bitrage relationships and, thus, arbitrage or relative value opportunities between the stock and the credit component ofthe bonds of the same firm. The general strategy is to buy low and sellhigh or buy the relatively undervalued asset and sell the relatively overvalued asset, while roughly maintaining market neutrality.More information in this area can be found in papers and presentations of Finkelstein, Lardy, et al. and the E2C (Equity to Credit) framework at CreditGrades.62Another example is the convertible bonds whereby the bond holdersare long call options on the underlying equity and can possibly strip offcheap equity volatilities. A further example is the counterparty creditrisk arbitrage that we discussed in the previous chapter whereby the market participants can arbitrage the funding cost between the cash (e.g.,bond) market and the derivatives market, among other opportunities.Other arbitrage opportunities include liquidity arbitrage wherebydealers make profits by turning illiquid assets into more liquid instruments by putting the illiquid assets into a pool that may be undivided ordivided into tranches.Instruments of this category include: ABS (Asset Backed Securities),MBS (Mortgage Backed Securities), CMO (Collateralized Mortgage Obligations), CLO (Collateralized Loan Obligations), CDO (CollateralizedDebt Obligations), synthetic CDO (or, typically, CSO or CollateralizedSwap Obligations), CD02 (or CDO of CDO), single tranche bespokeCSO. Putting the assets into a pool can diversify the specific or idiosyncratic risks of individual assets. Tranching can create different risk andreturn profiles from a pool. For instance, a senior tranche can have significantly reduced risk (such as the prepayment risk and the default risk).Particularly, one can create AAA (by S&P) or Aaa (by Moody’s) creditrated tranche or instrument fiom a pool of high yield or non-investmentgrade instruments.Using the same idea, an investment bank, typically A credit rated (by
62
Downloadable from:http://www.creditgrades.com/resources/pdf/CGtechdoc.pdf,http://www.creditgrades.com/resources/pdf/Finkelstein.pdf,http://www.creditgrades.com/resources/pdf/E2C_JPM_CDconference.pdf.
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S&P), can create AAA credit rated subsidiaries orwhich areoften required when trading with AAA or other high credit quality issuers or counterparties.A popular strategy in this category it to explore relative valuesamong different tranches. For instance, an investor can sell the creditprotection (or buy the credit risk) on the equity tranche of a CDO or CSOand buy the credit protection (or sell the credit risk) on the mezzaninetranche. This allows the investor to be roughly credit spread neutral andonly take the credit default risk.It is important to point out this strategy is highly dependent on theassumptions of the default correlation among the underlying creditnames in the CDO or CSO. It has been reported that various funds haveincurred significant losses due to the inaccurate or incorrect assumptionsof the correlation, and when the credit spread of the equity tranche widened, the credit spread of the mezzanine tranche did not widen as muchas predicted by the correlation assumption (used in the Copula model).The strategy of longing the bond and shorting the stock of the sameentity can also fail due to the inaccurate or incorrect assumptions of thecorrelation. A common challenge of this strategy is that the stock hasmuch higher price volatilities than the bond.2.4.9
Quasi-arbitrage Opportunities
As we discussed before, an arbitrage opportunity is whereby a portfoliowith zero present value (i.e., V(0)= 0 at time 0 or with no initial investment) and, through an SFTS, at a future time T, the portfolio has a positive expected value at time T (i.e., E[V(T)]> 0 or a profit) and zero probability for achieving negative values (i.e., P { V(7) < 0) = 0 or with norisk).A quasi-arbitrage opportunity is a slightly different situationwhereby a portfolio with zero present value (i.e., V(0) = 0 at time 0 orwith no initial investment) and, through an SFI'S, at a future time T, theportfolio has a positive expected value at time T (i.e,, E[V(T)]> 0 or aprofit) and non-zero probability for achieving negative values (i.e.,
63 This is often in the form of a DPC (Derivatives Product Company) sufficiently fundedto be rated as AAA or Aaa. The amount of capital needed is often determined by a Capital Adequacy Model approvedby rating agencies.
Sec. 2.4
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87
P ( V(7‘)c 0) > 0 or with some risks).
Exploring quasi-arbitrage opportunities is a critical part of the trading activities for various market participants including market makers ordealers and hedge funds.In a real market, when people talk about “arbitrage”, they typicallymean “quasi-arbitrage”. Stat arb (statistical arbitrage) is such an example, which we shall discuss more in the second part of this book.It would be interesting for the reader to think about the initial condition and the future outcome for hedging, speculation, buying or sellinginsurance, buying or selling lottery, gambling, etc.
