ies) and can be a significant percentage of the overall P&L of the derivatives desks, especiallyfor the interest rate derivatives desk, as it tends to have longer datedtrades and tighter bidask spread, and the commodities desk, as it tends tohave lower credit quality counterparties and longer dated trades.The total value of the net credit charges at the firm level can also besignificant, as derivatives are highly leveraged3' and large amount oftransactions in terms of notional can be executed with relatively less30 Such exposure reduction is less significant in the case of a cross-currency swap whichnormally involves changing of the principal at end (coupled with the additional FX volatility).3 1 For instance, one can enter a par swap with zero initial cost (except, in some cases,possibly with initial margins and collaterals).
30
Introduction to Counterparty Credit Risk
Chapter 1
amount of capital.
1.5
Prevailing Strategies for Counterparty Credit Risk Management
To completely eliminate one’s market risk on a trade (with one counterparty), one can enter an exact offsetting trade (with another counterparty). Without considering the counterparty credit risk, a dealer couldnot be happier, if he or she can do so and still captures some bidaskspread. However, this strategy alone would not work in general for managing the counterparty credit risk, since if one of the counterparties defaults, one can be left with obligations to the other counterparty. In otherwords, this strategy makes the dealer market risk neutral, but not creditrisk neutral.
Question 1.4. After one enters into offsetting trades with two separatecounterparties, what would happen exactly if one or both counterpartiesdefault?Counterparty credit risks are commonly managed by credit enhancement strategies, credit exposure modeling and management strategies, and net credit charge management strategies (including employingcredit derivatives). Credit enhancement strategies and credit exposureand modeling management strategies are the first line of defense in managing counterparty risks.Credit enhancement strategies include netting, margin or collateral,and credit contingency agreements, and mutual termination options, etc.If two parties have a netting agreement, then only the net MTM values of all the trades covered by the netting agreement is at risk when default happens. Since the trading parties often have many trades witheach other and often these trades can partially offset one another, nettingagreement can significantly reduce the exposure, and both potential andexpected credit loss. Such netting agreement (which is often in the formof a master agreement as opposed to trade-level agreement or trade confirm) can cover trades across different desks and even different productareas.The margin or collateral agreement allows the trading party who hasnet receivables (or positive MTM values or is owed money) to collect
Sec. 1.5
Prevailing Strategiesfor Counterparty Credit Risk Management
31
assets from the one with net payables as collaterals (based on pre-agreedtriggers), which, in the event of default, can be liquidated to reduce orcover the default loss. The party with weaker credit quality may alsoneed to post initial margins at the time when the trade is executed (andthe two parties may not have net receivables from each other) to coverpotential future market moves when the party with weaker credit qualitycan have net payables (or owe money). Similar to the netting agreement,such margin or collateral agreement (which is often in the form of a master agreement as opposed to trade-level agreement or trade confirm) cancover trades across different desks and even different product areas.
Question 1.5. There is a time lag between a collateral call and actuallyreceiving it. What effects does this have?The credit contingency agreement provides the trading parties options to take actions contingent upon credit events. For instance, oneparty can have the option to terminate the trades at their market values ifthe other party’s credit rating is below a certain level. Another exampleis the credit contingent collateral agreement which specifies the collateralthat one needs to post is a decreasing function of one’s credit rating, i.e.,more collateral if one’s credit rating is lowered.
Question 1.6, Why is the credit contingency option valuable and how toquantify it?The mutual termination agreement grants the trading parties optionsto terminate the trades at pre-specified dates at the market value. Thisessentially reduces the lifetime of the trades and the variance of theirMTM values, and thus reduces the credit exposures.
