Rational pricing is the assumption in financial economics

Financial economics

Financial Economics is the branch of economics concerned with "the allocation and deployment of economic resources, both spatially and across time, in an uncertain environment"....

 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 mechanics

Arbitrage

Arbitrage

In economics and finance, arbitrage is the practice of taking advantage of a price difference between two or more markets: striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices...

 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 general, arbitrage ensures that "the law of one price

Law of one price

The law of one price is an economic law stated as: "In an efficient market, all identical goods must have only one price."-Intuition:The intuition for this law is that all sellers will flock to the highest prevailing price, and all buyers to the lowest current market price. In an efficient market...

" 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

Law of one price

The law of one price is an economic law stated as: "In an efficient market, all identical goods must have only one price."-Intuition:The intuition for this law is that all sellers will flock to the highest prevailing price, and all buyers to the lowest current market price. In an efficient market...

").
Where this is not true, the arbitrageur will:

1. buy the asset on the market where it has the lower price, and simultaneously sell it (short
   Short selling
   In finance, short selling is the practice of selling assets, usually securities, that have been borrowed from a third party with the intention of buying identical assets back at a later date to return to that third party...
   
   ) 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.

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
   Short selling
   In finance, short selling is the practice of selling assets, usually securities, that have been borrowed from a third party with the intention of buying identical assets back at a later date to return to that third party...
   
   ) 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.

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

Risk-free interest rate

Risk-free interest rate is the theoretical rate of return of an investment with no risk of financial loss. The risk-free rate represents the interest that an investor would expect from an absolutely risk-free investment over a given period of time....

.

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
   Forward contract
   In finance, a forward contract or simply a forward is a non-standardized contract between two parties to buy or sell an asset at a specified future time at a price agreed today. This is in contrast to a spot contract, which is an agreement to buy or sell an asset today. It costs nothing to enter a...
   
   ) 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:

1. The arbitrageur agrees to pay for the asset on the future date (i.e. buys forward
   Forward contract
   In finance, a forward contract or simply a forward is a non-standardized contract between two parties to buy or sell an asset at a specified future time at a price agreed today. This is in contrast to a spot contract, which is an agreement to buy or sell an asset today. It costs nothing to enter a...
   
   ) and simultaneously sells (short
   Short selling
   In finance, short selling is the practice of selling assets, usually securities, that have been borrowed from a third party with the intention of buying identical assets back at a later date to return to that third party...
   
   ) 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

Backwardation

Normal backwardation, also sometimes called backwardation, is the market condition wherein the price of a forward or futures contract is trading below the expected spot price at contract maturity. The resulting futures or forward curve would typically be downward sloping , since contracts for...

.

Fixed income securities

Rational pricing is one approach used in pricing fixed rate bond

Fixed rate bond

In finance, a fixed rate bond is a type of debt instrument bond with a fixed coupon rate, as opposed to a floating rate note. A fixed rate bond is a long term debt paper that carries a predetermined interest rate...

s. Here, each cash flow can be matched by trading in (a) some multiple of a zero-coupon bond corresponding to the coupon date, and of equivalent credit worthiness (if possible, from the same issuer as the bond being valued) with the corresponding maturity, or (b) in a corresponding strip and ZCB.

Given that the cash flows can be replicated, the price of the bond must today equal the sum of each of its cash flows discounted at the same rate as each ZCB, as above. Were this not the case, arbitrage would be possible and would bring the price back into line with the price based on ZCBs; see Bond valuation: Arbitrage-free pricing approach

The pricing formula is as below, where each cash flow is discounted at the rate that matches the coupon date:

Price =

Often, the formula is expressed as , using prices instead of rates, as prices are more readily available.

See also Fixed income arbitrage

Fixed income arbitrage

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.e...

; Bond credit rating

Bond credit rating

In investment, the bond credit rating assesses the credit worthiness of a corporation's or government debt issues. It is analogous to credit ratings for individuals.-Table:...

.

Pricing derivatives

A derivative

Derivative (finance)

A derivative instrument is a contract between two parties that specifies conditions—in particular, dates and the resulting values of the underlying variables—under which payments, or payoffs, are to be made between the parties.Under U.S...

 is an instrument that allows for buying and selling of the same asset on two markets – the spot market

Spot price

The spot price or spot rate of a commodity, a security or a currency is the price that is quoted for immediate settlement . Spot settlement is normally one or two business days from trade date...

 and the derivatives market

Derivatives market

The derivatives market is the financial market for derivatives, financial instruments like futures contracts or options, which are derived from other forms of assets....