2.4.70 Why Should Derivatives Instruments ExistDerivatives instruments are powerful and useful instruments that allowone to tailor one’s investment risk and potential return according to one’sview of the market and serve the needs of hedgers, speculators, and arbitrageurs to maximize their objective functions, as shown by the examplesof the trading strategies presented in the previous sections.Nonetheless, an interesting question in passing is that if the derivatives instruments can be completely replicated by other instruments, whyshould they exist?There are actually many categories of reasons to justify the existenceand demand for derivatives instruments mostly stemming from the imperfections in the real market. Here we provide a few examples.As we discussed in the previous section, dealers have comparativeadvantages in lowering the costs associated with (dynamically) replicating or hedging derivatives instruments, especially when there is a good“two-way” market. These costs include transaction costs and costs forresources for trading, modeling, risk management, systems, etc., neededfor a derivatives business. Thus, it is often more cost effective for market users to enter into derivatives transactions with dealers than to (dynamically) replicate the derivatives instruments themselves.Often times, for the purpose of maintaining a client relationship, amarket participant would like to enter off-balance sheet derivativestransactions to sell the risk associated with an asset, but without physically selling the asset. This is the motivation for the popular total returnswap (TRS).Derivatives instruments also allow a market participant to short aninstrument without the risk of being short squeezed. Furthermore, de-
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rivatives allow a market participant to achieve significant leveraging forpotentially higher returns (with potentially higher risks).In addition, there are many cases whereby derivatives instrumentscannot be completely replicated by other instruments. These cases include the existence of un-hedgeable variables and derivatives instrumentson non-traded variables as we discussed before, as well as jumps withlarge unknown or stochastic jump sizes in the market.For instance, if one were to hedge the downward movement of astock (or the stock market), one can buy a put option on the stock (or astock index). Alternatively, one can dynamically replicate the same putoption by selling progressively more shares of the stock (or the stockindex) as the price of the stock (or the stock index) decreases. Withoutjumps, the real put option and the dynamically replicated put optionwould basically have the same payoff. However, in the presence ofdownward jumps in the price of the stock (or the stock index) with largeunknown or stochastic jump sizes, one simply would not have time tosell enough shares of the stock (or the stock index) to accurately replicatethe put option, and, thus, the dynamically replicated put option can havemuch less payoff than the real put option, and, consequently, one cansuffer significant loss.This has happened on the Black Monday, October 19, 1987, whenthe stock market suddenly crashed. On that day, the S&P 500 Index lost20.5% and the Dow Jones Industrial Average (DJIA) fell 22.6%, thelargest one-day decline in the stock market history.Ever since then, the fear of downward jumps in the stock market hasbeen largely responsible to the volatility skew as the put option writers ordealers demand higher (and proper) risk premium for bearing this jumprisk.One popular hedging strategy is to enter into a zero-cost collar bybuying a put option financed by selling a call option on the same stock(or the same stock index)As we pointed out before, the jumps in various economic indicatorscan possibly be hedged with derivatives instruments created by a paramutuel process, For more information, the reader can refer to Lange andEconomides (200 1) and http://www.longitude.com.An example of dealers for these derivatives trades can be found at:http://www2 .goldmansachs.com/econderivs/index.html.
Sec. 2.5
2.5
Martingale Arbitrage Modeling
89
Martingale Arbitrage Modeling
As we shall see, the Harrison-Pliska martingale no-arbitrage theoremtogether with the martingale representation and the change of probabilitymeasure, which we shall term as martingale arbitrage modeling, essentially is a model factory for creating various arbitrage pricing models.The general procedures are as follows. Firstly, based on the businessneeds, identify underlying assets or variables of interest that are stochastic and find the martingale relationships and the martingale probabilitymeasures, Secondly, use the martingale representation theorem to deriveSDEs (Stochastic Differential Equations) for the value of the underlyingassets or variables and apply change of probability measure and, ifneeded, derive the corresponding PDEs (Partial Differential Equations).Thirdly, apply analytical techniques and/or numerical techniques (suchas finite difference grid, tree, or Monte Carlo simulation) to solve theSDEs or the PDEs to obtain the PV and hedge ratios. The last step ofteninvolves a calibration process that ensures that the model can price simple and liquid instruments that are used as hedging or replication instruments in an SFTS.There should also be some model validation procedures to exam theimplied views in the model on, e.g., volatility skews and smiles, correlation, future or forward volatilities, and future or forward volatility skewsand smiles, and future or forward correlation. Some of them are the unhedgeable variables that we discussed earlier.We usually start with a model with various simplified assumptions,and then add more functionality to it to handle various market imperfections. As pointed out before, we also need to price the counterpartycredit risk, and possibly the unexpected risks, such as price of VAR, aswell as the liquidity risk, and the hedging cost, some of which shall bediscussed later on. In addition, we also need to handle the complicationsof the real market, such as the un-hedgeable variables according to themarket consensus.It is important to point out that aside from pricing, a more importantresponsibility of a model is to produce reasonable hedges ratios (forhedging and/or replication).