Question 1.7. What is the relationship between the variance of MTMvalues of the trade and the corresponding credit exposures?There are other strategies for reducing counterparty risks, such as byrestructuring or recouponing the trades that can be an alternative to themargin or collateral agreement and essentially turns part of the future
32
Introduction to Counterparty Credit Risk
Chapter 1
receivables into current cash payments.Credit exposure and modeling management strategies involve modeling or estimating potential exposures (PE) and setting a limit of allowedmaximum potential exposures (MPE) to each counterparty after considering the effect of the above strategies. Similar to market risk VAR(value at risk) for characterizing the market risk in a mathematical sense,PE of a portfolio is defined as its MTM values at a certain confidencelevel, e.g., at 95% level, which means at 95% chance the MTM of theportfolio would not exceed the PE.32 MPE is the maximum potentialexposure over the life of the portfolio. By setting the PE limit for eachcounterparty, one ensures not to be overly exposed to particular counterparties, and hopefully can have a diversified portfolio.
Question 1.8. Normally, the PE is lower considering the collateral effectthan the PE without collateral with everything else being the same. Arethere any exceptions to this statement?Question 1.9. What are the major differences between PE and marketrisk VAR and credit risk VAR?In order to better quantify whether or not one has a diversified portfolio, one needs to model and set limit for PE for each industry, each rating, each geographical region, so that one is not overly exposed to any ofthese categories.
Question 1.10. To compute the PE of an industry, can we simply aggregate the PEs of all the counterparties in that industry, and if so, why?There are various coherent risk measures that are additive.Another way of measuring the counterparty credit risk is the creditVAR and/or the economic capital, which is a step further than PE andmeasures the credit loss at a given percentile.PE is concerned with the positive MTM (or in the money) scenarios in which case onecan potentially suffer from credit risks and VAR is concerned with the negative (or out ofthe money) scenarios in which case one can potentially suffer from market risks. VARtends to have a short time horizon (such as one day and two weeks) and PE tends to havea time horizon up to the life time of the portfolio. Also see the answer to Question 1.9.32
Sec. 1.6
Wrong-way and Right-way Exposures or Trades
33
Other strategies of managing counterparty credit risk include favorable structuring of the cashflows and the timing of the cashflows in theunderlying trades. For instance, rather than entering a derivatives trade(that is unfunded), one can enter into a funded trade by issuing a structured note (including credit linked note) with the payoffs linked to thoseof the derivatives trade. This introduces two additional cashflows, one atthe start of the trade and one (the principal) at the maturity of the trade.This can often completely eliminate one’s counterparty credit risk, as onewould receive a premium close to the principal of the note at the start ofthe trade and would often have net payables (or owe money) going forward. This type of trade is feasible if one has a high credit rating in absolute terms or much higher credit rating relative to one’s counterparty.The funded placement of a CDO tranche uses the same strategy.In some cases, this funded trading is more difficult for one’s counterparty than a derivatives trade, which involves no or much less initialcashflows, which, in turn, is a reason for the popularity of the derivatives. But, in the funded trading, one’s counterparty does get the benefitof cheapening in the trade with zero or negative credit premium. Also, insome cases, one’s counterparty may not be allowed to engage in derivatives transactions.Similarly, as for the timing of the cashflows, in order to reduce one’scounterparty credit risk, one can structure the trades such that one willreceive the receivables sooner and pay the payables later. For instance,when buying an option, one can buy an option with premium paid in arrears or a contingent premium option.
