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. Financial economics is the branch of economics concerned with the workings of financial markets, such as the stock market, and the financing of companies. ... In economics, arbitrage is the practice of taking advantage of a state of imbalance between two (or possibly more) markets: a combination of matching deals are struck that exploit the imbalance, the profit being the difference between the market prices. ...

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 "locks in" a risk free profit without investing any of his own money. In economics, arbitrage is the practice of taking advantage of a state of imbalance between two or more markets: a combination of matching deals are struck that exploit the imbalance, the profit being the difference between the market prices. ...

In general, arbitrage ensures that "the law of one price" will hold; arbitrage also equalises the prices of assets with identical cashflows, 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: The law of one price is an economic law stated as: In an efficient market all identical goods must have only one price. ...

1. 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
2. deliver the asset to the buyer and receive that higher price
3. pay the seller on the cheaper market with the proceeds and pocket the difference.

To meet Wikipedias quality standards, this article or section may require cleanup. ...

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:

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

To meet Wikipedias quality standards, this article or section may require cleanup. ...

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. In finance, discounting is the process of finding the current value of an amount of cash at some future date, and along with compounding cash from the basis of time value of money calculations. ... The risk-free interest rate is the interest rate that it is assumed can be obtained by investing in financial instruments with no risk. ...

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:

1. The arbitrageur agrees to deliver the asset on the future date (i.e. sells forward) and simultaneously buys it today with borrowed money.
2. On the delivery date, the arbitrageur hands over the underlying, and receives the agreed price.
3. He then repays the lender the borrowed amount plus interest.
4. The difference between the agreed price and the amount owed is the arbitrage profit.

(b) where the discounted future price is lower than today's price: A forward contract is an agreement between two parties to buy or sell an asset (which can be of any kind) at a pre-agreed future point in time. ...

1. The arbitrageur agrees to pay for the asset on the future date (i.e. buys forward) and simultaneously sells the underlying today; he invests the proceeds.
2. On the delivery date, he cashes in the matured investment, which has appreciated at the risk free rate.
3. He then takes delivery of the underlying and pays the agreed price using the matured investment.
4. The difference between the maturity value and the agreed price is the arbitrage profit.

It will be noted that (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. A forward contract is an agreement between two parties to buy or sell an asset (which can be of any kind) at a pre-agreed future point in time. ... Backwardation, sometimes incorrectly referred to as backwardization, is the situation where futures contracts closer to expiration trade at higher prices than those that are far from expiration. ...

Fixed income securities

Fixed income securities have known cash flows (by definition). Further, each cash flow of a fixed income security can readily be matched by trading in some multiple of a risk free government issue Zero coupon bond with the corresponding maturity (or to a corresponding strip and ZCB). Hence, the price of any fixed income security, must today equal the sum of each of its cash flows discounted at the same rate as the corresponding government securities - i.e. the corresponding risk free rate. Were this not the case, arbitrage would be possible and would bring the price back into line with the price based on the government issued securities; see Fixed income arbitrage; Bond valuation. Fixed income refers to any type of investment that yields a regular (fixed) payment. ... Zero coupon bonds are bonds which do not pay periodic coupons, or so-called interest payments. ... Zero coupon bonds are bonds which do not pay periodic coupons, or so-called interest payments. ... The risk-free interest rate is the interest rate that it is assumed can be obtained by investing in financial instruments with no risk. ... Fixed income arbitrage is an investment strategy generally associated with hedge funds, which consists of the discovery and exploitation of inefficiencies in the pricing of bonds, i. ... Bond valuation is the process of determining the fair price of a bond. ...

The pricing formula is as below, where each cash flow is discounted at the rate which matches that of the corresponding government zero coupon instrument:

Price =

Pricing derivatives

A derivative is an instrument which 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. A derivative is a financial contract whose payoffs over time are derived from the performance of assets (such as commodities, shares or bonds), interest rates, exchange rates, or indices (such as a stock market index, consumer price index (CPI) or an index of weather conditions). ... The spot price of a commodity or a security or a currency is the price that is quoted for settlement (payment and delivery) of the transaction immediately. ... The derivatives markets are the financial markets for derivatives. ... Mathematical finance is the branch of applied mathematics concerned with the financial markets. ... The strike price, or exercise price, is a key variable in a derivatives contract between two parties. ... A reference rate is any publicly available quoted number or value that is used by the parties to a financial contract. ... The spot price of a commodity or a security or a currency is the price that is quoted for settlement (payment and delivery) of the transaction immediately. ...

see: Fundamental theorem of arbitrage-free pricing

In mathematical finance, a risk-neutral measure is a probability measure in which todays fair (i. ...