2.5.1
Harrison-Pliska Martingale No-arbitrage Theorem
The relationship between martingale and arbitrage pricing was formulated by Harrison and Kreps (1979) and Harrison and Pliska (1981), and
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can be summarized as the following Harrison-Pliska martingale noarbitrage theorem.As usual, we start with various simplified assumptions whereby themarket is frictionless, sufficiently liquid, and without counterparty creditrisk. A frictionless market is one with no transactions costs (e.g., bidaskspread and tax) and with no restrictions on short selling (including fundborrowing or bond short selling). A sufficiently liquid market is onewhereby arbitrarily large transactions of an instrument do not affect theprice of the instrument. We will then extend the theorem to includesome market imperfections, particularly, the counterparty credit risk.Researchers have conducted research on arbitrage pricing with variousmarket imperfections, such as transaction cost and liquidity risk (See,e.g., Cetin, Jarrow, and Protter, 2003). In this book, we shall discuss selected practical applications in the area of counterparty credit risk.The Harrison-Pliska martingale no-arbitrage theorem: The necessary and sufficient condition for no-arbitrage is the existence of at leastone equivalent martingale probability measure (equivalent to the realworld probability measure) such that the relative prices of all traded assets or instruments relative to a numeraire or deflator asset (that is atraded asset with strictly positive price) are martingales. These relativeprices shall also be termed as numeraire-normalized or numerairediscounted prices. Here we assume that all these traded assets pay nointermediate cashflows or income, such as dividends, coupons), earlyexercise, or knock-in and knock-out features. If these traded assets dopay intermediate cashflows, then we need to decompose them into portfolios of traded assets paying no intermediate cashflows or income,which we shall discuss more later on.For additional discussions on Harrison-Pliska martingale noarbitrage theorem, the reader can refer to, e.g., Duffie (1996) particularlyon the numeraire invariance. Here we focus more on practical applications of the theorem.More specifically, the Harrison-Pliska martingale no-arbitrage theorem states that the necessary and sufficient condition for no-arbitrage isthe existence of a traded asset or instrument N (as the numeraire withstrictly positive price) and an equivalent martingale measure PN(equivalent to the real world measure) such that the numerairenormalized or numeraire-discounted prices of all traded assets X are martingales, i.e.,
Sec. 2.5
Martingale Arbitrage Modeling
91
(2.15)where X and N do not pay intermediate cashflows or income, such asdividends and coupons in the time interval [s, t], X,and N I are prices of
X and N at time t, respectively, and the expectation E,” [...] is taken under the probability measure PN conditioned on the information availableup to time s. PN is referred to as the equivalent martingale or arbitragefree measure with respect to numeraire N.Intuitively speaking, Eq. (2.15) indicates that if the price of one unitof instrument X i s the same as the price of x units of the numeraire instrument N at time s, then going forward, the future values of one unit ofinstrument X , on the expected value sense, is also the same as the futurevalues of x units of the numeraire instrument N. Thus, one cannot long(or invest) one unit of instrument X and short (or fund by shorting) xunits of N, or vice versa, to make a riskless profit with probability of one.The equivalent martingale measure is closely related to the SFTS forreplicating or hedging one traded instrument with other traded instruments. The reader can gain more intuitions on this with the classical binomial approach presented in the next section.If the equivalent martingale measure is unique, then all the instruments can be completely replicated or hedged with other instruments,and thus the market is complete. In this case, if the price of an instrument violates Eq. (2.15), then there is an arbitrage opportunity as themarket participants can arbitrage between the instrument and its SFTSreplicating portfolio by buying the less expensive one and selling themore expensive one and make a profit with probability of one. Even ifthe equivalent martingale measure is unique for a given numeraire, itsassociated SFTS for replicating or hedging a particular traded instrumentis normally not unique, as the traded instrument normally can be replicated or hedged in different ways or with different replicating portfolios.On the other hand, if the equivalent martingale measure is notunique, then there are un-hedgeable andor un-diversifiable variables.Thus, not all the instruments can be completely replicated or hedged withother instruments and, therefore, the market is incomplete.
