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Rational pricing is the assumption in
financial economics Financial economics, also known as finance, is the branch of economics characterized by a "concentration on monetary activities", in which "money of one type or another is likely to appear on ''both sides'' of a trade". William F. Sharpe"Financia ...
that asset prices - and hence asset pricing models - will reflect the
arbitrage-free In economics and finance, arbitrage (, ) is the practice of taking advantage of a difference in prices in two or more markets; striking a combination of matching deals to capitalise on the difference, the profit being the difference between the ...
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 In economics and finance, arbitrage (, ) is the practice of taking advantage of a difference in prices in two or more markets; striking a combination of matching deals to capitalise on the difference, the profit being the difference between t ...
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 Normal backwardation, also sometimes called backwardation, is the market condition where the price of a commodity's forward or futures contract is trading below the ''expected'' spot price at contract maturity. The resulting futures or forward ...
.


Fixed income securities

:''See also
Fixed income arbitrage Fixed-income arbitrage is a group of market-neutral-investment strategies that are designed to take advantage of differences in interest rates between varying fixed-income securities or contracts (Jefferson, 2007). Arbitrage in terms of investment ...
;
Bond credit rating In investment, the bond credit rating represents the credit worthiness of corporate or government bonds. It is not the same as an individual's credit score. The ratings are published by credit rating agencies and used by investment professional ...
.'' Rational pricing is one approach used in pricing fixed rate bonds. Here, each cash flow on the bond can be matched by trading in either (a) some multiple of a
zero-coupon bond A zero coupon bond (also discount bond or deep discount bond) is a bond in which the face value is repaid at the time of maturity. Unlike regular bonds, it does not make periodic interest payments or have so-called coupons, hence the term zero- ...
, ZCB, corresponding to each coupon date, and of equivalent
credit worthiness A credit risk is risk of default on a debt that may arise from a borrower failing to make required payments. In the first resort, the risk is that of the lender and includes lost principal and interest, disruption to cash flows, and increased c ...
(if possible, from the same issuer as the bond being valued) with the corresponding maturity, or (b) in a strip corresponding to each coupon, and a ZCB for the return of principle on maturity. Then, 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 (per #Assets with identical cash flows). Were this not the case, arbitrage would be possible and would bring the price back into line with the price based on ZCBs. The mechanics are as follows. Where the price of the bond is misaligned with the present value of the ZCBs, the arbitrageur could: # finance her purchase of whichever of the bond or the sum of the ZCBs was cheaper # by short selling the other # and meeting her cash flow commitments using the coupons or maturing zeroes as appropriate # then, her profit would be the difference between the two values. The pricing formula is then P_0 = \sum_^T\frac, where each cash flow C_t\, is discounted at the rate r_t\, that matches the coupon date. Often, the formula is expressed as P_0 = \sum_^ T C(t) \times P(t), using prices instead of rates, as prices are more readily available. Rational pricing also applies to interest rate modeling more generally: yield curves must be arbitrage-free with respect to the prices of individual instruments. See Bootstrapping (finance) and
Multi-curve framework In finance, an interest rate swap (IRS) is an interest rate derivative (IRD). It involves exchange of interest rates between two parties. In particular it is a "linear" IRD and one of the most liquid, benchmark products. It has associations wi ...
.


Pricing derivatives

A
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
is an instrument that allows for buying and selling of the same asset on two markets – the
spot market The spot market or cash market is a public financial market in which financial instruments or commodities are traded for immediate delivery. It contrasts with a futures market, in which delivery is due at a later date. In a spot market, settle ...
and the
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. The market can be divided into two, that for exchange-traded derivatives a ...
.
Mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require ...
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 In finance, the strike price (or exercise price) of an option is a fixed price at which the owner of the option can buy (in the case of a call), or sell (in the case of a put), the underlying security or commodity. The strike price may be set ...
(or
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 pri ...
), and the spot price will be related such that arbitrage is not possible. See Fundamental theorem of arbitrage-free pricing.


