to the counterparty credit risk, except when the trade is fullycollateralized). Rather, the dealers do take limited positions (that aretypically limited by VAR, which, in turn, translates to limits in Delta,Gamma, and Vega, etc.). Good traders can profit from the proprietarytrading within such limits and make additional profits.
2.2
Subtleties in Arbitrage Pricing in Real Market
In the real market that is bound to have various imperfections, the situation becomes much more complicated. Here we discuss a few suchcomplications and subtleties, as well as how to deal with these subtleties.2.2.7
Counterparfy Credif Risk
As we discussed before, for instance, let’s consider the case wherebyParty A buys an asset or a financial instrument (such as an option) fromParty B at one price and simultaneously sells an identical asset or instrument to Party C at a higher price. Without the counterparty credit risk,this is an arbitrage opportunity for Party A with the price differential being the riskless profit. In the presence of the counterparty credit risk,however, as we pointed out earlier, the profit is not riskless any more,and the counterparty credit risk can have significant impact on arbitragepricing and hedging strategies.As usual, we shall proceed with our discussions first by assumingthat there is no counterparty credit risk, unless otherwise indicated. Weshall focus on the modeling of the counterparty credit risk in Chapter 8.
2.2.2
The Risk-free lnterest Rate
The arbitrage strategies or SFTS typically require borrowing and lendingmoney or fund. Text books often use the term “risk-free interest rate” orinterest rate without default risk to compute the cost and gain for suchborrowing and lending. But, practically, what is the “risk-free interestrate” in the real market? Is it the Treasury interest rate (or yield)? Another interesting and separate question is what strategies can a nongovernment entity employ to borrow close to the Treasury interest rate(or yield)?
46
Martingale Arbitrage Pricing in Real Market
Chapter 2
While Treasury instruments are valued by discounting at the corresponding Treasury interest rate (or yield), the Treasury interest rates often times cannot be exactly the “risk-free interest rate”.For instance, U.S. Treasury debt instruments (e.g., Treasury bills,notes, and bonds) are virtually defa~lt-free.~’But, the U.S. Treasury interest rate is not exactly the “risk-free interest rate”. This is due to thefact that the income from U.S. Treasury debt instruments ( e g , Treasurybills, notes, and bonds) is exempt from state and local taxes, though subject to the federal tax. For this reason, the U.S. Treasury interest rateshould actually be lower than the “risk-free interest rate” that is defaultfree, but with no tax benefit.In addition, many sovereign countries, especially those in the emerging market, are not default-free. Thus, the Treasury interest rates ofthese sovereigns should be higher than the “risk-free interest rate”.It turns out that for the purpose of derivatives pricing, there are manyinterest rates involved. Practically, in an SFTS for derivatives pricingand hedging, one needs to borrow and lend fund or money for buyingand selling the underlying assets, as well as for funding the derivativesinstrument itself. Thus, one does not need the “risk-free interest rate”per se; rather, one needs the effective financing or funding rates of theseassets or instruments.If a liquid rep0 (repurchase agreement) market is available for anasset, then the rep0 rate is the financing rate, which can have a range ofvalues and is only approximately the “risk-free interest rate”, if the underlying asset is approximately default-free and has no tax and otherbenefits. In a rep0 transaction, one party sells an asset to another partyon spot and promises to buy it back at a future date and at a predetermined (typically higher) price. The first party has a short spot positionand a long forward position, and effectively borrowed fund from the second party. The second party is said to have a reverse rep0 transaction.For more discussions on repo, the reader can refer to, e.g., Fleming andGarbade (2003), and Duffie (1 996).A rep0 transaction essentially is a fully secured or collateralized loan(with the underlying asset as the collateral) and thus the rep0 rate is sensitive to the credit quality and other characteristics (such as liquidity andvolatility) of the underlying asset. The borrower’s credit quality cancome in as a secondary effect for the joint default of the underlying assetIt is interesting to note that the CDS (Credit Default Swap) premium for credit protection on U.S. Treasury debt instruments seems to be about 1 bp per annum.
