No-Arbitrage & Replication

If two portfolios produce identical payoffs in every future state, they must have the same price today — the single principle behind all derivatives pricing.

1. The idea

An arbitrage is a trading strategy requiring zero (or negative) initial investment that produces a non-negative payoff in every state of the world and a strictly positive payoff in at least one state. Markets in equilibrium do not allow arbitrage to persist.

The no-arbitrage principle implies the Law of One Price: if portfolio A and portfolio B have identical payoffs at every future date and state, they must trade at the same price today, otherwise you could buy the cheap one, sell the expensive one, and lock in riskless profit.

Replication is the constructive use of this idea: to price an unknown asset, build a portfolio of known, tradeable instruments with the same future payoff, and set the unknown asset's price equal to the cost of that replicating portfolio.

2. Why it matters

Every pricing formula in this section — forwards, put-call parity, the binomial model — is a direct application of building a replicating portfolio and invoking no-arbitrage.

3. Core mathematics

Law of one price

Payoff_{A} = Payoff_{B} \forall states \Rightarrow Price_{A} = Price_{B}

Arbitrage definition

Cost \leq 0, Payoff \geq 0, Payoff > 0 somewhere

4. How to recognize it

Think of this when…

'Show that if the price were different, you could earn a riskless profit'
Any pricing question solvable by constructing a matching portfolio
Comparing two instruments with identical cash flows

5. Worked example

Worked example

A stock trades at $100. A forward contract to buy it in 1 year at $105 is available, and the 1-year risk-free rate is 4% (simple). Is there an arbitrage?

1
The no-arbitrage forward price is the cost of buying the stock today and financing it to year 1.
$F = S_{0} (1 + r) = 100 (1.04) = 104$
2
The quoted forward price (105) exceeds the fair value (104).
3
Arbitrage: sell the forward, borrow $100 to buy the stock now, deliver in a year for 105, repay 104, keep the $1 difference.
Answer: Yes — sell the overpriced forward and replicate the stock via borrowing, earning $1 riskless.

6. Common mistakes

Forgetting to account for financing (interest) costs when comparing prices over time
Ignoring transaction costs and dividends when they are specified in the problem
Assuming any price difference is arbitrage without checking payoffs actually match in every state

7. Quick check

The Law of One Price follows from:

An arbitrage requires:

8. Practice

7 QuantSolve questions use this concept.

Practice this concept

9. Related methods

No-Arbitrage Replication

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