Futures

In a
futures contract In finance, a futures contract (sometimes called a futures) is a standardized legal contract to buy or sell something at a predetermined price for delivery at a specified time in the future, between parties not yet known to each other. The asset ...
, 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 (the " asset with a known future-price", as above); see Spot–future parity. Thus, for a simple, non-dividend paying asset, the value of the future/forward, F(t)\,, will be found by accumulating the present value S(t)\, at time t\, to maturity T\, by the rate of risk-free return r\,. :F(t) = S(t)\times (1+r)^\, 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'': # The arbitrageur sells the futures contract and buys the underlying today (on the spot market) with borrowed money. # On the delivery date, the arbitrageur hands over the underlying, and receives the agreed forward price. # He then repays the lender the borrowed amount plus interest. # The difference between the two amounts is the arbitrage profit. * In the case where the forward price is ''lower'': # The arbitrageur buys the futures contract and sells the underlying today (on the spot market); he invests the proceeds. # On the delivery date, he cashes in the matured investment, which has appreciated at the risk free rate. # He then receives the underlying and pays the agreed forward price using the matured investment. short_the_underlying,_he_returns_it_now..html" ;"title="f he was short the underlying, he returns it now.">f he was short the underlying, he returns it now.# The difference between the two amounts is the arbitrage profit.


Swaps

Rational pricing underpins the logic of swap valuation. Here, two counterparties "swap" obligations, effectively exchanging
cash flow A cash flow is a real or virtual movement of money: *a cash flow in its narrow sense is a payment (in a currency), especially from one central bank account to another; the term 'cash flow' is mostly used to describe payments that are expected ...
streams calculated against a notional principal amount, and the value of the swap is the
present value In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has inte ...
(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 Swap (finance)#Valuation and Pricing. Once traded, swaps can (must) also be priced using rational pricing. The examples below are for
Interest rate swap In finance, an interest rate swap (IRS) is an interest rate derivative (IRD). It involves exchange of interest rates between two parties. In particular it is a "linear" IRD and one of the most liquid, benchmark products. It has associations with ...
s and is representative of pure rational pricing as it excludes
credit risk A credit risk is risk of default on a debt that may arise from a borrower failing to make required payments. In the first resort, the risk is that of the lender and includes lost principal and interest, disruption to cash flows, and increased ...
although the principle applies to any type of swap.


Valuation at initiation

Consider a fixed-to-floating Interest rate swap where Party A pays a fixed rate (" Swap 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: # Assume the position with the ''lower'' present value of payments, and borrow funds equal to this present value # Meet the cash flow obligations on the position by using the borrowed funds, and receive the corresponding payments—which have a higher present value # Use the received payments to repay the debt on the borrowed funds # 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

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 Bond or bonds may refer to: Common meanings * Bond (finance), a type of debt security * Bail bond, a commercial third-party guarantor of surety bonds in the United States * Chemical bond, the attraction of atoms, ions or molecules to form chemical ...
with the same schedule of payments. (Relatedly, given that their underlyings have the same cash flows, bond options and swaptions are equatable.) See Swap (finance)#Using bond prices.


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 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 In finance, a derivative is a contract that ''derives'' its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Derivatives can be use ...
, 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 Continuous production is a flow production method used to manufacture, produce, or process materials without interruption. Continuous production is called a continuous process or a continuous flow process because the materials, either dry bulk ...
. The examples below have shares as the underlying, but may be generalised to other instruments. The value of a
put option In finance, a put or put option is a derivative instrument in financial markets that gives the holder (i.e. the purchaser of the put option) the right to sell an asset (the ''underlying''), at a specified price (the ''strike''), by (or at) a ...
can be derived as below, or may be found from the value of the call using
put–call parity In financial mathematics, put–call parity defines a relationship between the price of a European call option and European put option, both with the identical strike price and expiry, namely that a portfolio of a long call option and a short pu ...
.


