Rational pricing

Overview

In financial economics, rational pricing is the assumption that asset pricing models must reflect the arbitrage-free price of the asset This argument underpins the fundamental theorem of asset pricing, and is a key aspect of mathematical finance, defining the pricing of derivatives and fixed income securities, and useful, also, in setting exchange rates, and in pricing shares, at the least aligning prices correctly. The essential argument is that where a mismatch exists between two or more markets, arbitrage will occur such that the arbitrageur makes a risk-free profit by purchasing and short-selling simultaneously in both markets. By doing so, the arbitrageur may deliver the purchased asset to the buyer, receiving that higher price, whilst paying the seller on the cheaper market with the proceeds and pocketing the difference. As such, the law of one price holds across trading exchanges; prices of assets with identical cash flows equalise; and the price of assets with known future cash flows may be calculated in advance. A fundamental distinction exists between theoretical pricing models, which derive fair-value mathematical relationships by assuming an arbitrage-free market, and arbitrage trading strategies, which market participants execute to monetize temporary real-world price discrepancies. The former are the focus of financial economics.

Categories of argument

Rational pricing arguments, as outlined, generally fall into two distinct categories. However, in many market applications — such as swaps and fixed-income securities — the model derivation directly mirrors the execution trade, meaning a strict binary classification does not always apply. Both categories rely on the arbitrage mechanics detailed below.

Theoretical models (derivation via Arbitrage-free constraints): Mathematical frameworks that derive exact fair value pricing formulas or relative price identities by assuming an arbitrage-free market. These models — such as the Black-Scholes option pricing formula, cost-of-carry futures pricing, or covered interest parity — use hypothetical cash-flow replication to show what an asset must cost to prevent deterministic arbitrage. Practical trading strategies (execution, given market inefficiencies): Practical trading methodologies used by market participants when market prices temporarily deviate from fair value. Strategies such as convertible arbitrage or dividend stripping do not derive market prices themselves; rather, they rely on long/short execution legs to monetize pricing discrepancies until convergence occurs.

Arbitrage mechanics

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.

The law of one price

The same asset must trade at the same price on all markets ("the law of one price"). Where this is not true, the arbitrageur will:

buy the asset on the market where it has the lower price, and simultaneously sell it (short) on the second market at the higher price deliver the asset to the buyer and receive that higher price pay the seller on the cheaper market with the proceeds and pocket the difference.

Assets with identical cash flows

Two assets with identical cash flows must trade at the same price. Where this is not true, the arbitrageur will:

sell the asset with the higher price (short sell) and simultaneously buy the asset with the lower price fund his purchase of the cheaper asset with the proceeds from the sale of the expensive asset and pocket the difference deliver on his obligations to the buyer of the expensive asset, using the cash flows from the cheaper asset.

An asset with a known future-price

An asset with a known price in the future must today trade at that price discounted at the risk free rate. Note that this condition can be viewed as an application of the above, where the two assets in question are the asset to be delivered and the risk free asset. (a) where the discounted future price is higher than today's price:

The arbitrageur agrees to deliver the asset on the future date (i.e. sells forward) and simultaneously buys it today with borrowed money. On the delivery date, the arbitrageur hands over the underlying, and receives the agreed price. He then repays the lender the borrowed amount plus interest. The difference between the agreed price and the amount repaid (i.e. owed) is the arbitrage profit. (b) where the discounted future price is lower than today's price:

The arbitrageur agrees to pay for the asset on the future date (i.e. buys forward) and simultaneously sells (short) the underlying today; he invests (or banks) the proceeds. On the delivery date, he cashes in the matured investment, which has appreciated at the risk free rate. He then takes delivery of the underlying and pays the agreed price using the matured investment. The difference between the maturity value and the agreed price is the arbitrage profit. Point (b) is only possible for those holding the asset but not needing it until the future date. There may be few such parties if short-term demand exceeds supply, leading to backwardation.

