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| + | '''Pricing racional''' é o pressuposto usado em finanças de que os preços dos [[activo]]s (e portanto, os modelos de preços) reflectem o preço ao qual é impossível a [[arbitragem]], visto que qualquer desvio deste preço será arbitrado até desaparecer. Este pressuposto é útil para avaliar instrumentos de rendimento fixo, particularmente [[obrigação|obrigações]], e é fundamental na avaliação de instrumentos [[derivados]]. |
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| − | '''Rational pricing''' é o presuposto usado em finanças de que os preços dos activos (e portanto, os modelos de preços) reflectem o preço ao qual é impossível a [[arbitragem]], visto que qualquer desvio deste preço será arbitrado até desaparecer. Este pressuposto é útil para preçar instrumentos de rendimento fixo, particularmente [[obrigações]], e é fundamental no preçar de instrumentos [[derivados]].
| + | ==Mecânica da Arbitragem== |
| | + | [[Arbitragem]] é a prática de retirar um benefício de uma ineficiência entre dois ou mais mercados. Onde esta ineficiência exista e possa ser explorada, a arbitragem obtém - após custos de transacção, armazenagem, transporte, dividendos, etc - um lucro sem risco e potencialmente sem investir nenhum capital. |
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| − | ==Arbitrage mechanics==
| + | De forma geral, a arbitragem garante que a "lei de um só preço" será mantida. A arbitragem também faz com que os preços de activos com os mesmos [[cash flows]] sejam iguais, e estabelece o preço de activos cujos cash flows futuros sejam conhecidos. |
| − | [[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. | + | |
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| − | 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.
| + | ===A lei de um só preço=== |
| | + | O mesmo activo tem que transaccionar ao mesmo preço em todos os mercados. Onde isto não for verdade, o arbitragista: |
| | + | #Comprará o activo no mercado que tem o preço mais baixo, e simultaneamente vende curto o activo no mercado com o preço mais elevado; |
| | + | #Entrega o activo ao comprador recebendo o preço mais elevado; |
| | + | #Paga o activo so vendedor ao preço mais baixo, e ganha a diferença. |
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| − | ===The law of one price=== | + | ===Activos com cash flows idênticos=== |
| − | The same asset must trade at the same price on all markets ("the [[law of one price]]").
| + | Dois activos com cash flows idênticos têm que transaccionar ao mesmo preço. Onde isto não for verdade, o arbitragista: |
| − | Where this is not true, the arbitrageur will:
| + | #Compra o activo com o preço mais baixo, e simultaneamente vende curto o activo com o preço mais elevado. |
| − | # buy the asset on the market where it has the lower price, and simultaneously sell it ([[short selling|short]]) on the second market at the higher price | + | #Financia a sua compra do activo mais barato com o resultado da venda do activo mais caro e fica com a diferença; |
| − | # deliver the asset to the buyer and receive that higher price | + | #Paga os cash flows ao comprador do activo mais caro, usando o cash flow do activo mais barato. |
| − | # pay the seller on the cheaper market with the proceeds and pocket the difference. | + | |
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| − | ===Assets with identical cash flows=== | + | ===Activo com preço futuro conhecido=== |
| − | Two assets with identical cash flows must trade at the same price.
| + | Um activo com um preço futuro conhecido, tem que transaccionar hoje a esse preço, descontado para o presente à [[taxa de juro sem risco]]. |
| − | Where this is not true, the arbitrageur will:
| + | |
| − | # sell the asset with the higher price ([[short selling|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.
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| − | | + | |
| − | ===An asset with a known future-price===
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| − | An asset with a known price in the future, must today trade at that price [[discount]]ed at the [[Risk-free interest rate|risk free rate]].
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| − | 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.
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| − | (a) where the discounted future price is ''higher'' than today's price:
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| − | # The arbitrageur agrees to deliver the asset on the future date (i.e. [[Forward contract|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 owed is the arbitrage profit.