Arbitrage free pricing

Here, the future payoff is "locked in" using either "delta hedging" or the "
replicating portfolio In mathematical finance, a replicating portfolio for a given asset or series of cash flows is a portfolio of assets with the same properties (especially cash flows). This is meant in two distinct senses: static replication, where the portfolio has ...
" 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 or Calls may refer to: Arts, entertainment, and media Games * Call, a type of betting in poker * Call, in the game of contract bridge, a bid, pass, double, or redouble in the bidding stage Music and dance * Call (band), from Lahore, Paki ...
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 In finance, delta neutral describes a portfolio of related financial securities, in which the portfolio value remains unchanged when small changes occur in the value of the underlying security. Such a portfolio typically contains options and their ...
). 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. # Solve for Δ such that: #: value of position in one period = Δ × ''S up'' - max (''S up'' – strike price, 0) = Δ × ''S down'' - max (''S down'' – strike price, 0) # 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. # Solve simultaneously for Δ and B such that: #* Δ × ''S up'' - B × (1 + r) = \max (0, ''S up'' – strike price) #* Δ × ''S down'' - B × (1 + r) = \max (0, ''S down'' – strike price) # Solve for the value of the call, using Δ and B, where: #* call = Δ × ''S current'' - B Note that there is no discounting here 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 at each decision point, or whether, instead, a premium over risk free, differing by state, would be required. The best example of this would be under real options analysisSee Ch. 23, Sec. 5, in: Frank Reilly, Keith Brown (2011). "Investment Analysis and Portfolio Management." (10th Edition). South-Western College Pub. where managements' actions actually change the risk characteristics of the project in question, and hence the
Required rate of return The discounted cash flow (DCF) analysis is a method in finance of valuing a security, project, company, or asset using the concepts of the time value of money. Discounted cash flow analysis is widely used in investment finance, real estate devel ...
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'')...". See Real options valuation#Technical considerations. (Another case where the modelling assumptions may depart from rational pricing is the valuation of employee stock options.)


Risk neutral valuation

Here the value of the option is calculated using the
risk neutrality In economics and finance, risk neutral preferences are preferences that are neither risk averse nor risk seeking. A risk neutral party's decisions are not affected by the degree of uncertainty in a set of outcomes, so a risk neutral party is indif ...
assumption. Under this assumption, the "
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
" (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 The risk-free rate of return, usually shortened to the risk-free rate, is the rate of return of a hypothetical investment with scheduled payments over a fixed period of time that is assumed to meet all payment obligations. Since the risk-free ...
. # Solve for p #: under risk-neutrality, 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): #: \begin S &= \frac \\ &= \frac \\ \Rightarrow p &= \frac\\ \end # 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: #: \begin C &= \frac \\ &= \frac \\ \end


=The risk neutrality assumption

= 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 In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or '' equivalent martingale measure'') is a probability measure such that each share price is exactly equal to the discounted expectation of the share price u ...
as opposed to the actual
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
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.


Pricing shares

The
arbitrage pricing theory In finance, arbitrage pricing theory (APT) is a multi-factor model for asset pricing which relates various macro-economic (systematic) risk variables to the pricing of financial assets. Proposed by economist Stephen Ross in 1976, it is widely beli ...
(APT), a general theory of asset pricing, has become influential in the pricing of
shares In financial markets, a share is a unit of equity ownership in the capital stock of a corporation, and can refer to units of mutual funds, limited partnerships, and real estate investment trusts. Share capital refers to all of the shares of ...
. APT holds that the expected return of a financial asset can be modelled as a linear function of various
macro-economic Macroeconomics (from the Greek prefix ''makro-'' meaning "large" + ''economics'') is a branch of economics dealing with performance, structure, behavior, and decision-making of an economy as a whole. For example, using interest rates, taxes, an ...
factors, where sensitivity to changes in each factor is represented by a factor specific
beta coefficient In finance, the beta (β or market beta or beta coefficient) is a measure of how an individual asset moves (on average) when the overall stock market increases or decreases. Thus, beta is a useful measure of the contribution of an individual ...
: :E\left(r_j\right) = r_f + b_F_1 + b_F_2 + ... + b_F_n + \epsilon_j :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_ is the sensitivity of the asset to factor k, :* and \epsilon_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), 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 In finance, arbitrage pricing theory (APT) is a multi-factor model for asset pricing which relates various macro-economic (systematic) risk variables to the pricing of financial assets. Proposed by economist Stephen Ross in 1976, it is widely beli ...
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: # Today: short sell the ''portfolio'' and buy the mispriced-asset with the proceeds. # 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: # Today: short sell the mispriced-asset and buy the ''portfolio'' with the proceeds. # 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 In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a well-diversified portfolio. The model takes into ac ...
(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 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.


No-arbitrage pricing under systemic risk

Classical valuation methods like the Black-Scholes model or the Merton model cannot account for systemic counterparty risk which is present in systems with financial interconnectedness. More details regarding risk-neutral, arbitrage-free asset and derivative valuation can be found in the
systemic risk In finance, systemic risk is the risk of collapse of an entire financial system or entire market, as opposed to the risk associated with any one individual entity, group or component of a system, that can be contained therein without harming the ...
article (see also valuation under systemic risk).


