Binomial Option Pricing: Replication, Carry, Early Exercise, and Convergence

For educational purposes only; not investment advice. Investing may result in loss.

Direct answer

A binomial model values an option on a discrete state lattice, not by forecasting that the underlying will literally make only two moves. At each node, it chooses an up factor u or down factor d, prices terminal contractual cash flows, and works backward. The same node value can be checked with a locally replicating stock-and-cash portfolio when the one-factor market is complete.

For a step of length Delta t, separate the financing growth factor B = exp(r Delta t) from the underlying carry growth factor G = exp((r - q) Delta t). Under a continuous proportional yield q, the no-arbitrage pricing weight is

p = (G - d) / (u - d), subject to d < G < u.

European continuation is V = [p V_u + (1 - p) V_d] / B. At an allowed holder-exercise node, American or Bermudan value is V = max(H,C), where H is immediate exercise value and C is continuation value. This framework produces a model value per underlying unit; contract multiplier, quantity, official settlement, bid-ask execution, fees, margin, and credit remain separate.

Mechanism

1. Lock the claim and clock. Record exact underlying and option series, call or put, strike, valuation timestamp, expiration and day count, exercise dates and style, official settlement, multiplier, deliverable, currency, dividends, borrow, adjustment terms, and output units.
2. Choose and document the lattice. Set N, Delta t = T / N, and a parameterization. A common CRR choice is u = exp(sigma sqrt(Delta t)) and d = 1 / u; alternative trees can match different moments and converge differently.
3. Separate carry and discounting. Build B = exp(r Delta t) from the model-consistent zero rate and G = exp((r - q) Delta t) from continuous carry, then calculate p = (G - d) / (u - d). A known cash dividend requires an explicit ex-dividend node treatment and may break simple recombination.
4. Build states and terminal claims. In a constant-factor recombining tree, S(i,j) = S0 u^j d^(i-j). Apply the contract’s call, put, cash, physical, adjusted-deliverable, barrier, average, or other terminal rule to the official state variable; path dependence needs extra state.
5. Induct backward and test exercise. Discount the risk-neutral child value at every node. For European claims use continuation; for American claims compare H and C at every permitted time; for Bermudan claims compare only on contractual exercise dates and honor notice and settlement timing.
6. Validate replication and numerics. At each one-step node, calculate Delta = (V_u - V_d) / (S_u - S_d) and cash position b = (u V_d - d V_u) / [B(u - d)]. Check state payoffs, price bounds, put-call relations, step doubling, odd-even oscillation, alternative trees, independent code, and exercise boundaries.
7. Map model output to the account. Calibrate curves, dividends, borrow, and a strike-expiry volatility surface; convert per-unit value through actual multiplier and quantity; compare executable bid and ask; stress discrete hedging, gaps, costs, liquidity, margin, assignment, settlement, tax, and model version before use.

Worked examples

• One-period replication. Let S0 = $100, u = 1.20, d = 0.80, B = G = 1.05, and a European call have K = $100. Then p = (1.05 - 0.80) / (1.20 - 0.80) = 0.625; terminal payoffs are $20/$0; and C0 = [0.625 x $20 + 0.375 x $0] / 1.05 = $11.904762. Replication gives Delta = ($20 - $0) / ($120 - $80) = 0.50 and b = -$38.095238. The up-state portfolio pays $60 - $40 = $20, the down state pays $40 - $40 = $0, and initial cost is $50 - $38.095238 = $11.904762.
• American put and early exercise. Use a two-step tree with S0 = $100, K = $100, u = 1.20, d = 0.80, and B = G = 1.05, so p = 0.625. Terminal put payoffs at stock values $144/$96/$64 are $0/$4/$36. At the first up node, continuation is [0.625 x $0 + 0.375 x $4] / 1.05 = $1.428571, above intrinsic $0. At the first down node, continuation is [0.625 x $4 + 0.375 x $36] / 1.05 = $15.238095, below immediate exercise $20, so exercise. American value is $7.993197; European value is $6.292517; the discrete early-exercise premium is $1.700680.
• Carry and nonmonotone convergence. Let S0 = $100, K = $100, T = 1, continuously compounded r = 5%, continuous yield q = 2%, and annualized sigma = 20%. With CRR, a one-step European call is $11.073541, two steps give $8.342293, three give $9.855621, four give $8.760327, and 64 steps give $9.196691, approaching the matched Black-Scholes-Merton value $9.227006. More steps improve the limit under the model but need not move monotonically or repair wrong cash-flow inputs.
• Model value versus executable account dollars. Reuse the first example’s per-share value $11.904762, but suppose the executable market is $11.60 bid / $12.20 ask. Buying one standard contract with M = 100 and Q = 1 costs $12.20 x 100 = $1,220, versus model amount $11.904762 x 100 = $1,190.4762, a $29.5238 difference before fees. Immediate sale at the bid returns $11.60 x 100 = $1,160, a $60 round-trip loss. The model Delta = 0.50 maps to 50 shares for this assumed multiplier, but discrete rehedging, gaps, borrow, and execution prevent a guaranteed replication result.