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| − | | + | |
| − | (b) where the discounted future price is ''lower'' than today's price:
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| − | # The arbitrageur agrees to pay for the asset on the future date (i.e. [[Forward contract|buys forward]]) and simultaneously sells ([[Short selling|short]]) the underlying today; he invests 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.
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| − | # The difference between the maturity value and the agreed price is the arbitrage profit.
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| − | | + | |
| − | 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]].
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| − | ==Fixed income securities==
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| − | Rational pricing is one approach used in pricing [[fixed rate bond]]s. Here, each cash flow can be matched by trading in some multiple of a "risk free" government issue [[zero coupon bond]] with the corresponding maturity, or in a corresponding [[Zero coupon bond#Strip bonds|strip]] and ZCB.
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| − | 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 the corresponding government securities - i.e. the corresponding [[risk free rate]] (here, assuming similar [[credit worthiness]]). 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.
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| − | The pricing formula is as below, where each cash flow <math>C_t\,</math> is discounted at the rate <math>r_t\,</math> which matches that of the corresponding government zero coupon instrument:
| + | |
| − | :Price = <math> P_0 = \sum_{t=1}^T\frac{C_t}{(1+r_t)^t}</math>
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| − | | + | |
| − | Often, the formula is expressed as <math> P_0 = \sum_{t=1}^ T C(t) \times P(t)</math>, using prices instead of rates, as prices are more readily available.
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| − | :''See [[Fixed income arbitrage]]; [[Bond valuation]]; [[Bond credit rating]].''
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| − | ==Pricing derivatives==
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| − | A [[derivative (finance)|derivative]] is an instrument which allows for buying and selling of the same asset on two markets – the [[spot price|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.
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| − | :''see: [[Fundamental theorem of arbitrage-free pricing]]''
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| − | ===Futures===<!-- This section is linked from [[Contango]] -->
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| − | 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, <math>F(t)\,</math>, will be found by discounting the present value <math>S(t)\,</math> at time <math>t\,</math> to maturity <math>T\,</math> by the rate of risk-free return <math>r\,</math>.
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| − | | + | |
| − | :<math>F(t) = S(t)\times (1+r)^{(T-t)}\,</math>
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| − | This relationship may be modified for storage costs, dividends, dividend yields, and convenience yields; see [[Futures contract#Pricing|futures contract pricing]].
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| − | Any deviation from this equality allows for arbitrage as follows.
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| − | *In the case where the forward price is ''higher'':
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| − | # 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.
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| − | | + | |
| − | *In the case where the forward price is ''lower'':
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| − | # The arbitrageur buys the futures contract and sells the underlying today (on the spot market); he invests the proceeds.
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| − | # 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. [If he was [[short selling|short]] the underlying, he returns it now.]
| + | |
| − | # The difference between the two amounts is the arbitrage profit.
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| − | ===Options===
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| − | 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 (finance)|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''.
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| − | 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 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 value]]s at the later two nodes.
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| − | 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 [[Black-Scholes#The Black.E2.80.93Scholes PDE|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 [[Code (computer programming)|coded]] correctly), while Black-Scholes, in fact, models a [[Continuous-time Markov process|continuous process]].
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| − | 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]].
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| − | ====Arbitrage free pricing====
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| − | 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.
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| − | =====Delta hedging=====<!-- This section is linked from [[Black–Scholes]] -->
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| − | It is possible to create a position consisting of '''Δ''' [[call option|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 certain 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.
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| − | 1) Solve for Δ such that:
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| − | : value of position in one period = ''S up'' - Δ × (''S up'' – strike price ) = ''S down'' - Δ × (''S down'' – strike price)
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| − | 2) solve for the value of the call, using Δ, where:
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| − | : value of position today = value of position in one period ÷ (1 + r) = ''S current'' – Δ × value of call
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| − | =====The replicating portfolio=====
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| − | 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.