1.6
Wrong-way and Right-way Exposures or Trades
Credit exposure and modeling and management strategies also involvecategorizing exposures into wrong-way exposure and right-way exposureand providing certain quantification for them. These are essentially related to the correlation between the market risk processes and the creditrisk processes.Wrong-way exposure or trade refers to a situation whereby the counterparty exposure and the counterparty credit quality have a negativecorrelation, e.g., when one has more exposures to one’s counterparty iswhen one’s counterparty is more likely to default. Thus, wrong-way exposure results larger expected credit loss (as compared to the case of zerocorrelation). Equivalently, this is a situation whereby the counterparty
34
Introduction to Counterparty Credit Risk
Chapter 1
exposure and the counterparty credit spread have a positive correlation.This is another significant challenge in the modeling and management ofthe counterparty credit risk.Right-way exposure or trade refers to an opposite situation wherebythe exposure and the counterparty credit quality have a positive correlation, e.g., when one has more exposures to one’s counterparty is whenone’s counterparty is less likely to default. Thus, right-way exposureresults in smaller expected credit loss (as compared to the case ofzero correlation). Equivalently, this is a situation whereby the counterparty exposure and the counterparty credit spread have a negativecorrelation.There are cases in which wrong-way exposures and right-wayexposures can be easily identified. However, there are cases inwhich wrong-way exposures and right-way exposures can be counterintuitive and their identification requires close examination of thesituation.For instance, for a commodity producer, is it more or less likelyto default when the price of its commodity rises? A dealer who hastrades with the commodity producer would often need to face thisquestion.On the surface, if the commodity price rises, the commodity producer should be better off. But, is this always true? A typical situation isfor the commodity producer to short forward contracts33on its commodities to hedge the potential downside movement of the price of its futurecommodity productions. The dealer on the other hand would long theseforward contracts.In one instance, a commodity producer almost went default when itscommodity price rose. It was due to that that, as its commodity pricerose, the commodity producer owed more money on its hedges or theshort forward contracts and had to come up with capital to meet the margin or collateral calls.
33 Or paying floating on a commodity swap which is a series of forward contracts.
Sec. 1.7
Introduction to Modeling and Pricing of Counterparty Credit Risk
35
Question 1.11. What are some simple examples of wrong-way exposures and right-way exposures? What about an emerging marketsovereign entity entering a cross currency swap paying its own currencyand receiving a stronger currency?
1.7
Introductionto Modeling and Pricing of Counterparty CreditRisk
In essence, pricing the counterparty credit risk is the same as pricing defaultable OTC derivatives or the premium of the embedded default options in OTC derivatives, which is a relatively new area of the derivatives modeling.As we pointed out earlier, these default options, though embedded,are real trades, and thus need to be priced and hedged properly. Just as aCDS is for buying and selling credit protections on a defaultable bond,the default options in OTC derivatives are for buying and selling creditprotections on these defaultable OTC derivatives.To price the counterparty credit risk or to compute the creditchargebenefit or CVA, it typically starts with modeling of loss givendefault (LGD) and default probabilities. More specifically, it involvesmodeling market exposures (including the modeling of collateral strategies) and credit spread or default probabilities, and, ideally, also creditrating transitions, default recoveries, as well as correlation between themarket process and the credit processes.A common subtlety is that pricing the counterparty credit risk isreally pricing options on a basket of defaultable derivatives in a counterparty portfolio because of the netting agreement and the asymmetrictreatment of default (except for extinguishing trades). As we pointed outpreviously, the prevailing trade-level based derivatives models (or micromodels) normally do not and often cannot price the specific counterpartycredit risk, even though they can price the expected market risks (whichinclude the credit risks of reference obligations in credit derivatives) andsome generic counterparty credit risk by discounting at LIBOR. Therefore, additional portfolio-based derivatives models (or macro models) areneeded for pricing the specific counterparty credit risk.
36
Introduction to Counterparty Credit Risk
Chapter 1
Furthermore, the trades in a counterparty portfolio can be modeledby different pricers, booked in different systems, from different desks, oreven from different product areas. This presents significant challenges tothe counterparty credit risk modeling and the system.More details shall be presented in Chapter 8.