2.5.2
Martingale Derivatives Pricing in a Binomial Economy
In order for the reader to build intuitions on martingale modeling, here
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we present analyses on martingale derivatives pricing in a binomialeconomy.A binomial economy is a simplified economy such that once itreaches any allowed state ( W, ) at any given time t, in the next time stept+dt, it can only reach two possible (children) states (say, an up ( o&~, )and a down (ci&) state), with urepresenting the set of all economicvariables, possibly including the history of these variables. The state utand its children statesand &At form a building block for theeconomy, which, in turn, can have a number of such building blocks. Asthe size of the time step dt approaches to 0, the binomial economy can bemade to converge to the continuous time economy under various technical conditions in the sense of weak c ~ n v e r g e n c e This. ~ ~ is the essence ofa binomial model.For simplicity and easiness to establish intuition, we only considerone such building block with the next time step t+dt coinciding with theexpiration date T of the derivatives instrument.Here we also assume the derivatives instrument is a simple contingent claim whereby there are no intermediate cashflows and there is onlyone cashflow at the expiration date T. In general, we assume that all theassets and instruments in this section do have intermediate cashflowsbefore the expiration date T.Let S, and C, denote the time t price of the underlying asset (or,more generally, a traded hedge instrument) and the derivatives claim atthe state q , and S; and C,X denote the time T price of the underlyingasset and the derivatives instrument at the state a$, where x = u or d forthe up and down state. These are shown schematically below. The terminal payoff C,X of the derivatives instrument is usually known andfinding the time t value of the derivatives instrument C, is the goal ofthe derivatives pricing.
&1 More precisely, multi-dimensional binomial building blocks are needed for modelingthe real economy with each dimension representing one source of economic uncertainty.We use 1-dimensional binomial economy here for simplicity so that intuitions and someinsights can be gained easily.
See. 2.5
93
Martingale Arbitrage Modeling
(a) The states of the
underlying asset.
(b) The states of thederivatives instrument.
Figure 2.1. The real world process of the underlying asset (a) and the payoffs ofthe derivatives claim (b) in a binomial economy. The time t value of the derivatives claim C, is yet to be determined.
Obviously, the transition probabilities, p,, and pd, to the up and downstate satisfy
o<pu<l, o < p d < 1 , Pu+Pd=l.
(2.16)
These probabilities and the economic states define the real worldprobability measure P . The conditional expected value (conditioned onthe information available up to time t) under this measure P is definedas(2.17)where YTrepresents the value any economic variable, such as ST and CT,at time T.From the previous discussions, we know that the price of a derivatives instrument is not determined by the above expected value, butrather by arbitrage pricing. We will proceed to find the arbitrage price ofthe derivatives instrument in question in this binomial economy settingthrough portfolio replication.To order to proceed further, we also need a traded numeraire instrument (with strictly positive price and no intermediate cashflows beforethe expiration date 2“). Intuitively, the numeraire instrument is for funding and investment purpose in an SFTS. In other words, one is long (orbuys) the derivatives receivables, then one needs to short the numeraireinstrument to fund the derivatives receivables. On the other hand, if oneis short (or sells) the derivatives receivables, then one needs to invest the
94
Martingale Arbitrage Pricing in Real Market
Chapter 2
proceeds in the numeraire instrument. Common examples of the numeraire instrument include money market accounts, zero-coupon bonds,and annuities.Similar to before, let N1 denote the time t price of the numeraireinstrument and N i denote the time T price of the numeraire instrumentat the state o$ ,where x = u or d for the up and down state.With all these, we can easily replicate the derivatives instrument withthe underlying or hedge instrument and the numeraire instrument througha buyhell and hold SFTS. This means that we need to find the numberof units or the amount, 4sand &, for the underlying or hedge instrumentand the numeraire instrument that can replicate the terminal payoffs ofthe derivatives instrument, i.e.,(2.18)
or equivalently,(2.19)= S $ / N : , then S and N are not independent, and we needIfanother instrument to replicate the derivatives instrument. So, withoutlosing generality, we consider the case that S; / N ; # S,d I N $ . Solvingthe above equations in this case yields
4s =
(2.20)
4 N =c;/N;-$Ss;/N;
The reader may recognize that +s is really the Delta hedge ratio.Thus, the time t arbitrage price of the derivatives instrument is givenby the time t price of its replication portfolio, i.e.,
Cl = 4SSt + 4”t
-
(2.2 1)
We can rewrite the above equation so that it is closer to the HarrisonPliska martingale relationship, i.e.,
Sec. 2.5
Martingale Arbitrage Modeling
95
(2.22)where(2.23)q u =l-qdThe reader may recognize that the above equations are just simple linearinterpolation of the numeraire-normalizedprices.In order for there to be no-arbitrage between the underlying or hedgeinstrument and the numeraire instrument, we mush haveOIq, 51, 0 5 % 5 1 .