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| − | 1) Solve simultaneously for Δ and B such that:
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| − | : i) Δ × ''S up'' - B × (1 + r) = [[Option time value#Intrinsic value|MAX]] ( 0, ''S up'' – strike price )
| + | |
| − | : ii) Δ × ''S down'' - B × (1 + r) = [[Option time value#Intrinsic value|MAX]] ( 0, ''S down'' – strike price )
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| − | 2) solve for the value of the call, using Δ and B, where:
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| − | : call = Δ × ''S current'' - B
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| − | ====Risk neutral valuation====<!-- This section is linked from [[Black–Scholes]] -->
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| − | Here the value of the option is calculated using the [[Risk-neutral measure|risk neutrality]] assumption. Under this assumption, the “[[expected value]]” (as opposed to "locked in" value) is [[discounted]]. The expected value is calculated using the [[Option time value#Intrinsic value|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 interest rate|risk free rate]].
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| − | 1) solve for p
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| − | : for no arbitrage to be possible in the share, today’s price must represent its expected value discounted at the risk free rate:
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| − | :S = [ p × (up value) + (1-p) ×(down value) ] ÷ (1+r) = [ p × S × u + (1-p) × S × d ] ÷ (1+r)
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| − | :then, p = [(1+r) - d ] ÷ [ u - d ]
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| − | 2) solve for call value, using p
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| − | : for no arbitrage to be possible in the call, today’s price must represent its expected value discounted at the risk free rate:
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| − | :Option value = [ p × Option up + (1-p)× Option down] ÷ (1+r)
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| − | := [ p × (''S up'' - strike) + (1-p)× (''S down'' - strike) ] ÷ (1+r)
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| − | =====The risk neutrality assumption=====
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| − | 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.
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| − | ===Swaps===
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| − | Rational pricing underpins the logic of [[Swap (finance)|swap]] valuation. Here, two [[Counterparty|counterparties]] "swap" obligations, effectively exchanging [[cash flow]] streams calculated against a notional [[:wikt:principal|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.
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| − | ====Valuation at initiation====
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| − | To be arbitrage free, the terms of a swap contract are such that, initially, the [[Net present value|''Net'' present value]] of these future cash flows is equal to zero; see [[Swap (finance)#Valuation|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 [[Arbitrage]]ur, C, could:
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| − | # assume the position with the ''lower'' present value of payments, and borrow funds equal to this present value
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| − | # meet the cash flow obligations on the position by using the borrowed funds, and receive the corresponding payments - which have a higher present value
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| − | # use the received payments to repay the debt on the borrowed funds
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| − | # pocket the difference - where the difference between the present value of the loan and the present value of the inflows is the arbitrage profit.
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| − | ====Subsequent valuation====
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| − | 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]]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 (finance)|Bond]] with the same schedule of payments.
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| − | ==Pricing shares==
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| − | The [[Arbitrage pricing theory]] (APT), a general theory of asset pricing, has become influential in the pricing of [[stock|shares]]. APT holds that the [[expected return]] of a financial asset, can be modelled as a [[linear function]] of various [[macroeconomics|macro-economic]] factors, where sensitivity to changes in each factor is represented by a factor specific [[beta coefficient]]:
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| − | :<math>E\left(r_j\right) = r_f + b_{j1}F_1 + b_{j2}F_2 + ... + b_{jn}F_n + \epsilon_j</math>
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| − | | + | |
| − | :where
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| − | :* <math>E(r_j)</math> is the risky asset's expected return,
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| − | :* <math>r_f</math> is the [[risk free rate]],
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| − | :* <math>F_k</math> is the macroeconomic factor,
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| − | :* <math>b_{jk}</math> is the sensitivity of the asset to factor <math>k</math>,
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| − | :* and <math>\epsilon_j</math> is the risky asset's idiosyncratic random shock with mean zero.