Chapter 2
MartingaleArbitrage Pricingin Real Market
Arbitrage pricing theory has been extensively applied in finance engineering for derivatives pricing, ever since Fisher Black, Myron Scholes,and Robert Merton first applied the risk neutral or preference-free derivatives valuation to option pricing by explicitly constructing an arbitrage orreplicating strategy, or, equivalently, the risk neutral probability measure.Early pioneering work include: Black and Scholes (1973), Merton(1973), and Ross (1976).The martingale no-arbitrage theory (Harrison and Kreps 1979, Harrison and Pliska 1981) is an important development in the arbitrage pricingtheory. As we shall see, the Harrison-Pliska martingale no-arbitragetheorem together with martingale representation and change of probability measure (which we shall term collectively as martingale arbitragemodeling) essentially is a model factory for creating various arbitragepricing models. With this approach, we can start the modeling conveniently relying on the existence of an equivalent martingale probabilitymeasure.In this chapter, we shall jump start on the martingale arbitrage modeling and give a review on various concepts and theories. Our primarygoal, however, is how to apply these concepts and theories in practiceand how to address various subtle issues in the real market. These subtleties include using the correct risk-free interest rate, modeling of defaultable derivatives, modeling and hedging with the volatility skew andsmile.We shall also present examples of simple arbitrage strategies and
38
Martingale Arbitrage Pricing in Real Market
Chapter 2
derivatives pricing without the need of using Stochastic Calculus to helpthe readers build intuitions in this area.Readers who would like to become experts on stochastic calculusmay want to consult stochastic calculus books with applications to finance in parallel, such as those by Karatzas and Shreve (1999) andShreve (2004a, 2004b).
2.1
Basics of Arbitrage
2.1.1
Arbitrage Opportunity and Arbifrage Pricing
Intuitively speaking, an arbitrage opportunity is an opportunity for oneto make riskless profits with no net investment.For instance, let’s consider the case whereby Party A buys an assetor a financial instrument34(such as an option) from Party B at one priceand simultaneously sells an identical asset or instrument to Party C at ahigher price. This back-to-back transaction for Party A is the same as theintermediation of OTC derivatives by Party A. Without the counterpartycredit risk, this is an arbitrage opportunity for Party A with the price differential being the riskless profit (with no net investment).In a way, this is similar to the market making process of a dealer bybuying low and selling high with the price differential (or the bidaskspread) being the profit, except that the dealers do house significantamount of risks and provide values or liquidities to the financial market.It is important to point out that, in the presence of the counterpartycredit risk, however, as we discussed in the previous chapter, the profit isnot riskless any more. For instance, in the above example, if the party,assuming to be Party B, who owes money to Party A defaults, then PartyA would incur default loss and yet still owe money to Part C. In general,the loss given default is orders of magnitude greater than the profit derived from the broker’s bidask spread (that typically does not considerthe counterparty credit risk).In some cases, e.g., when high yield or non-investment grade counterparties are involved, even the expected default loss (that is typically34 By financial instrument, we mean a financial asset, such as an obligation or an option
including OTC derivatives contracts. By asset, we mean real (tangible) assets as well asfinancial instruments (or financial assets). For the simplicity of exposition, when real(tangible) assets are used in financial transactions for the purpose of hedging, arbitraging,and speculating, etc., we shall also refer them as financial instruments and use the termfinancial instrument and the term asset interchangeably.