(2.24)
This means that qu and q d can be interpreted as probabilities. Furthermore, there is another important condition, i.e., ifp, = 0 (or l), then q, =0 (or l), and vice versa. This means that they agree on what is possibleand what is not possible. The details are left to the reader as a simpleexercise.All these mean that qu and q d together with the state variables definea probability measure that is equivalent to the real world probabilitymeasure P . We shall denote this new probability measure as PN (withrespect to numeraire N).Based on Eq. (2.22), it interesting to see that the numerairenormalized derivatives price C, INT satisfy the following martingalerelationship(2.25)where the expectation is taken the probability measure P,,,. Thus, P,,,issame as the equivalent martingale or arbitrage-free measure referencedthe Harrison-Pliska martingale no-arbitrage theorem, and we have obtained an example of such probability measure by construction.So far, we have ensured that there is no-arbitrage among the underlying or hedge instrument, the numeraire instrument, and the derivativesinstrument. In the same derivatives instrument can be hedged or replicated by many hedge instruments (which also be derivatives instruments
96
Martingale Arbitrage Pricing in Real Market
Chapter 2
themselves). Thus, to ensure no-arbitrage among all these instruments,intuitively speaking, the probability measure (or qu and q d ) need to beindependent of the hedge instruments. Since they are independent of thederivatives instrument according to Eq. (2.23), they can only depend onthe numeraire instrument.Since the underlying or hedge instrument can be conceptuallythought of as a derivatives instrument on itself (such as a zero-strike forward contract), then we should have(2.26)
which can be easily verified by Eq. (2.22) and (2.23).However, in a real market, it often necessary for the derivatives instrument and the underlying instrument to have different numeraires (ordifferent funding) in the martingale relationships, which we shall discussmore later on.2.5.3
Harrison-Pliska Martingale No-arbitrage Theorem for Assets withhtermediate Cashflows or h o m e
The Harrison-Pliska no-arbitrage theorem can also be applied to thecases where the assets X and N do pay intermediate cashflows or income,such as dividends (including foreign interest rate payments) or coupons.Commodities that incur storage costs, convenience yield, etc., can betreated as assets paying (negative or positive) dividends or coupons.Some of the credit risk and liquidity risk can also be handled by a dividend yield.In general, if assets X and N do pay intermediate cashflows or income, then they can be decomposed into portfolios of assets paying nocoupons before certain time. Thus, the martingale relationship can begeneralized to(2.27)where X, and Nt are prices of X and N at time t, respectively, including all the cashflows paid on or after time T and excluding all thecashflows before time T. The above equation is very useful for pricingsimple swaps and bonds and their European style derivatives. In general,this is useful for handling intermediate cashflows or income that are notpath (or history) dependent.
Sec. 2.5
Martingale Arbitrage Modeling
97
Another approach is to include all the cashflows or income (ratherthan excluding them). This is useful for handling intermediate cashflowsor income that are path (or history) dependent. Bermudan or Americanstyle derivatives typically have path-dependent and stochastic intermediate cashflows or income that depend on whether or not an option hasbeen exercised or an event has occurred in the past. Sometimes, ratherthan modeling the value of an asset, it is more convenient to model thevalue of the portfolio that includes the asset and all the intermediatecashflows or income generated by the asset. This is also the essence ofan SFTS. Assume X , is the time t value of such portfolio and the numeraire asset N does not pay intermediate cashflows or income, then X ,satisfies the original Harrison-Pliska martingale relationship or Eq.(2.15).A specific case is that the numeraire asset N does not pay intermediate cashflows, but the numerator asset X , such as a stock, pays a dividendcharacterized by a continuous dividend yield d, .Define the dividend multiplier as(2.28)
then X , 0, includes all the intermediate dividend income up to time t andthus should be used in the martingale relationship, i.e.,X,D,/N, = E," [ X,D,/N,]
2.5.4
(0 5 s It IT ) .
