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Option Replicating Portfolios: State Payoffs and No-Arbitrage Value

Build a one-period option replica from stock and cash, verify every terminal state, and separate an exact model hedge from a real-world approximation.

Updated

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

Direct answer

A replicating portfolio combines traded instruments so its payoff equals an option’s payoff in every state represented by the model. In a one-period binomial model, stock plus risk-free borrowing or lending can exactly replicate a European option. If the option and replica have identical state-contingent payoffs, the law of one price gives them the same value today; otherwise one could buy the cheaper payoff and sell the dearer one.

This is a conditional model result, not a claim that the option’s market quote must equal one calculated number. Exact replication depends on the specified states, available instruments, financing, distributions, contract terms, and frictionless execution. With jumps, spreads, discrete rebalancing, funding constraints, or early exercise, a live hedge is normally approximate.

Solve state payoffs, not a memorized formula

Let today’s stock price be S0. At the next date it is either Su or Sd, with Su > Sd. The option pays Cu or Cd. Let R be the gross risk-free growth factor for the period. Assume the stock pays no intervening cash distribution; if it does, include that cash flow consistently. A portfolio holding Δ shares and a present cash position B0 has terminal values:

Δ × Su + B0 × R = Cu

Δ × Sd + B0 × R = Cd

Subtracting the equations gives:

Δ = (Cu - Cd) / (Su - Sd)

Then:

B0 = (Cu - Δ × Su) / R

option value today = Δ × S0 + B0

A negative B0 means borrowing. The calculation must verify both terminal states; merely matching an expected or average payoff is not replication.

Before pricing the option, check that the stock-and-cash market itself satisfies:

Sd < R × S0 < Su

This condition is equivalent to the risk-neutral weight below satisfying 0 < q < 1. If it fails, the model already admits an arbitrage between stock and cash. When it holds, the replica price can be cross-checked with:

q = (R × S0 - Sd) / (Su - Sd)

option value = [q × Cu + (1 - q) × Cd] / R

This q is a no-arbitrage pricing weight inside the model, not necessarily a forecast of the real-world probability of an up move.

In a multi-period tree, solve the portfolio again at each node. A self-financing hedge changes its stock and cash holdings using only value already inside the strategy, with no unexplained external deposit or withdrawal. Continuous-time models idealize this rebalancing; real hedges trade at discrete times and prices.

One-period call replica

Suppose:

  • S0 = $100;
  • next-period stock is Su = $120 or Sd = $80;
  • a European call has strike $100, so Cu = $20 and Cd = $0;
  • the period’s gross risk-free factor is R = 1.02.

Stock holding:

Δ = ($20 - $0) / ($120 - $80) = 0.5 share

Present cash position:

B0 = ($20 - 0.5 × $120) / 1.02 = -$39.2157

Replica cost:

0.5 × $100 - $39.2157 = $10.7843

Verify each state:

State Stock component Cash repayment Portfolio payoff Call payoff
Up $60 -$40 $20 $20
Down $40 -$40 $0 $0

Risk-neutral cross-check:

q = (1.02 × $100 - $80) / ($120 - $80) = 0.55

[$20 × 0.55 + $0 × 0.45] / 1.02 = $10.7843

For scale, if the claim uses a 100-share multiplier, the model value is $1,078.43 per contract and the initial stock component is 50 shares, paired with model borrowing of $3,921.57. Standard U.S. equity options ordinarily represent 100 shares, but corporate actions can adjust the deliverable. These remain theoretical quantities before spreads, financing differences, margin, fees, dividends, and integer-share constraints.

If the market call is offered at $11.20, the $0.4157 gap from the frictionless replica is not automatically an executable arbitrage. The option and every hedge leg must trade at the relevant bid or ask and size; financing, stock borrow, exercise style, distributions, margin, and unwind costs must also be included.

Replication limits and risks

  • State-model risk: the stock can finish outside or between the assumed branches.
  • Jump risk: prices can move before the hedge is rebalanced.
  • Discrete-hedging risk: continuous adjustment is unavailable in practice.
  • Transaction costs: repeated trading, spreads, fees, and market impact consume theoretical value.
  • Financing risk: actual borrowing and lending rates differ from the model rate.
  • Distribution and borrow risk: dividends, other distributions, and stock availability alter carrying economics.
  • Early-exercise risk: an American option requires exercise decisions at intermediate nodes.
  • Volatility-surface risk: real options across strikes and maturities do not follow one fixed volatility input.
  • Liquidity risk: the option or hedge may not trade at required size and time.
  • Contract risk: exercise style, multiplier, deliverable, settlement, and adjustments must match the modeled payoff.
  • Model-arbitrage confusion: a theoretical price difference is not automatically a riskless executable trade.

Common misconceptions

  • “Replication matches the average payoff.” It must match every modeled state.
  • “Delta stays 0.5 until expiration.” In a multi-period or live market, Delta changes and requires rebalancing.
  • “Risk-neutral probability is the true probability.” It is a pricing weight under model assumptions.
  • “The replica eliminates all risk.” Model, jump, execution, funding, and contract risks remain.
  • “A theoretical difference is free arbitrage.” Every leg, size, time, financing term, and cost must be executable.
  • “Borrowing is just a notation.” It is an economic cash position with funding and margin constraints.
  • “European and American options replicate identically.” Early exercise adds decisions at intermediate nodes and changes the valuation problem.
  • “One-period replication proves a continuous hedge works exactly.” Real rebalancing is discrete and path-dependent.

Authoritative sources

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