. Mathematical finance

Mathematical finance

Mathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical...

 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

Strike price

In options, the strike price is a key variable in a derivatives contract between two parties. Where the contract requires delivery of the underlying instrument, the trade will be at the strike price, regardless of the spot price of the underlying instrument at that time.Formally, the strike...

 (or reference rate

Reference rate

A reference rate is a rate that determines pay-offs in a financial contract and that is outside the control of the parties to the contract. It is often some form of LIBOR rate, but it can take many forms, such as a consumer price index, a house price index or an unemployment rate...

), and the spot price

Spot price

The spot price or spot rate of a commodity, a security or a currency is the price that is quoted for immediate settlement . Spot settlement is normally one or two business days from trade date...

 will be related such that arbitrage is not possible.

see: Fundamental theorem of arbitrage-free pricing

Fundamental theorem of arbitrage-free pricing

The fundamental theorems of arbitrage/finance provide necessary and sufficient conditions for a market to be arbitrage free and for a market to be complete. An arbitrage opportunity is a way of making money with no initial investment without any possibility of loss...

Futures

In a futures contract

Futures contract

In finance, a futures contract is a standardized contract between two parties to exchange a specified asset of standardized quantity and quality for a price agreed today with delivery occurring at a specified future date, the delivery date. The contracts are traded on a futures exchange...

, for no arbitrage to be possible, the price paid on delivery (the forward price

Forward price

The forward price is the agreed upon price of an asset in a forward contract. Using the rational pricing assumption, for a forward contract on an underlying asset that is tradeable, we can express the forward price in terms of the spot price and any dividends etc...

) 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

Future value

Future value is the value of an asset at a specific date. It measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming a certain interest rate, or more generally, rate of return; it is the present value multiplied by the accumulation...

 of the underlying

Underlying

In finance, the underlying of a derivative is an asset, basket of assets, index, or even another derivative, such that the cash flows of the derivative depend on the value of this underlying...

 discounted at the risk free rate (the "asset with a known future-price", as above). Thus, for a simple, non-dividend paying asset, the value of the future/forward, , will be found by accumulating the present value at time to maturity by the rate of risk-free return .


This relationship may be modified for storage costs, dividends, dividend yields, and convenience yields; see futures contract pricing.

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
   Short selling
   In finance, short selling is the practice of selling assets, usually securities, that have been borrowed from a third party with the intention of buying identical assets back at a later date to return to that third party...
   
    the underlying, he returns it now.]
4. The difference between the two amounts is the arbitrage profit.

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

Option (finance)

In finance, an option is a derivative financial instrument that specifies a contract between two parties for a future transaction on an asset at a reference price. The buyer of the option gains the right, but not the obligation, to engage in that transaction, while the seller incurs the...

 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 value

Intrinsic value

Intrinsic value can refer to:*Intrinsic value , of an option or stock.*Intrinsic value , of a coin.*Intrinsic value , in ethics and philosophy.*Intrinsic value , in philosophy....

s 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

Put option

A put or put option is a contract between two parties to exchange an asset, the underlying, at a specified price, the strike, by a predetermined date, the expiry or maturity...

 can be derived as below, or may be found from the value of the call using put-call parity.

Arbitrage free pricing

Here, the future payoff is "locked in" using either "delta hedging" or the "replicating portfolio

Replicating portfolio

In the valuation of a life insurance company, the actuary considers a series of future uncertain cashflows and attempts to put a value on these cashflows...

" 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 shares and 1 call

Call option

A call option, often simply labeled a "call", is a financial contract between two parties, the buyer and the seller of this type of option. The buyer of the call option has the right, but not the obligation to buy an agreed quantity of a particular commodity or financial instrument from the seller...

 sold, 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 certain value corresponds to the forward price above ("An asset with a known future price"), 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.

1. Solve for Δ such that:

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

1. Solve simultaneously for Δ and B such that:
   • Δ × S up - B × (1 + r) =  ( 0, S up – strike price )
   • Δ × S down - B × (1 + r) =  ( 0, S down – strike price )
2. Solve for the value of the call, using Δ and B, where:
   • call = Δ × S current - B

Note that there is no discounting - the interest rate appears only as part of the construction. This approach is therefore used in preference to others where it is not clear whether the risk free rate may be applied as the discount rate

Discount rate

The discount rate can mean*an interest rate a central bank charges depository institutions that borrow reserves from it, for example for the use of the Federal Reserve's discount window....

 at each decision point, or whether, instead, a premium over risk free would be required. The best example of this would be under Real options analysis

Real options analysis

Real options valuation, also often termed Real options analysis, applies option valuation techniques to capital budgeting decisions. A real option itself, is the right — but not the obligation — to undertake some business decision; typically the option to make, abandon, expand, or contract a...