Futures

In a futures contract, for no arbitrage to be possible, the price paid on delivery (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. Thus, for a simple, non-dividend paying asset, the value of the future/forward, , will be found by discounting the present value at time to maturity by the rate of risk-free return . In finance, a futures contract is a standardized contract, traded on a futures exchange, to buy or sell a certain underlying instrument at a certain date in the future, at a pre-set price. ... The forward price is the agreed upon price of an asset in a forward contract. ... Future value measures what money is worth at a specified time in the future assuming a certain interest rate. ... In finance, an underlying is an investment from which a derivative security is derived. ...

This relationship may be modified for storage costs, dividends, dividend yields, and convenience yields; see futures contract pricing. In finance, a futures contract is a standardized contract, traded on a futures exchange, to buy or sell a certain underlying instrument at a certain date in the future, at a pre-set price. ...

Any deviation from this equality allows for arbitrage as follows.

• In the case where the forward price is higher:

1. The arbitrageur sells the futures contract and buys the underlying today (on the spot market) with borrowed money.
2. On the delivery date, the arbitrageur hands over the underlying, and receives the agreed forward price.
3. He then repays the lender the borrowed amount plus interest.
4. The difference between the two amounts is the arbitrage profit.

• In the case where the forward price is lower:

1. The arbitrageur buys the futures contract and sells the underlying today (on the spot market); he invests the proceeds.
2. On the delivery date, he cashes in the matured investment, which has appreciated at the risk free rate.
3. He then receives the underlying and pays the agreed forward price using the matured investment. [If he was short the underlying, he returns it now.]
4. The difference between the two amounts is the arbitrage profit.

Spot can refer to: Look up spot in Wiktionary, the free dictionary. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

Options

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 which either "locks in" or "infers" this value; both approaches deliver identical results. Methods which lock-in future cash flows assume arbitrage free pricing, and those which infer expected value assume risk neutral valuation. In finance, an option is a contract whereby one party (the holder or buyer) has the right but not the obligation to exercise a feature of the contract (the option) on or before a future date (the exercise date or expiry). ...

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. Given these two states, the "arbitrage free" approach creates a position which will have 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. In finance, an underlying is an investment from which a derivative security is derived. ... In finance, the binomial options model provides a generalisable numerical method for the valuation of options. ... Intrinsic value can refer to: Intrinsic value (finance), of an option or stock. ...

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 increasingly 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 Black-Scholes model, often simply called Black-Scholes, is a model of the varying price over time of financial instruments, and in particular stocks. ... In finance, the binomial options model provides a generalisable numerical method for the valuation of options. ... The Black-Scholes model, often simply called Black-Scholes, is a model of the varying price over time of financial instruments, and in particular stocks. ... In computer programming, the word code refers to instructions to a computer in a programming language. ... In probability theory, a continuous-time Markov process is a stochastic process { X(t) : t ≥ 0 } that enjoys the Markov property and takes values from amongst the elements of a discrete set called the state space. ...

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. A put option (sometimes simply called a put) is a financial contract between two parties, the buyer and the seller of the option. ... In financial mathematics, put-call parity defines a relationship between the price of a European call option and a European put option - both with the identical strike price and expiry. ...

Arbitrage free pricing

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.

Delta hedging

It is possible to create a position consisting of Δ calls sold and 1 share, such that the position’s value will be identical in the S up and S down states, and hence known with certainty (see Delta hedging). This value corresponds to the forward price above, and as above, for no arbitrage to be possible, the present value of the position must be its expected future value discounted at the risk free rate, r. The value of a call is then found by equating the two. A call option is a financial contract between two parties, the buyer and the seller of this type of option. ... Delta hedging is the process of setting or keeping the delta of a portfolio of financial instruments zero, or as close to zero as possible - where delta is the sensitivity of the value of a derivative to changes in the price of its underlying instrument; see Hedge (finance). ...

1) Solve for Δ such that:

value of position in one period = S up - Δ × (S up – strike price ) = S down - Δ × (S down – strike price)

2) solve for the value of the call, using Δ, where:

value of position today = value of position in one period ÷ (1 + r) = S current – Δ × value of call

The replicating portfolio

It is possible to create a position consisting of Δ shares and $B borrowed at the risk free rate, which will produce identical cash flows to one option on the underlying share. The position created is known as a "replicating portfolio" since its cash flows replicate those of the option. As shown, 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.

