Arbitrage and No-Arbitrage
No-arbitrage is the pricing consistency rule: you cannot get something for nothing at scale.
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).
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.
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.
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.