40
Sec. 2.2
Subtleties in Arbitrage Pricing in Real Market
47
and the borrower. All these risks can be reduced by the initial margin,the “hair cut” on the collaterals, and the margin calls, which essentiallyreduces the loan amount (or value) or increases the collateral amount (orvalue) and, in turn,has an impact on the effective financing rate.If the underlying asset is default-free and has no liquidity risk (suchas a Treasury instrument), then the rep0 transaction essentially hasroughly the same risk as the transaction in the underlying asset, and therep0 rate should be close to the yield of the underlying asset.This also answered our previous question of what strategies can anon-government entity employ to borrow close to the U.S. Treasury interest rate (or yield). In other words, through a rep0 transaction, an entitycan buy a Treasury debt instrument by borrowing fund close to theTreasury yield, regardless of its credit quality. Also, through a reverserep0 transaction on a Treasury debt instrument, an entity can lend fundclose to the US. Treasury interest rate (or yield) to a counterparty virtually without exposing itself to the counterparty credit risk.Similarly, various other non-Treasury assets can have their own financing rates from their perspective rep0 markets. These rates are higherthan the corresponding U S . Treasury rep0 rates reflecting the credit risksand the liquidity risks in these assets, and, possibly, the borrower’s creditquality can come in as a secondary effect for the joint default of the underlying asset and the borrower, as indicated earlier. Thus, these interestrates are not exactly risk-free any more.In general, the Treasury debt of government of a country (or a sovereign entity) can have default risk, especially for emerging market countries, which often have high (implied) default probabilities. Most sovereign defaults involve defaults on debt denominated in foreign currencies.Some involve defaults on debt denominated in the local or domestic currency.41For derivatives instruments, currently there are no liquid rep0 markets, Thus, they are typically funded through unsecured borrowing andthe financing rates highly depend on the credit quality of the borrower.The common approach in practice is to use a reference interest rate,4 ’ Though the government can print more money denominated in its own domestic currency, if needed, this essentially converts the default risk (to the creditors or debt holders)to the inflation risk in the entire country and the devaluation of the domestic currency.More information on sovereign defaults can be found from the website of Standard &Poor’s or http://www.standardandpoors.com(particularly with a search using sovereignand default as the key words).
48
Martingale Arbitrage Pricing in Real Market
Chapter 2
typically a rate in the swap market, such as LIBOR (possibly plus aspread), as the financing rate in the trade-level arbitrage pricing (or micro) models. This captures some generic counterparty credit risks andsome liquidity risks. The specific counterparty credit risk not priced inby this approach is priced by portfolio-based macro models as creditpremium (the net credit charge) or CVA. As we pointed out in the previous chapter, the PV of a trade from the trade-level model should beadjusted by the credit premium (the net credit charge) or CVA (and possibly by other quantities, as such the unexpected risk premium).Some of the strategies that we discussed in the previous chapter arefor the purpose of creating a synthetic rep0 market for OTC derivatives.For exchange traded options, it may be possible to imply the riskfree interest rate from the put-call parity.In summary, in a single model for pricing a derivatives instrument,there can be more than one interest rates simultaneously involved, e.g.,corresponding to the credit qualities and other characteristics of the instruments and entities involved. In some cases, we may need one interestrate or index for computing the coupon payment, another interest rate forthe growth or discounting of the underlying instrument payoffs, and yetanother one for the growth or discounting of the derivatives instrumentpayoffs. For instance, when pricing an OTC option on a Treasury issue,we need to use the corresponding Treasury yield for the growth or discounting of the underlying Treasury payoff, but to use LIBOR (possiblywith a spread) to discount the option payoff.Then, the question arises as for how to model them consistently.One simple approach to model them consistently is to chose one rate (asa primary rate or the state variable) to model and assume all other rates tobe deterministic (possibly linear or approximately linear) functions of theprimary rate.42 This shall be addressed in details later on.In general, the interest rate (or the financing rate) that one needs topay in unsecured borrowing by, e.g., issuing a bond or CP (commercialpaper), highly depends on one’s credit quality. How to borrow at lowerinterest rate or lend at higher interest rate is the goal of funding cost arbitrage.One typical situation of funding cost arbitrage is in the case of buying an asset that has no rep0 market. More specifically, rather than buyIn general, this does not work for pricing non-linear derivatives (such as options) onthe spread between the interest rates and non-linear credit derivatives, which requires themodeling of the spread dynamics, among other quantities.