1) Solve simultaneously for Δ and B such that:

i) Δ × S up - B × (1 + r) = (S up – strike price )

ii) Δ × S down - B × (1 + r) = max(0,(S down – strike price ))

2) solve for the value of the call, using Δ and B, where:

call = Δ × S current - B

Risk neutral valuation

Here the value of the option is calculated using the risk neutrality assumption. Under this assumption, the “expected value” (as opposed to "locked in" value) is discounted. The expected value is calculated using the intrinsic values from the later two nodes: “Option up” and “Option down”, with u and d as price multipliers as above. These are then weighted by their respective probabilities: “probability” p of an up move in the underlying, and “probability” (1-p) of a down move. The expected value is then discounted at r, the risk free rate. In mathematical finance, a risk-neutral measure is a probability measure in which todays fair (i. ... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical... In finance, discounting is the process of finding the current value of an amount of cash at some future date, and along with compounding cash form the basis of time value of money calculations. ... Option Value In finance, the value of an option consists of two components, its intrinsic value and its time value. ... The risk-free interest rate is the interest rate that it is assumed can be obtained by investing in financial instruments with no risk. ...

1) solve for p

for no arbitrage to be possible in the share, today’s price must represent its expected value discounted at the risk free rate:

S = [ p × (up value) + (1-p) ×(down value) ] ÷ (1+r) = [ p × S × u + (1-p) × S × d ] ÷ (1+r)

then, p = [(1+r) - d ] ÷ [ u - d ]

2) solve for call value, using p

for no arbitrage to be possible in the call, today’s price must represent its expected value discounted at the risk free rate:

Option value = [ p × Option up + (1-p)× Option down] ÷ (1+r)

= [ p × (S up - strike) + (1-p)× (S down - strike) ] ÷ (1+r)

The risk neutrality assumption

Note that above, the risk neutral formula does not refer to the volatility of the underlying – 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. Volatility is the standard deviation of the change in value of a financial instrument with a specific time horizon. ... In mathematical finance, a risk-neutral measure is a probability measure in which todays fair (i. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...

Swaps

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. In finance a swap is a derivative, where two counterparties exchange one stream of cash flows against another stream. ... A counterparty is a legal and financial term. ... In finance, cash flow refers to the amounts of cash being received and spent by a business during a defined period of time, sometimes tied to a specific project. ... The present value of a future cash flow is the nominal amount of money to change hands at some future date, discounted to account for the time value of money. ...

To be arbitrage free, the terms of a swap contract are such that, initially, the NPV of these future cash flows is equal to zero; see swap valuation. For example, consider a fixed-to-floating Interest rate swap where Party A pays a fixed rate, and Party B pays a floating rate. Here, the fixed rate would be such that the present value of future fixed rate payments by Party A is equal to the present value of the expected future floating rate payments (i.e. the NPV is zero). Were this not the case, an Arbitrageur, C, could: Net present value is a form of calculating discounted cash flow. ... In finance a swap is a derivative, where two counterparties exchange one stream of cash flows against another stream. ... In the field of derivatives, a popular form of swap is the interest rate swap, in which one party exchanges a stream of interest for another stream. ... In economics, arbitrage is the practice of taking advantage of a state of imbalance between two or more markets: a combination of matching deals are struck that exploit the imbalance, the profit being the difference between the market prices. ...

1. assume the position with the lower present value of payments, and borrow funds equal to this present value
2. meet the cash flow obligations on the position by using the borrowed funds, and receive the corresponding payments - which have a higher present value
3. use the received payments to repay the debt on the borrowed funds
4. pocket the difference - where the difference between the present value of the loan and the present value of the inflows is the arbitrage profit.

Once traded, swaps can also be priced using rational pricing. For example, an interest rate swap can be decomposed into a series of Forward rate agreements; i.e. we have two instruments with identical cashflows, and therefore, as above, arbitrage free pricing must apply. Similarly, the "receive-fixed" leg of a swap, can be valued by comparison to a Bond with the same schedule of payments. This article needs to be cleaned up to conform to a higher standard of quality. ... Cashflow can refer to: For cashflow with relation to finance, see cash flow. ... In finance, a bond is a debt security, in which the issuer owes the holders a debt and is obliged to repay the principal and interest (the coupon). ...