Model and lifecycle checklist

• Match exact underlying, option root, call or put, strike, expiration, exercise style, settlement, multiplier, currency, and deliverable.
• Freeze valuation timestamp, calendar, time zone, year fraction, number of steps, node dates, and business-day treatment.
• State the tree parameterization and whether its moments, carry, skew, or other calibration targets match the intended model.
• Keep B for discounting separate from G for underlying carry when dividends, foreign rates, convenience yield, or borrow matter.
• Require d < G < u; a p outside [0,1] is a failed no-arbitrage setup, not a forecast probability.
• Treat continuous yield q separately from known discrete, special, uncertain, or path-dependent distributions.
• Verify terminal payoff against official settlement, cash or physical delivery, adjusted contracts, and corporate actions.
• Add sufficient state for barriers, averages, lookbacks, multiple assets, stochastic volatility, rates, credit, or other path dependence.
• Apply early exercise only to the contractual right holder and only at valid dates before applicable notice cutoffs.
• Distinguish American, Bermudan, and European exercise opportunity sets from cash, physical, or futures settlement.
• Record every node where H exceeds C, and test dividend, rate, borrow, and numerical sensitivity of the exercise boundary.
• Verify node replication and state payoffs where the market assumptions imply completeness; do not assume costless continuous hedging in practice.
• Check no-arbitrage bounds, parity, monotonicity, convexity, and limiting cases before trusting a price.
• Compare several step counts, odd and even sequences, step doubling, alternative parameterizations, and an independent implementation.
• Do not interpret oscillation, a stable printed decimal, or agreement between related code paths as proof of accuracy.
• Calibrate the full relevant volatility surface and curves; one historical volatility or one at-the-money quote is not universal.
• Separate model value, theoretical midpoint, exchange mark, official settlement, executable bid, executable ask, and liquidation price.
• Convert per-unit outputs with actual multiplier and quantity; adjusted and nonstandard contracts may not use 100.
• Stress bid-ask, partial fills, discrete hedge timing, jumps, halts, liquidity, funding, borrow, margin, assignment, and broker liquidation.
• Preserve inputs, market-data source, corporate-action and dividend version, code and solver version, convergence evidence, approvals, fees, tax, and final account reconciliation.

Common misconceptions

• “Risk-neutral probability predicts the next move.” p is a model-consistent pricing weight under specified tradable and no-arbitrage assumptions, not a physical forecast.
• “The risk-free growth factor always belongs in both formulas.” Discounting uses B, while the underlying pricing weight uses carry factor G; they coincide only under the simplified zero-yield setup.
• “More steps guarantee a correct price.” Refinement addresses discretization under one model; convergence can oscillate and cannot fix bad dividends, volatility, settlement, or exercise terms.
• “American value is just European value plus a fixed premium.” Early-exercise value is state-, date-, rate-, dividend-, borrow-, and model-dependent and can be zero.
• “Replication makes the screen price risk free.” The proof assumes model completeness and frictionless trading; executable spreads, discrete hedges, gaps, funding, margin, tax, and settlement remain.

Related topics

• American and European Options
• Black-Scholes Model
• Implied Volatility

Authoritative sources

• Option Pricing: A Simplified Approach - Elsevier; establishes the CRR discrete replication and risk-neutral framework under its assumptions, not current market inputs or an execution price.
• Two-State Option Pricing - Wiley; develops a two-state option-pricing approach, not a claim that real prices follow only two outcomes or that every incomplete market is uniquely priced.
• Binomial Models for Option Valuation - Examining and Improving Convergence - Taylor & Francis; supports oscillation, parameterization, and convergence comparisons under studied models, not universal error bounds for every payoff.
• Pricing the American Put Option: A Detailed Convergence Analysis for Binomial Models - Elsevier; supports American-put convergence and acceleration pitfalls under analyzed trees, not all exercise products or market prices.
• American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods - Oxford University Press; supports American-option bounds and numerical comparisons under specified assumptions, not a broker exercise policy or guaranteed accuracy.
• Characteristics and Risks of Standardized Options - The Options Clearing Corporation; covers standardized-option rights, exercise, assignment, settlement, adjustments, and risks, not a binomial calibration or strategy recommendation.
• Industry Services - The Options Clearing Corporation; documents an OCC proprietary variation of CRR for specified delta services, not the exact implementation in this article or a universal market price.
• Options Calculator - Cboe Global Markets; provides current educational theoretical-price and Greek tooling, not an executable quote, independent validation, or assurance that its assumptions match a particular contract.