 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. Here, in the above formulae, we then have: "Δ × S up - B × (1 + r up)..." and "Δ × S down - B × (1 + r down)..." .

Risk neutral valuation

Here the value of the option is calculated using the risk neutrality

Risk-neutral measure

In mathematical finance, a risk-neutral measure, is a prototypical case of an equivalent martingale measure. It is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing, which implies that in a complete market a derivative's price is the discounted...

 assumption. Under this assumption, the “expected value

Expected value

In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

” (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

Risk-free interest rate

Risk-free interest rate is the theoretical rate of return of an investment with no risk of financial loss. The risk-free rate represents the interest that an investor would expect from an absolutely risk-free investment over a given period of time....

.

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 (i.e., the share price is a Martingale

   Martingale (probability theory)

   In probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...

   
   ):

   

   1. 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:

      
      

      The risk neutrality assumption

      
      Note that above, the risk neutral formula does not refer to the volatility

      Volatility (finance)

      In finance, volatility is a measure for variation of price of a financial instrument over time. Historic volatility is derived from time series of past market prices...

      
       of the underlying – p as solved, relates to the risk-neutral measure

      Risk-neutral measure

      In mathematical finance, a risk-neutral measure, is a prototypical case of an equivalent martingale measure. It is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing, which implies that in a complete market a derivative's price is the discounted...

      
       as opposed to the actual probability distribution

      Probability distribution

      In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

      
       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. See Fundamental theorem of arbitrage-free pricing

      Fundamental theorem of arbitrage-free pricing

      The fundamental theorems of arbitrage/finance provide necessary and sufficient conditions for a market to be arbitrage free and for a market to be complete. An arbitrage opportunity is a way of making money with no initial investment without any possibility of loss...

      
      .
      

      Swaps

      Rational pricing underpins the logic of swap

      Swap (finance)

      In finance, a swap is a derivative in which counterparties exchange certain benefits of one party's financial instrument for those of the other party's financial instrument. The benefits in question depend on the type of financial instruments involved...

      
       valuation. Here, two counterparties

      Counterparty

      A counterparty is a legal and financial term. It means a party to a contract. A counterparty is usually the entity with whom one negotiates on a given agreement, and the term can refer to either party or both, depending on context....

      
       "swap" obligations, effectively exchanging cash flow

      Cash flow

      Cash flow is the movement of money into or out of a business, project, or financial product. It is usually measured during a specified, finite period of time. Measurement of cash flow can be used for calculating other parameters that give information on a company's value and situation.Cash flow...

      
       streams calculated against a notional principal amount, and the value of the swap is the present value

      Present value

      Present value, also known as present discounted value, is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk...

      
       (PV) of both sets of future cash flows "netted off" against each other.
      

      Valuation at initiation

      To be arbitrage free, the terms of a swap contract are such that, initially, the Net present value

      Net present value

      In finance, the net present value or net present worth of a time series of cash flows, both incoming and outgoing, is defined as the sum of the present values of the individual cash flows of the same entity...

      
       of these future cash flows is equal to zero; see swap valuation. For example, consider the valuation of a fixed-to-floating Interest rate swap

      Interest rate swap

      An interest rate swap is a popular and highly liquid financial derivative instrument in which two parties agree to exchange interest rate cash flows, based on a specified notional amount from a fixed rate to a floating rate or from one floating rate to another...

      
       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:

      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

      
      

      Subsequent valuation

      Once traded, swaps can also be priced using rational pricing. For example, the Floating leg of an interest rate swap can be "decomposed" into a series of forward rate agreement

      Forward rate agreement

      In finance, a forward rate agreement is a forward contract, an over-the-counter contract between parties that determines the rate of interest, or the currency exchange rate, to be paid or received on an obligation beginning at a future start date. The contract will determine the rates to be used...

      
      s. 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

      Bond (finance)

      In finance, a bond is a debt security, in which the authorized issuer owes the holders a debt and, depending on the terms of the bond, is obliged to pay interest to use and/or to repay the principal at a later date, termed maturity...

      
       with the same schedule of payments. (Relatedly, given that their underlying

      Underlying

      In finance, the underlying of a derivative is an asset, basket of assets, index, or even another derivative, such that the cash flows of the derivative depend on the value of this underlying...

      
      s have the same cash flows, bond option

      Bond option

      In finance, a bond option is an option to buy or sell a bond at a certain price on or before the option expiry date. These instruments are typically traded OTC....

      
      s and swaption

      Swaption

      A swaption is an option granting its owner the right but not the obligation to enter into an underlying swap. Although options can be traded on a variety of swaps, the term "swaption" typically refers to options on interest rate swaps....

      
      s are equatable.)
      

