# Option Replicating Portfolios: State Payoffs and No-Arbitrage Price

Construct a one-period option replica from stock and borrowing, verify every terminal state, and understand why real dynamic hedges depart from the model.

Canonical: https://wiki.fcontext.com/options/option-replication-portfolio/
Fact checked: 2026-07-22

> For educational purposes only; not investment advice.

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## Direct answer

A replicating portfolio uses traded instruments to produce the same future payoff as an option in every state represented by a 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 payoffs, no-arbitrage reasoning assigns them the same current value.

This is a conditional pricing result. Exact replication depends on the model's states, trading opportunities, financing, distributions, contract terms, and frictionless execution assumptions. In a real market with jumps, spreads, discrete hedging, borrow limits, and early exercise, a hedge is normally approximate.

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## Solve state payoffs, not a memorized formula

Let today's stock price be `S0`. At the next date it is either `Su` or `Sd`. The option pays `Cu` or `Cd`. Let `R` be the gross risk-free growth factor for the period. A portfolio holding `Δ` shares and 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. Matching only an expected payoff is not replication.

The same price can be cross-checked with risk-neutral probability:

`q = (R × S0 - Sd) / (Su - Sd)`

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

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

In a multi-period tree, the portfolio is re-solved at each node. A **self-financing** hedge changes stock and cash holdings using only value already inside the strategy, without unexplained external deposits or withdrawals. Continuous-time option models idealize this rebalancing; real hedges are discrete.

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## 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`

With a standard 100 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`. These are theoretical quantities before bid-ask 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 tradable arbitrage. Both option and hedge must be executable at relevant sides and sizes; financing, stock borrow, exercise style, dividends, and unwind costs must be included.

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## Replication limits and risks

- **State-model risk:** the actual stock can finish outside or between 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:** the trader's borrowing and lending rates differ from the model rate.
- **Dividend and borrow risk:** 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:** multiplier, deliverable, cash settlement, and adjustments must match the modeled payoff.
- **Model-arbitrage confusion:** a theoretical price difference is not a riskless executable trade.

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## 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, financing, and contract risks remain.
- **“A theoretical difference is free arbitrage.”** All legs, sizes, timing, financing, and costs 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 changes the valuation problem.
- **“One-period replication proves a continuous hedge works exactly.”** Real rebalancing is discrete and path-dependent.

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## Related topics

- [Binomial option pricing](/options/binomial-option-pricing/)
- [Delta hedging](/options/delta-hedging/)
- [No-arbitrage option bounds](/options/no-arbitrage-option-bounds/)
- [Self-financing hedge](/options/self-financing-hedge/)

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## Authoritative sources

- [Characteristics and Risks of Standardized Options](https://www.theocc.com/company-information/documents-and-archives/options-disclosure-document) — OCC
- [The Pricing of Options and Corporate Liabilities](https://doi.org/10.1086/260062) — Black and Scholes, *Journal of Political Economy*
- [Theory of Rational Option Pricing](https://doi.org/10.2307/3003143) — Merton, *The Bell Journal of Economics and Management Science*
- [Option Pricing: A Simplified Approach](https://doi.org/10.1016/0304-405X(79)90015-1) — Cox, Ross, and Rubinstein, *Journal of Financial Economics*