Sec. 2.1
Basics ofArbitrage
39
much smaller than the loss given default) can be much greater than theprofit derived from the broker bidask spread. Such default risk needs tobe priced in and hedged and the real riskless profit (or loss) should be theprice differential less the credit charge, as we discussed in the previouschapter.Aside from the back-to-back transactions, a more common arbitragestrategy is to replicate or hedge (statically or dynamically) an instrumentwith other more liquid instruments using static and/or dynamic tradingstrategies, and then either buy the instrument low and sell the replicationportfolio high or vice versa. This is a typical market making process,which we can also term as buy low hedge high and sell high hedge low.The profits of dealers mainly come from the bidask spread, rather thanbetting on the favorable market movements.We shall proceed with our discussions first by assuming that there isno counterparty credit risk, unless otherwise indicated. The counterpartycredit risk can be mitigated by employing credit enhancement strategiesand credit hedging strategies, which we have discussed in the previouschapter. We shall focus on the modeling of the counterparty credit riskin Chapter 8.Under various idealized assumptions, such as a frictionless market(where there is no liquidity risk, no bidask spread, no restrictions inshort selling, etc.), if the replication portfolio can perfectly replicate theoriginal instrument (by replicating all the contingent or non-contingentcashflows of the original instrument), then the arbitrage price of theoriginal instrument is the market price of the replication portfolio regardless of the risk preference of the market participants. This is the basicidea of risk neutral or preference-free derivatives valuation or pricing.3sIn other words, if an instrument is not traded at its arbitrage price,then there would be arbitrage opportunities, and if an instrument is tradedat its arbitrage price, then there would be no arbitrage. Thus, arbitragepricing is also often termed as no-arbitragepricing. But arbitrage is thekey. Arbitrage would lead to less or no arbitrage, which is an indicationof an efficient market.In practice, no-arbitrage means that the price is within the bidaskspread. The profit resulting from the bidask spread is typically only35Theoretically, the preference-free or risk-neutral valuation has a slightly greater scopethan the arbitrage pricing through replication. More specifically, the preference-free orrisk-neutral valuation can still be valid, even if the perfect replication is not available, aslong as the risks are completely diversifiable.
40
Martingale Arbitrage Pricing in Real Market
Chapter 2
available to dealers with large and diversified portfolios due to theircomparative advantage of having lower hedging or replication costs thanother market participants.The comparative advantage partially comes from the fact that adealer with a large and diversified portfolio will seldom need to hedge onan individual instrument or trade basis, as a new trade is often naturallypartially hedged or offset by other existing instruments or trades in theportfolio. This is particularly true if there is a good “two-way” market.This is also important for researches on the hedging cost.While arbitrage pricing is the definitive technique for derivativespricing for market making purposes, it is important to point out that othermarket participants can use non-arbitraging models, such as forecastingmodels, to identify various investment opportunities, such as coupon enhancement, funding cheapening, and relative value opportunities. Theseopportunities are speculative in nature and involve risk-taking. A significant amount of trading activities is resulted from the different viewsof arbitrage models and non-arbitrage or forecasting models.Due to the arbitrage nature, before we apply an arbitrage model toprice a derivatives instrument, we really need to understand first how tohedge the derivatives instrument. In other words, we need to first makesure that the model prices all the liquid replication or hedge instrumentscorrectly to the market. This essentially is the model calibration.While we can easily apply the risk neutral or preference-free pricingmethodology to the process of a traded underlying asset, it may not beeasy to apply such methodology directly to the process of a non-tradedunderlying variables, such as interest rates, credit spreads, volatilities,inflation rates, and mortgage prepayment rates. Thus, in the latter case,the model calibration is a generic way of deriving the arbitrage processesof these variables.There are a few more steps that we need to undertake before a modelcan produce a reasonable price that can be traded in the market and reasonable sensitivities or hedge ratios.For instance, derivatives instruments, especially complex ones, candepend on some of such variables as volatility skew and smile, correlation,forward volatility, forward volatility skew and smile, forward correlation, volatility of volatility, jumps with stochastic jump sizes, correlationskew, etc. Many of these variables are un-hedgeable, i.e., they cannotbe hedged with liquid instruments or cannot be hedged with low cost. Inthis case, other factors, such as the consensus or the risk preference of
