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Option Replicating Portfolios: State Payoffs and No-Arbitrage Price

For educational purposes only; not investment advice.

Direct answer

Section titled “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.

Solve state payoffs, not a memorized formula

Section titled “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.

One-period Call replica

Section titled “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.

Replication limits and risks

Section titled “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.

Common misconceptions

Section titled “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.
Section titled “Related topics”

Authoritative sources

Section titled “Authoritative sources”