42
Sec. 2.2
Sublieties in Arbitrage Pricing in Real Market
49
ing the asset funded with unsecured borrowing, one can enter a total return swap on the asset with a lower funding cost. Another strategy is thatone can issue a structured note (such as a callable range accrual note)with various exotic features for significant coupon enhancement and thenone can swap it with a dealer for achieving lower funding cost than issuing a plain vanilla bond. The reader may want to think about the rationale for this situation.Yet another strategy for funding cost saving is that rather than buying an instrument outright, one can sell a put option on the instrument.This way, one can capture the premium of the put option and possiblybuy the same instrument at a lower price (or the strike price).The reader may want to think about whether this is an arbitrage strategy or a speculation strategy, and the risks involved in this strategy andhow to cope or (partially) hedge such risks. This strategy is often usedwhen a company tries to buy back its own stock shares. One interestingimplication of this particular strategy is that the put option is a wrongway trade (or exposure) to the option writer and needs to be priced properly.It is also important to point out that, unless otherwise indicated, theinterest rate that we work with is the nominal interest rate (which, unlessunder special situations, should not be negative), and it is an aggregationof the real interest rate (which can be negative) and the inflation rate(which can also be negative).Fleming and Garbade (2004) provides a discussion of negativenominal financing rate under special situations, which may be consideredas a convenience yield that is not explicitly priced in.
2.2.3
BidlAsk Spread
In the presence of bidask spread in the real market, no-arbitrage roughlymeans the price is within the corresponding bidask spread. If the marketis also complete, the price of an arbitrage pricing model, unless otherwise indicated, is approximately the mid-market price (i.e., the arithmeticaverage of the bid and ask prices), and then the bidask spread is added infor the dealer to make a profit. Part of the bid/ask spread is obtained bythe model and part is obtained from the market consensus.The typical arbitrage strategy of a dealer is to buy (or sell) an instrument from (or to) a market user or another dealer and then sell (or buy)the replication portfolio or instruments to (or from) other market users ordealers so as to earn the bidask spread through buying low and selling
50
Martingale Arbitrage Pricing in Real Market
Chapter 2
high. The bidask spread essentially allows the dealer to cover variouscosts, such as the replication or hedging cost (from the bidask spread ofthe hedge instruments), the premium of the unexpected risk, and to makea profit.It is important to point out that a dealer’s bidask spread should alsoinclude the price of the specific counterparty credit risk. For simplicity,unless indicated otherwise, we do not include the price of the specificcounterparty credit risk in the bid/ask spread, which is the same as howbrokers quote the bidask spread. We shall discuss the bidask spreadincluding the price of the specific counterparty credit risk separately,which essentially is the same as pricing the specific counterparty creditrisk or CVA that can be significantly outside of the broker bidaskspread.The bidask spread (excluding the price of the specific counterpartycredit risk or CVA) is typically determined by the market consensus. Itis typically measured by certain number of units of the sensitivities orhedge ratios. For instance, for linear instruments, such as swaps, thebidask spread is measured by Delta (or PVO143). For nonlinear instruments, such as caps/floors, swaptions, and callable or other exotic derivatives, the bid/ask spread is measured by Vega which is equivalent to aspread on the volatilities. The bidask spread premium is also oftenstructured into the trade as a funding spread (i-e., the spread in the coupon of the funding leg).Often times, the bidask spread of an instrument may not be enoughto cover all the costs involved if one were to completely hedge this instrument on a stand-alone basis. This may seem to be puzzling. But inreality, a dealer with a large and diversified portfolio will seldom need tohedge on an individual instrument or trade basis. A new trade is oftennaturally partially hedged or offset by other existing trades in the portfolio. This is particularly true if there is a good “two-way” market. Forcredit derivatives, there is also a portfolio diversification benefit for thedefault risk.Also, a dealer often does not completely hedge away some of therisks, such as the Vega risk, the Gamma, and the cross Gamma risk, inorder to save some hedging cost and to take on some positions. This essentially is proprietary trading or speculation within market making,which needs to be conducted within certain risk limits or VAR (Value at43 PVOl stands for the PV change of an instrument with respect to the 1 basis point (bp)parallel shift of the yield curve.