Sec. 2.1
Basics ofArbifrage
41
the market participants and supply and demand, would come into in thepicture in the determination of price (and hedging) of the derivatives instruments. This calls for additional model calibration or secondarymodel calibration that we shall discuss more later on.It is interesting to note in passing that although the dealers try to bemarket neutral for their trading portfolios through arbitraging and hedging, typically, they can only afford to be approximately Delta neutral orthe first order market neutral, but not completely neutral in other Greeksor sensitivities, such as Vega, due to high hedging costs.Furthermore, the amount of business or the business flow, which is asignificant factor governing the amount of trading profit, is not marketneutral, not even Delta neutral. Regardless whether a dealer is long orshort volatility (or Vega) in hisher portfolio, typically, the dealer’s business flow is a long position in volatility, i.e., benefits from increasingvolatility. Intuitively, this is due to that with higher volatilities, therewould be more investment (or speculation) opportunities for investors toexplore and more risks for hedgers to hedge, and hence there would bemore trading opportunities and more business flows.Similarly, a dealer’s business flow is also sensitive to other marketvariables, such as the slope of the term structure of underlying assets orstate variables, e.g., forward interest rates and forward FX rates. Undercertain circumstances, it may be meaningful for one to partially hedgeone’s business flow or the P&L (profit and loss), e.g., for reducing thevolatility in one’s earnings, especially if such hedging is inline withone’s view of speculation if one were to conduct proprietary trading activities. An example of such hedging strategy may be to assume someshort positions in volatility. During the period of later 2005 and early2006, the volatility in the equity market (e.g., characterized by the VIXindex) decreased dramatically, which had caused the reduction of profitfor equity derivatives dealers, as well as that of funds based on equitystat arb (statistical arbitrage).The derivatives business of a dealer is typically structured into different desks (for handling products with increasing complexity levels anddecreasing liquidity), such as the flow desk for managing underlying orlinear trades (such as forwards and swaps), the plain vanilla options desk,exotic derivatives desk, and hybrid derivatives desk36 for managingtrades involving multiple product areas, such equity, interest rate, credit,FX hybrid derivatives. This, as we shall discuss more, has some implica-
’‘ The later two desks are also referred to as structured trading desk.
42
Martingale Arbitrage Pricing in Real Market
Chapter 2
tions on hedging strategies, arbitrage pricing, risk management, as wellas on how we think about some of the concepts.In the next few sections, we shall revisit many of the above discussions in more details.2.1.2
Self Financing Trading Strategies and Arbitrage
More accurate definition of arbitrage opportunities and arbitrage pricingthrough replication and hedging, particularly dynamic replication andhedging, requires the concept of self financing trading strategy.A selffinancing trading strategy (SFTS) is a trading strategy suchthat once a portfolio is established, no money is put into or taken out ofthe portfolio.SFTS has important theoretical and practical implications. Any trading strategy can be considered to be an SFTS respect to the trading portfolio that includes all the cashflows or income that it generates. For example, a simple buy-and-hold trading strategy involving a coupon bondwould be an SFTS if the coupon cashflows are reinvested and included inthe trading portfolio. It would not be an SFTS, if the coupon cashflowsare excluded from the trading portfolio.More specifically, let’s consider a trading strategy (4 ( t ) ,5 ( 1 ) ) involving trading n financial instruments where @j ( t ) is the number ofunits (or the number of shares or the notional amount) at time t of the ithinstrument whose time-t unit price or value is 5 ( t ) , The typical application of this in practice is to hedge or replicate a derivatives instrument.The total value of the portfolio at time t is
Under proper technical conditions, an SFTS is equivalent to
2 6 (+qj( t )= 0 .
