56.83, 116.18-100=16.18, max(86.07-100,0)=0, max(63.76-100,0)=0

Backward induction:
  t=0.50: (0.5073*56.83 + 0.4927*16.18)*exp(-0.0125) = 36.74
          (0.5073*16.18 + 0.4927*0)*exp(-0.0125) = 8.11
          (0.5073*0 + 0.4927*0)*exp(-0.0125) = 0
  t=0.25: (0.5073*36.74 + 0.4927*8.11)*exp(-0.0125) = 22.35
          (0.5073*8.11 + 0.4927*0)*exp(-0.0125) = 4.06
  t=0:    (0.5073*22.35 + 0.4927*4.06)*exp(-0.0125) = 13.18

Tree price: $13.18
BSM price: $13.15 (close even with just 3 steps)

American Put Early Exercise

Same parameters but pricing a put (K=100):

Put payoffs at T: 0, 0, max(100-86.07,0)=13.93, max(100-63.76,0)=36.24

Backward induction with early exercise check:
  t=0.50 (S=74.08):
    Continuation: (0.5073*13.93 + 0.4927*36.24)*exp(-0.0125) = 24.60
    Exercise: 100-74.08 = 25.92
    Value = max(24.60, 25.92) = 25.92 ***EARLY EXERCISE***

  t=0.50 (S=100): Continuation=6.99, Exercise=0, Value=6.99
  t=0.50 (S=135): Continuation=0, Exercise=0, Value=0

American put premium over European: $0.35 (from early exercise value)

Convergence Analysis

European call (S=100, K=100, r=5%, sigma=30%, T=1):
BSM exact = $14.23

N=10:   $14.35 (error = +$0.12)
N=50:   $14.25 (error = +$0.02)
N=100:  $14.24 (error = +$0.01)
N=200:  $14.23 (error < $0.005)
N=500:  $14.23 (error < $0.002)

Richardson extrapolation (N=100, N=101): $14.231 (error < $0.001)
Leisen-Reimer (N=51): $14.231 (error < $0.001)

Quality Gate

• CRR parameters must satisfy: 0 < p < 1 (verify risk-neutral probability is valid)
• Tree must recombine: verify u*d = 1 within numerical precision
• American option price must be >= European option price (verify early exercise premium is non-negative)
• American call on non-dividend stock must equal European call (verify no early exercise)
• Convergence must be verified: increase N until price changes by less than $0.01 between successive N
• Use at least N=200 steps for production pricing; N=500+ for precise benchmark values
• Early exercise boundary must be smooth and monotonic (irregular boundary suggests implementation error)
• Discrete dividend handling must correctly adjust stock prices at ex-dividend nodes
• Validate against BSM for European options: tree price must converge to BSM within 0.1%
• Richardson extrapolation or Leisen-Reimer should be used when computational time is limited