(2.29)
Foundation for Arbitrage Pricing
The Harrison-Pliska no-arbitrage theorem provides a foundation for arbitrage pricing. For instance, suppose X is a simple derivatives asset or aEuropean style derivatives instrument with only one pre-specified payment X , (that can depend on future states). Then, we can get its PV ortoday's price through
(2.30)as long as we know the stochastic or dynamic processes of X , and N , ,and the measure PN . Finding the stochastic or dynamic processes willbe the tasks of the following sections.
Martingale Arbitrage Pricing in Real Market
98
Chapter 2
Often times, a convenient choice of numeraire asset is a default-freezero-coupon bond with maturity of T whose time t price ist 5 2'). In this case, the arbitrage pricing can be simplifiedP ( t , T ) (0 Ito
xo = P(oyr)E,' [ X T ]
(0 It 5 2')
y
(2.31)
as P ( T , T ) = 1 (which is an assumption that we shall revisit in a realmarket).As pointed out in the previous section that derivatives pricing involving assets with intermediate cashflows or income can also be achievedwith the Harrison-Pliska no-arbitrage theorem.2.5.5
Examples of Martingales and Equivalent Martingale Measures
A consequence of martingale arbitrage derivatives pricing is the arbitrage-free valuation. Thus, the aforementioned equivalent martingalemeasures (with respect to the numeraire N) are also referred to as arbitrage-fiee measures (with respect to the numeraire N).The martingale measure with respect to the money market account,8( t ) = exp
( r du) as the numeraire is referred to as the spot arbi(u)
trage-free measure Q , and the martingale measure with respect to thezero-coupon bond P(t,T) as the numeraire is referred to as the forwardarbitrage-free measure (of T ) PT ,The martingale measure with respect to the annuityN8;-lP(
Aj,N(t)=
Is) as the numeraire is referred to as the annuity
t ,T .
j=i+l
arbitrage-free measure PA(i,N).There is also a rolling spot arbitrage-j?ee measure or spot LIBORmeasure where the numeraire is a series of zero-coupocbonds. The numeraire initially is the bond with the shortest maturity, and as the timeevolves and the bond with the shortest maturity matures, then the bondwith the next shortest maturity becomes the numeraire, and so on.Aside from the general martingale relationships that we discussedpreviously, some quantity is naturally expressed as a ratio of two tradedassets and, thus, is naturally a martingale under the measure with respectto the numeraire asset. Examples of such quantities are as follows.
Sec. 2.5
99
Martingale Arbitrage Modeling
The forward price F, ( t , T ) of a traded asset X, with no cashflowsbefore time t, i.e.,
F, ( t , T )= X , / P ( t , T )
(0 I t I T ) ,
(2.32)
(0 I s I t I T ) .
(2.33)
(oltlq),
(2.34)
is a PT-martingale. In other words,
F, ( s , T )= E,' [F, ( t , T ) ]The forward LIBOR Lj ( t ) given by
l+s,L;(t)=P(t,q)lP(t,q+,)
is a PT+,-martingale.The forward swap rate given by
is a PA,,N-martingale.The above approach is more or less to find a martingale quantity orvariable given a probability measure (by constructing the ratio of twotraded assets). We can also find the martingale measure for a given variable. For instance, we can ask what is the martingale measure for thetime t futures price @(t,T) with maturity T and continuous settlement?Unlike the forward price, it turns out to be the spot arbitrage-free measure Q . In other words,
@(s,T)=E:[@(t,T)]
(OIsItIT).
(2.36)
It is worthwhile mentioning that the futures price itself is not a tradedasset, but a futures contract (whose payoff depends on htures price) is.A futures contract can have cashflows at every settlement time and has aPV of zero at every settlement time. In other words,
O=EF
[J:;is;
&@(t,T)
]
(OlsItIT).
(2.37)
We can see that Eq. (2.36) is a sufficient condition of Eq. (2.37),since
100
Martingale Arbitrage Pricing in Real Market
EF [@(t+ At, T)-@(t,T ) ]= 0
(0ItIt + At I T ) ,
Chapter 2
(2.38)
thus,
=O(0I t I t
+ At I T )
It can be proved that Eq. (2.36) is also a necessary condition of Eq.(2.37).Another useful martingale relationship is a special case of Eq. (2.15)by letting X t = P( t ) , i.e.,P ( s ) / N , = E," [m/N,]( 0 5 s 5 t > .
(