(2.2)
i=l
This means that with an SFTS, the proceeds from selling an instrument(or reducing the number of units of an instrument) need to be invested inbuying other instruments, and vice versa.Under various assumptions, the above equation is equivalent to
Sec. 2.1
43
Basics ofArbitrage
d V ( t )= 2 4 (t)d? ( t )
(2.3)
i=l
This means that the change of the total value (with respect to time t) onlydepends on the change of the value (with respect to time t) of each of theinstruments involved, but does not depend on the change of the numberof units.In practice, we tend to think about the above equation as approximation to a discrete-time process, as one can only rebalance one’s hedgingor replication portfolio on a discrete-time basis, typically daily and, asneeded, intraday due to new trades, large market movement (typicallytriggered by some events), or when the risk limit is exceeded.While the above equations are quite intuitive in discrete time, thereare subtleties in the continuous time limit. Interested readers can alsorefer to Musiela and Rutkowsky (Musiela and Rutkowsky, 1997) for fi,lrther discussions. It is also interesting to think about what if jump stochastic processes are involved in the continuous time limit.As usual, by stochastic processes, we mean semi-martingales (satisfying various technical conditions) defined on a usual filtered probability
,P). For more details, the reader can refer to, e.g.,space (Q,{T}r=oKaratzas and Shreve (1999) and Shreve (2004a, 2004b).With SFTS, we can define an arbitrage opportunity more accurately.More specifically, an arbitrage opportunity refers to the following situation:A portfolio with zero initial value (i.e., V ( 0 )= 0 at time 0 or with noinitial investment) becomes zero or positive in value at a future time T(i.e., V ( T )2 0 ) with a positive expected value at time T (i.e.,
E [ V ( T ) ] > 0 or a profit) through SFTS.Equivalently, at time T, the portfolio has non-zero probability forachieving positive values (i.e., P{ V ( T )> 01 > 0 ) and zero probabilityfor achieving negative values (i.e., P{V ( T )< O} = 0 or with no risk).In other words, in this case V (0) = 0 is a traded price, but it is different from (and less than) the arbitrage price. Thus, there is an arbitrageopportunity.The reader may want to take a few minutes to think about the situa-
44
Martingale Arbitrage Pricing in Real Market
Chapter 2
tion whereby the portfolio has a non-zero initial value before reading on.In this case, the key is to form a new portfolio with zero initial valueso that we can apply the above statement. More specifically, if the portfolio has a positive initial value, then it means that we will need to pay toacquire the portfolio (or to fund it by shorting, e g , a zero-coupon bond),whereas if the portfolio has a negative initial value, then it means that wewill get paid to acquire the portfolio (and we can invest the proceeds in,e.g., a zero-coupon bond). Thus, we can form a new portfolio with zeroinitial value by combining the original portfolio and the funding or investment portfolio, and apply the statement for the case of zero initialvalue.With above definition of arbitrage, we can make some of our earlierstatements more accurate. In other words, a general approach of obtaining the arbitrage price of an instrument is to statically or dynamicallyreplicate the instrument with other instruments through an SFTS.More specifically, to replicate an instrument, one needs to replicateits future cashflows and its positions in other instruments. If the replication is perfect or exact37and obtained through an SFTS, then the presentvalue of the instrument should be the same as the present value of thereplicating portfolio, which is an arbitrage price of the in~trurnent.~~Otherwise, there would be arbitrage opportunities between the instrument and its replicating portfolio, as one can long the less expensive oneand short the more expensive one to generate riskless profit.If an instrument can be replicated by more than one replicating portfolios, the present values of all the portfolios and the instrument shouldbe the same to avoid arbitrage opportunities.Such portfolio replications can be used to create perfect hedges39,which, theoretically, is desirable by the dealers whose profits mainlycome from the bid/ask spread, rather than the favorable market move~~~~~~
37 Theoretically, a perfect replication also needs to exactly replicate the payoffs associated with the counterparty credit risk. However, involving the counterparty credit riskmakes the situation much more complicated. So, here we assume that the counterpartycredit risk is not significant and we shall address it separately later on. The credit risk inthe underlying instruments, on the other hand, can be incorporated. With the development of credit derivatives, the credit risk in the underlying instruments becomes part ofthe market risk.38 Here we assume that the bidask spread is zero.39 Here, again, we assume that the counterparty credit risk is not significant and we shalladdress it separately later on. Under this assumption, a perfect hedge refers to a marketneutral position that is free of all market risks (including the credit risk in the underlyinginstruments).
Sec. 2.2
Subtleties in Arbitrage Pricing in Real Market
45
ments.In the real market, however, it is normally very difficult and expensive to achieve perfect replication or hedging (even with back-to-backtrades due