Pricing shares

The Arbitrage pricing theory (APT), a general theory of asset pricing, has become influential in the pricing of shares. 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: Arbitrage pricing theory (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. ... In financial terminology, stock is the capital raised by a corporation, through the issuance and sale of shares. ... The expected gain (or expected return) is the weighted-average most likely outcome in gambling, probability theory, economics or finance. ... A linear function is a mathematical function term of the form: f(x) = m x + c where c is a constant. ... Macroeconomics is the economics sub-field of study that considers aggregate behavior, and the study of the sum of individual economic decisions. ... The Beta coefficient, or financial elasticity (sensitivity of the asset returns to market returns, relative volatility), is a key parameter in the Capital asset pricing model (CAPM). ...

where

• E(r_j) is the risky asset's expected return,
• r_f is the risk free rate,
• F_k is the macroeconomic factor,
• b_jk is the sensitivity of the asset to factor k,
• and ε_j is the risky asset's idiosyncratic random shock with mean zero.

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. If the price diverges, arbitrage should bring it back into line. Here, to perform the arbitrage, the investor “creates” a correctly priced asset (a synthetic asset) being a portfolio which has the same net-exposure to each of the macroeconomic factors as the mispriced asset but a different expected return; see the APT article for detail on the construction of the portfolio. The arbitrageur is then in a position to make a risk free profit as follows: The risk-free interest rate is the interest rate that it is assumed can be obtained by investing in financial instruments with no risk. ... In finance, discounting is the process of finding the current value of an amount of cash at some future date, and along with compounding cash from the basis of time value of money calculations. ... In economics, arbitrage is the practice of taking advantage of a state of imbalance between two or more markets: a combination of matching deals are struck that exploit the imbalance, the profit being the difference between the market prices. ... Arbitrage pricing theory (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. ...

• Where the asset price is too low, the portfolio should have appreciated at the rate implied by the APT, whereas the mispriced asset would have appreciated at more than this rate. The arbitrageur could therefore:

1. Today: short sell the portfolio and buy the mispriced-asset with the proceeds.
2. At the end of the period: sell the mispriced asset, use the proceeds to buy back the portfolio, and pocket the difference.

• Where the asset price is too high, the portfolio should have appreciated at the rate implied by the APT, whereas the mispriced asset would have appreciated at less than this rate. The arbitrageur could therefore:

1. Today: short sell the mispriced-asset and buy the portfolio with the proceeds.
2. At the end of the period: sell the portfolio, use the proceeds to buy back the mispriced-asset, and pocket the difference.

Note that under "true arbitrage", the investor locks-in a guaranteed payoff, whereas under APT arbitrage, the investor locks-in a positive expected payoff. The APT thus assumes "arbitrage in expectations" - i.e that arbitrage by investors will bring asset prices back into line with the returns expected by the model. To meet Wikipedias quality standards, this article or section may require cleanup. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

The Capital asset pricing model (CAPM) is an earlier, (more) influential theory on asset pricing. Although based on different assumptions, the CAPM can, in some ways, be considered a "special case" of the APT; specifically, the CAPM's Securities market line represents a single-factor model of the asset price, where Beta is exposure to changes in value of the Market. The capital asset pricing model (CAPM) is used in finance to determine a theoretically appropriate required rate of return (and thus the price if expected cash flows can be estimated) of an asset given that assets non-diversifiable risk. ... Capital Market Line Modern portfolio theory (MPT) proposes how rational investors will use diversification to optimize their portfolios, and how an asset should be priced given its risk relative to the market as a whole. ...

See also

• Efficient market hypothesis
• Fair value
• Fundamental theorem of arbitrage-free pricing
• Homo economicus
• List of valuation topics
• Rational choice theory
• Rationality
• Volatility arbitrage

This article or section seems not to be written in the formal tone expected of an encyclopedia entry. ... Definition Fair value, also called fair price, is a concept used in finance and economics. ... In mathematical finance, a risk-neutral measure is a probability measure in which todays fair (i. ... Homo economicus, or Economic man, is the concept in some economic theories of man as both rational and It is a term used for an approximation or model of Homo sapiens that acts to obtain the highest possible well-being for himself given available information about opportunities and other constraints... What follows is a list of over 250 Wikipedia articles on finance topics. ... It has been suggested that this article or section be merged with Decision theory. ... In philosophy, the word rationality has been used to describe numerous religious and philosophical theories, especially those concerned with truth, reason, and knowledge. ... Volatility arbitrage, a. ...

External links

• Risk Neutral Pricing in Discrete Time (PDF), Prof. Don M. Chance
• The Idea Behind Arbitrage Pricing, Quantnotes
• The Notion of Arbitrage and Free Lunch in Mathematical Finance, Prof. Walter Schachermayer
• Option Valuation in the Binomial Model, Prof. Ernst Maug
• Pricing Futures and Forwards by Arbitrage Argument, Quantnotes
• The relationship between futures and spot prices, The Investment Analysts Society of South Africa