      Pricing shares

      The arbitrage pricing theory

      Arbitrage pricing theory

      In finance, arbitrage pricing theory is a general theory of asset pricing that holds that the expected return of a financial asset can be modeled as a linear function of various macro-economic factors or theoretical market indices, where sensitivity to changes in each factor is represented by a...

      
       (APT), a general theory of asset pricing, has become influential in the pricing of shares

      Stock

      The capital stock of a business entity represents the original capital paid into or invested in the business by its founders. It serves as a security for the creditors of a business since it cannot be withdrawn to the detriment of the creditors...

      
      . APT holds that the expected return

      Expected return

      The expected return is the weighted-average outcome in gambling, probability theory, economics or finance.It isthe average of a probability distribution of possible returns, calculated by using the following formula:...

      
       of a financial asset, can be modelled as a linear function

      Linear function

      In mathematics, the term linear function can refer to either of two different but related concepts:* a first-degree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....

      
       of various macro-economic

      Macroeconomics

      Macroeconomics is a branch of economics dealing with the performance, structure, behavior, and decision-making of the whole economy. This includes a national, regional, or global economy...

      
       factors, where sensitivity to changes in each factor is represented by a factor specific beta coefficient

      Beta coefficient

      In finance, the Beta of a stock or portfolio is a number describing the relation of its returns with those of the financial market as a whole.An asset has a Beta of zero if its returns change independently of changes in the market's returns...

      
      :
      
      

      where

      • is the risky asset's expected return,
      • is the risk free rate,
      • is the macroeconomic factor,
      • is the sensitivity of the asset to factor ,
      • and 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), a portfolio with the same net-exposure to each of the macroeconomic factors as the mispriced asset but a different expected return. See the arbitrage pricing theory article for detail on the construction of the portfolio. The arbitrageur is then in a position to make a risk free profit as follows:
      

      • 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
         Short selling
         In finance, short selling is the practice of selling assets, usually securities, that have been borrowed from a third party with the intention of buying identical assets back at a later date to return to that third party...
         
          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
         Short selling
         In finance, short selling is the practice of selling assets, usually securities, that have been borrowed from a third party with the intention of buying identical assets back at a later date to return to that third party...
         
           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.
      
      The capital asset pricing model

      Capital asset pricing model

      In finance, the capital asset pricing model is used to determine a theoretically appropriate required rate of return of an asset, if that asset is to be added to an already well-diversified portfolio, given that asset's non-diversifiable risk...

      
       (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.
      

      See also

      • Efficient market hypothesis
        Efficient market hypothesis
        In finance, the efficient-market hypothesis asserts that financial markets are "informationally efficient". That is, one cannot consistently achieve returns in excess of average market returns on a risk-adjusted basis, given the information available at the time the investment is made.There are...
        
        
      • Fair value
        Fair value
        Fair value, also called fair price , is a concept used in accounting and economics, defined as a rational and unbiased estimate of the potential market price of a good, service, or asset, taking into account such objective factors as:* acquisition/production/distribution costs, replacement costs,...
        
        
      • Fundamental theorem of arbitrage-free pricing
        Fundamental theorem of arbitrage-free pricing
        The fundamental theorems of arbitrage/finance provide necessary and sufficient conditions for a market to be arbitrage free and for a market to be complete. An arbitrage opportunity is a way of making money with no initial investment without any possibility of loss...
        
        
      • Homo economicus
        Homo economicus
        Homo economicus, or Economic human, is the concept in some economic theories of humans as rational and narrowly self-interested actors who have the ability to make judgments toward their subjectively defined ends...
        
        
      • List of valuation topics
      • Rational choice theory
        Rational choice theory
        Rational choice theory, also known as choice theory or rational action theory, is a framework for understanding and often formally modeling social and economic behavior. It is the main theoretical paradigm in the currently-dominant school of microeconomics...
        
