“If there is no arbitrage, every asset has one uniquely determined price.”
Not quite.Uniqueness requires enough traded instruments to replicate the claim; incomplete markets can admit a range of arbitrage-free prices.
Foundations
Arbitrage and No-Arbitrage
No-arbitrage is the pricing consistency rule: you cannot get something for nothing at scale.
Mental model
Prices cannot offer a risk-free gain once identical cash flows are compared on the same terms.
Arbitrage is a zero-cost strategy with non-negative payoff and positive payoff with some probability.
No-arbitrage forces relationships between prices (spot, forwards, options).
Explore the mechanism
Which price relationships must hold when two strategies deliver the same future cash flows?
In a simple one-period world with spot S₀ and interest rate r, a no-arbitrage forward price is F* = S₀(1 + r). If the observed forward Fₘ deviates from F*, a cash-and-carry strategy can lock in a riskless gain per unit.
Numbers
Spot S₀ ≈ 100.00
Rate r ≈ 5.0%
No-arbitrage forward F* ≈ 105.00
Market forward Fₘ ≈ 105.00
Mispricing Fₘ − F* ≈ 0.00 (within tolerance)
Interpretation
The blue line marks the no-arbitrage forward F* implied by spot and the interest rate. When the orange market forward Fₘ sits exactly on that line (up to small noise), there is no free lunch: buying or selling spot and forwards just transfers value through time.
Move Fₘ away from F*: the green arrow shows the arbitrage direction and the bar shows the profit per unit of a simple cash-and-carry strategy. In practice, transaction costs, funding constraints and limits widen the band, but the core idea remains: no-arbitrage is a consistency rule linking prices across time and instruments.
UNDERSTAND
What to notice
Most pricing models are ‘no-arbitrage’ models: they are consistency engines, not forecasts.
In a simple forward/spot/rate setting, no-arbitrage pins the forward price to a narrow band; outside that band, cash-and-carry strategies generate riskless profit.
Where it appears
- checking derivative prices against replicating portfolios
- detecting inconsistent forward and spot quotes
- setting economic bounds in pricing tests
Where the intuition breaks
What this view leaves out
- No-arbitrage restricts relative prices but does not by itself identify real-world probabilities or expected returns.
Common mistakes
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Arbitrage and No-Arbitrage
Arbitrage and No-ArbitrageYOU ARE HERE
Foundation ofBinomial Option Pricing (CRR)The CRR tree prices derivatives by replication and no-arbitrage at each step.
Foundation ofBlack–Scholes Option PricingBlack–Scholes valuation rests on dynamic replication and the exclusion of arbitrage.
Foundation ofForwards and Futures Pricing (Cost of Carry)Cost-of-carry pricing is derived by comparing replicating cash-and-carry strategies under no-arbitrage.
Foundation ofPut–Call ParityPut–call parity is a no-arbitrage identity linking calls, puts, the underlying and discounting.
Foundation ofRisk-Neutral Measure (Intuition)Risk-neutral pricing is meaningful because no-arbitrage permits consistent valuation under an equivalent pricing measure.
Validated bySanity Checks: Monotonicity and BoundsMonotonicity and bound checks test whether prices respect basic no-arbitrage restrictions.
Foundation of · Binomial Option Pricing (CRR)
The CRR tree prices derivatives by replication and no-arbitrage at each step.
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