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Binomial Option Pricing: Replication, Risk-Neutral Valuation, and Early Exercise

For educational purposes only; not investment advice.

Direct answer

Section titled “Direct answer”

The binomial option-pricing model divides time into steps and assumes that the underlying moves to an up or down value at each step. Option payoffs are calculated at final nodes and worked backward to today. The model makes replication and early-exercise decisions visible rather than hiding them inside a closed-form formula.

It is a numerical framework, not a prediction that prices literally make only two moves. More steps create a finer approximation, but a larger tree does not repair incorrect volatility, dividends, rates, exercise terms, or settlement inputs.

Risk-neutral backward induction

Section titled “Risk-neutral backward induction”

Let one-step up and down multipliers be u and d, and let the risk-free growth factor for a step be R. When d < R < u, the no-arbitrage risk-neutral up probability is:

p = (R − d) / (u − d)

For European node values V_u and V_d, the current value is:

V_0 = [pV_u + (1 − p)V_d] / R

p is a pricing weight, not the observed or forecast probability of an up move. If R lies outside (d, u), the simple tree violates its no-arbitrage setup.

At every American-style node, compare immediate exercise with continuation:

V = max(intrinsic value, discounted expected continuation value)

Discrete dividends, changing rates, volatility term structures, and multiple exercise dates can be incorporated by adjusting the tree. Path-dependent contracts may require extra state variables, causing the tree to grow rapidly.

One-period call example

Section titled “One-period call example”

Suppose a stock is $100, moves to $120 or $80 in one period, the strike is $100, and the risk-free growth factor is 1.05. The call’s terminal payoffs are $20 and $0.

p = (1.05 − 0.80) / (1.20 − 0.80) = 0.625

C_0 = [0.625 × $20 + 0.375 × $0] / 1.05 ≈ $11.90

Replication gives the same answer. The hedge ratio is (20 − 0) / (120 − 80) = 0.5 share. Borrowing about $38.10 today grows to $40 and makes the portfolio pay $20 in the up state and $0 in the down state. Its initial cost is $50 − $38.10 = $11.90.

This example excludes dividends, transaction costs, spreads, taxes, and discrete trading. Those omissions explain why a model value is a benchmark rather than a guaranteed execution price.

Model checklist

Section titled “Model checklist”
  • Match the contract’s exercise style, expiration, settlement, multiplier, and corporate-action terms.
  • Convert annual rates, dividends, and volatility to the same step length and compounding convention.
  • Confirm d < R < u and that terminal payoffs are coded correctly.
  • Increase step counts and inspect convergence; oscillation can persist near strikes or barriers.
  • Test alternative tree parameterizations instead of trusting one discretization.
  • For American options, record nodes where exercise dominates continuation and test dividend assumptions.
  • Compare output with market bounds and an independent method; investigate violations before use.
  • Add executable bid-ask spreads, fees, hedging frequency, and liquidity to any trading conclusion.

Common misconceptions

Section titled “Common misconceptions”
  • “The risk-neutral probability is the chance the stock rises.” It is a no-arbitrage pricing weight.
  • “More steps guarantee accuracy.” They address discretization, not bad inputs or a wrong model.
  • “The tree predicts each path.” It values contingent cash flows across modeled states.
  • “American value always exceeds European value by a lot.” The early-exercise premium may be zero.
  • “A model price should equal the market midpoint.” Supply, spread, skew, and model conventions differ.
  • “Replication is costless in practice.” Trading is discrete and incurs spread, fees, funding, and gap risk.
Section titled “Related topics”

Authoritative sources

Section titled “Authoritative sources”