Arbitrage is the practice of taking advantage of a state of imbalance between two (or possibly more) markets.
the arbitrageur can "lock in" a risk-free profit by purchasing and selling simultaneously in both markets.
Where this is not true, the arbitrageur will: Two assets with identical cash flows must trade at the same price.
Per the logic outlined, rational pricing applies also to interest rate modeling more generally.
Here, yield curves in entirety must be arbitrage-free with respect to the prices of individual instruments.
Were this not the case, the ZCBs implied by the curve would result in quoted bond-prices, e.g., differing from those observed in the market, presenting an arbitrage opportunity.
See Bootstrapping (finance) and Multi-curve framework for the methods employed, and Model risk for further discussion.
In other words, the rational forward price represents the expected future value of the underlying discounted at the risk free rate (the "asset with a known future-price", as above); see Spot–future parity.
Here, two counterparties "swap" obligations, effectively exchanging cash flow streams calculated against a notional principal amount, and the value of the swap is the present value (PV) of both sets of future cash flows "netted off" against each other.
Similarly, the "receive-fixed" leg of a swap can be valued by comparison to a bond with the same schedule of payments.
(Relatedly, given that their underlyings have the same cash flows, bond options and swaptions are equatable.)
In an option contract, however, exercise is dependent on the price of the underlying, and hence payment is uncertain.
Option pricing models therefore include logic that either "locks in" or "infers" this future value; both approaches deliver identical results.
To do this, (in their simplest, though widely used form) both approaches assume a "binomial model" for the behavior of the underlying instrument, which allows for only two states – up or down.
Here, the value of the share in the up-state is S × u, and in the down-state is S × d (where u and d are multipliers with d < 1 < u and assuming d < 1+r < u; see the binomial options model).
The assumption of binomial behaviour in the underlying price is defensible as the number of time steps between today (valuation) and exercise increases, and the period per time-step is correspondingly short.
Here, the future payoff is "locked in" using either "delta hedging" or the "replicating portfolio" approach.
As shown above ("Assets with identical cash flows"), in the absence of arbitrage opportunities, since the cash flows produced are identical, the price of the option today must be the same as the value of the position today.
The best example of this would be under real options analysis[1] where managements' actions actually change the risk characteristics of the project in question, and hence the Required rate of return could differ in the up- and down-states.
(Another case where the modelling assumptions may depart from rational pricing is the valuation of employee stock options.)
Note that above, the risk neutral formula does not refer to the expected or forecast return of the underlying, nor its volatility – p as solved, relates to the risk-neutral measure as opposed to the actual probability distribution of prices.
Nevertheless, both arbitrage free pricing and risk neutral valuation deliver identical results.
In fact, it can be shown that "delta hedging" and "risk-neutral valuation" use identical formulae expressed differently.
APT holds that the expected return of a financial asset can be modelled as a linear function of various macro-economic factors, where sensitivity to changes in each factor is represented by a factor specific beta coefficient: The model derived rate of return will then be used to price the asset correctly – the asset price should equal the expected end of period price discounted at the rate implied by model.
The arbitrageur is then in a position to make a risk free profit as follows: Note that under "true arbitrage", the investor locks-in a guaranteed payoff, whereas under APT arbitrage, the investor locks-in a positive expected payoff.
Although based on different assumptions, the CAPM can, in some ways, be considered a "special case" of the APT; specifically, the CAPM's security market line represents a single-factor model of the asset price, where beta is exposure to changes in the "value of the market" as a whole.