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| − | 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 [[discount]]ed 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 [[Arbitrage pricing theory#Arbitrage mechanics|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:
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| − | *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:
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| − | #Today: [[short selling|short sell]] the ''portfolio'' and buy the mispriced-asset with the proceeds.
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| − | #At the end of the period: sell the mispriced asset, use the proceeds to buy back the ''portfolio'', and pocket the difference.
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| − | *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:
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| − | #Today: [[short selling|short sell]] the mispriced-asset and buy the ''portfolio'' with the proceeds.
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| − | #At the end of the period: sell the ''portfolio'', use the proceeds to buy back the mispriced-asset, and pocket the difference.
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| − | 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.
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| − | 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 [[Modern portfolio theory#Securities Market Line|Securities market line]] represents a single-factor model of the asset price, where Beta is exposure to changes in value of the Market.
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| | ==Ver também== | | ==Ver também== |
| | + | *[[Arbitragem]] |
| | *[[Teoria do mercado eficiente]] | | *[[Teoria do mercado eficiente]] |
| | *[[Fair value]] | | *[[Fair value]] |
| − | *[[Teorema dos preços arbitrage-free]] | + | *[[Teorema dos preços livres de arbitragem]] |
| | *[[Homo economicus]] | | *[[Homo economicus]] |
| − | *[[Arbitragem de volatilidade]] | + | *[[Delta hedging]] |
| | | | |
| | ==Links relevantes== | | ==Links relevantes== |
| | *[http://cepa.newschool.edu/het/essays/sequence/arbitpricing.htm Pricing by Arbitrage], The History of Economic Thought Website | | *[http://cepa.newschool.edu/het/essays/sequence/arbitpricing.htm Pricing by Arbitrage], The History of Economic Thought Website |
| | *[http://www.quantnotes.com/fundamentals/basics/arbitragepricing.htm The Idea Behind Arbitrage Pricing], Quantnotes | | *[http://www.quantnotes.com/fundamentals/basics/arbitragepricing.htm The Idea Behind Arbitrage Pricing], Quantnotes |
| − | *[http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN96-02.pdf Risk Neutral Pricing in Discrete Time] ([[portable document format|PDF]]), Prof. Don M. Chance | + | *[http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN96-02.pdf Risk Neutral Pricing in Discrete Time] (PDF), Prof. Don M. Chance |
| | *[http://www-personal.umich.edu/~shumway/courses.dir/f872.dir/noarb.pdf No Arbitrage in Continuous Time], Prof. Tyler Shumway | | *[http://www-personal.umich.edu/~shumway/courses.dir/f872.dir/noarb.pdf No Arbitrage in Continuous Time], Prof. Tyler Shumway |
| | *[http://www.fam.tuwien.ac.at/~wschach/pubs/preprnts/prpr0118a.pdf The Notion of Arbitrage and Free Lunch in Mathematical Finance], Prof. Walter Schachermayer | | *[http://www.fam.tuwien.ac.at/~wschach/pubs/preprnts/prpr0118a.pdf The Notion of Arbitrage and Free Lunch in Mathematical Finance], Prof. Walter Schachermayer |
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| − | [[Categoria:Teorias]] | + | |
| | + | [[Categoria:Teorias]][[Categoria:Arbitragem]] |
De forma geral, a arbitragem garante que a "lei de um só preço" será mantida. A arbitragem também faz com que os preços de activos com os mesmos cash flows sejam iguais, e estabelece o preço de activos cujos cash flows futuros sejam conhecidos.
O mesmo activo tem que transaccionar ao mesmo preço em todos os mercados. Onde isto não for verdade, o arbitragista:
Dois activos com cash flows idênticos têm que transaccionar ao mesmo preço. Onde isto não for verdade, o arbitragista:
Um activo com um preço futuro conhecido, tem que transaccionar hoje a esse preço, descontado para o presente à taxa de juro sem risco.
