Рациональное ценообразование в финансовой экономике

Assumption in financial economics

Rational pricing is the assumption in financial economics that asset prices – and hence asset pricing models – will reflect the arbitrage free price of the asset as any deviation from this price will be "arbitraged away". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments.

Arbitrage is the practice of taking advantage of a state of imbalance between two (or possibly more) markets. Where this mismatch can be exploited (i. e. after transaction costs, storage costs, transport costs, dividends etc.) the arbitrageur can "lock in" a risk free profit by purchasing and selling simultaneously in both markets. In general, arbitrage ensures that "the law of one price" will hold; arbitrage also equalises the prices of assets with identical cash flows, and sets the price of assets with known future cash flows.

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. Investment banks, and other market makers here, thus invest considerable resources in "curve stripping". See Bootstrapping (finance) and Multi curve framework for the methods employed, and Model risk for further discussion.

A derivative is an instrument that allows for buying and selling of the same asset on two markets – the spot market and the derivatives market. Mathematical finance assumes that any imbalance between the two markets will be arbitraged away. Thus, in a correctly priced derivative contract, the derivative price, the strike price (or reference rate), and the spot price will be related such that arbitrage is not possible. See Fundamental theorem of arbitrage free pricing.

Rational pricing underpins the logic of swap valuation. 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. To be arbitrage free, the terms of a swap contract are such that, initially, the Net present value of these future cash flows is equal to zero; see Once traded, swaps can (must) also be priced using rational pricing. The examples below are for interest rate swaps and are representative of pure rational pricing as these exclude credit risk although the principle applies to any type of swap.

The Floating leg of an interest rate swap can be "decomposed" into a series of forward rate agreements. Here, since the swap has identical payments to the FRA, arbitrage free pricing must apply as above – i. e. the value of this leg is equal to the value of the corresponding FRAs. 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.) See further under .

As above, where the value of an asset in the future is known (or expected), this value can be used to determine the asset's rational price today. 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. Methods that lock in future cash flows assume arbitrage free pricing, and those that infer expected value assume risk neutral valuation. 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. If S is the current price, then in the next period the price will either be S up or S 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). Then, given these two states, the "arbitrage free" approach creates a position that has an identical value in either state – the cash flow in one period is therefore known, and arbitrage pricing is applicable. The risk neutral approach infers expected option value from the intrinsic values at the later two nodes. Although this logic appears far removed from the Black–Scholes formula and the lattice approach in the Binomial options model, it in fact underlies both models; see The Black–Scholes PDE. 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. The Binomial options model allows for a high number of very short time steps (if coded correctly), while Black–Scholes, in fact, models a continuous process. The examples below have shares as the underlying, but may be generalised to other instruments. The value of a put option can be derived as below, or may be found from the value of the call using put–call parity.

Here, the future payoff is "locked in" using either "delta hedging" or the "replicating portfolio" approach. As above, this payoff is then discounted, and the result is used in the valuation of the option today.

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. Given this equivalence, it is valid to assume "risk neutrality" when pricing derivatives. A more formal relationship is described via the fundamental theorem of arbitrage free pricing.