        
      • Rationality
        Rationality
        In philosophy, rationality is the exercise of reason. It is the manner in which people derive conclusions when considering things deliberately. It also refers to the conformity of one's beliefs with one's reasons for belief, or with one's actions with one's reasons for action...
        
        
      • Risk-neutral measure
        Risk-neutral measure
        In mathematical finance, a risk-neutral measure, is a prototypical case of an equivalent martingale measure. It is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing, which implies that in a complete market a derivative's price is the discounted...
        
        
      • Volatility arbitrage
        Volatility arbitrage
        In finance, volatility arbitrage is a type of statistical arbitrage that is implemented by trading a delta neutral portfolio of an option and its underlier. The objective is to take advantage of differences between the implied volatility of the option, and a forecast of future realized volatility...
        
        

      
      

      External links

      Arbitrage free pricing

      • Pricing by Arbitrage, The History of Economic Thought Website
      • The Idea Behind Arbitrage Pricing, Samy Mohammed, Quantnotes
      • "The Fundamental Theorem" of Finance; part II. Prof. Mark Rubinstein
        Mark Rubinstein
        Mark Edward Rubinstein is a leading financial economist and financial engineer. He is currently Professor of Finance at the Haas School of Business of the University of California, Berkeley, where he is involved in teaching courses in the , an academic program that is focused on equipping...
        
        , Haas School of Business
        Haas School of Business
        The Walter A. Haas School of Business, also known as the Haas School of Business or simply Haas, is one of 14 schools and colleges at the University of California, Berkeley....
        
        
      • Elementary Asset Pricing Theory, Prof. K. C. Border California Institute of Technology
        California Institute of Technology
        The California Institute of Technology is a private research university located in Pasadena, California, United States. Caltech has six academic divisions with strong emphases on science and engineering...
        
        
      • The Notion of Arbitrage and Free Lunch in Mathematical Finance, Prof. Walter Schachermayer
      • Risk Neutral Pricing in Discrete Time (PDF
        Portable Document Format
        Portable Document Format is an open standard for document exchange. This file format, created by Adobe Systems in 1993, is used for representing documents in a manner independent of application software, hardware, and operating systems....
        
        ), Prof. Don M. Chance
      • No Arbitrage in Continuous Time, Prof. Tyler Shumway

      
      
      Risk neutrality and arbitrage free pricing

      • Risk-Neutral Probabilities Explained. Nicolas Gisiger
      • Risk-neutral Valuation: A Gentle Introduction, Part II. Joseph Tham Duke University
        Duke University
        Duke University is a private research university located in Durham, North Carolina, United States. Founded by Methodists and Quakers in the present day town of Trinity in 1838, the school moved to Durham in 1892. In 1924, tobacco industrialist James B...
        
        

      
      
      Application to derivatives

      • Option Valuation in the Binomial Model (archived
        Internet Archive
        The Internet Archive is a non-profit digital library with the stated mission of "universal access to all knowledge". It offers permanent storage and access to collections of digitized materials, including websites, music, moving images, and nearly 3 million public domain books. The Internet Archive...
        
        ), Prof. Ernst Maug, Rensselaer Polytechnic Institute
        Rensselaer Polytechnic Institute
        Stephen Van Rensselaer established the Rensselaer School on November 5, 1824 with a letter to the Rev. Dr. Samuel Blatchford, in which van Rensselaer asked Blatchford to serve as the first president. Within the letter he set down several orders of business. He appointed Amos Eaton as the school's...
        
        
      • Pricing Futures and Forwards by Arbitrage Argument, Quantnotes
      • The relationship between futures and spot prices, Investment Analysts Society of Southern Africa
        Investment Analysts Society of Southern Africa
        The Investment Analyst's Society of Southern Africa is the liaison body for the financial analyst profession in South Africa. It is based in Johannesburg South Africa, with members from Cape Town, Durban and throughout the region.-Membership:...
        
        
      • The illusions of dynamic replication, Emanuel Derman
        Emanuel Derman
        Emanuel Derman is a South African-born academic, businessman and writer. He is best known as a quantitative analyst, and author of the book My Life as A Quant: Reflections on Physics and Finance....
        
         and Nassim Taleb
        Nassim Taleb
        Nassim Nicholas Taleb is a Lebanese American essayist whose work focuses on problems of randomness and probability. His 2007 book The Black Swan was described in a review by Sunday Times as one of the twelve most influential books since World War II....
        
        
      • Swaptions and Options, Prof. Don M. Chance

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