Fixed-income securities

Under rational pricing models, as outlined, two assets with identical cash flows must trade at the same price. Where this is not true, an arbitrageur will short the asset with the higher price and simultaneously buy the asset with the lower price. The sale of the higher priced asset funds his purchase of the cheaper asset, and the purchase of the cheaper asset allows him to deliver on his obligations to the buyer. Thus, the arbitrageur earns a risk-free profit. In the case of fixed-income securities, since (1) an arbitrageur could reconstruct the cashflows of any instrument using (multiples or fractions of e.g.) zero coupon bonds, so (2) the above arbitrage relationship applies, but now linking interest rates and market-prices. The pricing formula for a fixed-income security is thus:

P

0

=

∑

t = 1

T

C

t

( 1 +

r

t

)

t

{\displaystyle P_{0}=\sum _{t=1}^{T}{\frac {C_{t}}{(1+r_{t})^{t}}}}

where each cash flow

C

t

{\displaystyle C_{t}\,}

is discounted at the rate

Futures

As above, an asset with a known price in the future must today trade at that price discounted at the risk free rate. Thus, in a futures contract, for no arbitrage to be possible, the forward price must be the same as the cost, including interest, of buying and storing the asset. In other words, the rational forward price represents the expected future value of the underlying discounted at the risk free rate, a condition known as spot–future parity. Then, for a simple, non-dividend paying asset, the value of the future/forward,

F ( t )

{\displaystyle F(t)\,}

, will be found by accumulating the present value

S ( t )

{\displaystyle S(t)\,}

at time

t

{\displaystyle t\,}

to maturity

T

{\displaystyle T\,}

by the rate of risk-free return

r

{\displaystyle r\,}

:

F ( t ) = S ( t ) × ( 1 + r

)

( T − t )

{\displaystyle F(t)=S(t)\times (1+r)^{(T-t)}\,}

Swaps

Rational pricing underpins, also, the logic of swap valuation. Here, the instrument consists of two "legs", and to be arbitrage free, the terms of a swap contract are such that, initially, the net present value of both legs is equal. This must hold, since, if the two legs had differing values initially, an arbitrageur could: assume the position with the lower present value of payments; borrow funds equal to this present value to meet the cash flow obligations on the position; receive the corresponding payments - which have a higher present value - and use these to repay the debt on the borrowed funds. In the case of an interest rate swap, for example,

where one party pays a fixed rate, and the other a floating rate, at initiation:

P

IRS

=

P

fixed

−

P

float

{\displaystyle P_{\text{IRS}}=P_{\text{fixed}}-P_{\text{float}}}

P

fixed

= N R

∑

i = 1

n

1

d

i

Options

Under rational pricing assumptions, in a correctly priced options contract, the derivative premium (price), the strike price, and the spot price will be related such that arbitrage is not possible. Here, corresponding to the stereotypical arbitrage arguments above, so three single-period pricing formulae may be derived: delta hedging, the "replicating portfolio", and risk neutral valuation. Although these approaches result in different formulae they are, in fact, equivalent economically,

and it is then valid to assume "risk neutrality" when pricing derivatives generally (outline follows). A more formal relationship is described at the fundamental theorem of arbitrage-free pricing. The classic derivation

assumes a "binomial" behavior of the underlying instrument, where we have only two states – up or down: i.e. if S is the current price, then in the next period the price will either be

S

up

{\displaystyle S_{\text{up}}}

or

S

down

{\displaystyle S_{\text{down}}}

, and

S

u

= S × u

{\displaystyle S_{u}=S\times u}

, and

S

d

= S × d

{\displaystyle S_{d}=S\times d}

Delta hedging

It is possible to create a position consisting of Δ shares and 1 call option sold, such that the position's value, i.e. its "intrinsic value", will be identical in the S up and S down states, and hence known with certainty, and the above relationship applies. Under this approach - known as Delta hedging - we solve for Δ such that:

V

p

= Δ ×

S

u

− max (

S

u

− k , 0 )

= Δ ×

S

d

− max (

S

d

− k , 0 )

{\displaystyle {\begin{aligned}V_{p}&=\Delta \times S_{u}-\max(S_{u}-k,0)\\&=\Delta \times S_{d}-\max(S_{d}-k,0)\\\end{aligned}}}

where

V

Risk neutral pricing

Option pricing models - as outlined - more typically assume risk neutrality, where the option value is the discounted expected value of the end of period intrinsic values - i.e. that the underlying share price is a "Martingale". Under this assumption:

S

=

p ×

S

u

+ ( 1 − p ) ×

S

d

1 + r

⇒ p

=

( 1 + r ) − d

u − d

{\displaystyle {\begin{aligned}S&={\frac {p\times S_{u}+(1-p)\times S_{d}}{1+r}}\\\Rightarrow p&={\frac {(1+r)-d}{u-d}}\\\end{aligned}}}

where p (under more formal treatments, q) is the probability of an up move in the underlying, (1-p) is the probability of a down move, and r is the risk-free rate. Then, to return an option value consistent with this share price behaviour, we must say:

C

=

p ×

C