See also

* *
Contingent claim analysis In finance, a contingent claim is a derivative whose future payoff depends on the value of another “underlying” asset,Dale F. Gray, Robert C. Merton and Zvi Bodie. (2007). Contingent Claims Approach to Measuring and Managing Sovereign Credit R ...
*
Covered interest arbitrage Covered interest arbitrage is an arbitrage trading strategy whereby an investor capitalizes on the interest rate differential between two countries by using a forward contract to ''cover'' (eliminate exposure to) exchange rate risk. Using forward ...
*
Efficient-market hypothesis The efficient-market hypothesis (EMH) is a hypothesis in financial economics that states that asset prices reflect all available information. A direct implication is that it is impossible to "beat the market" consistently on a risk-adjusted bas ...
* Fair value * Fundamental theorem of arbitrage-free pricing * Homo economicus * List of valuation topics *
No free lunch with vanishing risk No free lunch with vanishing risk (NFLVR) is a no-arbitrage argument. We have ''free lunch with vanishing risk'' if by utilizing a sequence of time self-financing portfolios, which converge to an arbitrage strategy, we can approximate a self-fina ...
*
Rational choice theory Rational choice theory refers to a set of guidelines that help understand economic and social behaviour. The theory originated in the eighteenth century and can be traced back to political economist and philosopher, Adam Smith. The theory postula ...
*
Rationality Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an ab ...
*
Risk-neutral measure In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or '' equivalent martingale measure'') is a probability measure such that each share price is exactly equal to the discounted expectation of the share price u ...
* Volatility arbitrage *
Systemic risk In finance, systemic risk is the risk of collapse of an entire financial system or entire market, as opposed to the risk associated with any one individual entity, group or component of a system, that can be contained therein without harming the ...
*Yield curve / interest rate modeling: ** ** Bootstrapping (finance) **
Multi-curve framework In finance, an interest rate swap (IRS) is an interest rate derivative (IRD). It involves exchange of interest rates between two parties. In particular it is a "linear" IRD and one of the most liquid, benchmark products. It has associations wi ...


References

{{reflist


External links

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Pricing by Arbitrage
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Mark Rubinstein Mark Edward Rubinstein (June 8, 1944 – May 9, 2019) was a leading financial economist and financial engineer. He was ''Paul Stephens Professor of Applied Investment Analysis'' at the Haas School of Business of the University of California, Be ...
,
Haas School of Business The Walter A. Haas School of Business, also known as Berkeley Haas, is the business school of the University of California, Berkeley, a public research university in Berkeley, California. It was the first business school at a public universit ...

Elementary Asset Pricing Theory
Prof. K. C. Border
California Institute of Technology The California Institute of Technology (branded as Caltech or CIT)The university itself only spells its short form as "Caltech"; the institution considers other spellings such a"Cal Tech" and "CalTech" incorrect. The institute is also occasional ...

The Notion of Arbitrage and Free Lunch in Mathematical Finance
Prof. Walter Schachermayer
No Arbitrage in Continuous Time
Prof. Tyler Shumway ;Risk neutrality and arbitrage free pricing
Risk Neutral Pricing in Discrete Time
(
PDF Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems. ...
), Prof. Don M. Chance
Risk-Neutral Probabilities Explained
Nicolas Gisiger
Risk-neutral Valuation: A Gentle IntroductionPart II
Joseph Tham
Duke University Duke University is a private research university in Durham, North Carolina. Founded by Methodists and Quakers in the present-day city of Trinity in 1838, the school moved to Durham in 1892. In 1924, tobacco and electric power industrialist Jam ...
;Application to derivatives
Option Valuation in the Binomial Model
(
archived An archive is an accumulation of historical records or materials – in any medium – or the physical facility in which they are located. Archives contain primary source documents that have accumulated over the course of an individual o ...
), Prof. Ernst Maug,
Rensselaer Polytechnic Institute Rensselaer Polytechnic Institute () (RPI) is a private research university in Troy, New York, with an additional campus in Hartford, Connecticut. A third campus in Groton, Connecticut closed in 2018. RPI was established in 1824 by Stephen Van ...

Pricing Futures and Forwards by Arbitrage Argument
Quantnotes

Investment Analysts Society of Southern Africa
The illusions of dynamic replication
Emanuel Derman and Nassim Taleb
Swaptions and Options
Prof. Don M. Chance Pricing Finance theories Mathematical finance Arbitrage Financial economics