Sec. 2.2
Subtleties in Arbitrage Pricing in Real Market
51
Risk) limits. This makes dealers more advantageous than other marketparticipants and some arbitrage prices are only afforded by the dealers.In other words, if one were to model the bidask spread, it involvesmuch more than the modeling of the hedging cost.This also justifies a merit of derivatives instruments. In other words,while other market participants can replicate a derivatives instrumentthemselves, it is often more cost effective to buy (or sell) the derivativesinstrument from (or to) dealers. There are many other reasons justifyingthe demand on derivatives instruments.2.2.4
Un-hedgeable Variables
In the real market, often times, a derivatives instrument cannot be hedgedor replicated perfectly with other more liquid instruments, i.e., the liquidunderlying instruments and the liquid European options on the underlying instruments, in which case the market is not complete.A typical situation is that the derivatives instrument can depend onvariables that cannot be fully determined given the price information ofall the liquid instruments. We shall term these variables un-hedgeablevariables, as they cannot be hedged by the liquid instruments (or cannotbe hedged with low cost).In this case, we cannot simply use the arbitrage pricing. Other factors, such as the consensus or the risk preference of the market participants and supply and demand, would come into in the picture in the determination of price (and hedging) of the derivatives instrument^.^^ Thiscalls for additional model calibration or secondary model calibration,which we shall discuss in the next section.Examples of un-hedgeable variables include correlation, future orforward correlation, future or forward volatilities, future or forward volatility skews and smiles, volatility of volatility, jumps with stochasticjump sizes, correlation skew, etc.The volatility skew and smile both refer to the strike dependency ofthe implied Black volatility that is widely observed in various financialmarkets, such as the equity, FX, and fixed-income markets. Roughlyspeaking, the volatility skew refers to the monotonic strike dependencyof the implied Black volatility (that can be produced by a CEV45model)As we pointed out before, theoretically, the preference-free or risk-neutral valuationcan still be applied in this case, if the un-hedgeable risks are completely diversifiable.45 CEV stands for Constant Elasticity of Variance.
44
52
Martingale Arbitrage Pricing in Real Market
Chapter 2
( e g , higher implied volatility at lower strike), whereas the volatilitysmile refers to a non-monotonic dependency (that cannot be produced bya CEV model) (e.g., higher implied volatility at both lower and higherstrikes). In other words, the volatility skew and smile differ in the functional form of their strike dependency of the implied Black volatility.Let’s take a look at an example of a Bermudan option whereby theoption holder can exercise the option at a set of pre-specified dates and atpre-specified strikes.While the Vega risk of a Bermudan option can be largely hedged bya set of European options, it cannot be completely hedged by such European options. In other words, different models (such as those with different volatility structures or mean reversion) calibrating to the same setof European options can give rise to different prices for the same Beraudan option. This means that the Bermudan option depends on variablesthat are un-hedgeable with respect to European options, and, to that extent, its market price is partially determined by, e.g., market consensusand supply and demand.The fundamental reason is that the Bermudan option price or its exercise boundary is determined by conditional expectations (i.e., expectations conditioned on future time and states), based on which the optionholder needs to make a decision to exercise or to continue, and the European option prices do not directly depend on these conditional expectations. More specifically, these conditional expectations are determinedby conditional quantities, such as, the future or forward volatilities andthe future or forward volatility skews and smiles that are examples of theun-hedgeable variables in this case. On the other hand, the Europeanoption prices are determined by the unconditional expectations (is., expectations conditioned on the information available today, which, in turn,are determined by the unconditional or term quantities, such as the Blackor term volatilities (i.e., the equivalent or some average volatilities over aperiod of time) and the term volatility skews and smiles.Consequently, the European option prices cannot fully determine theconditional quantities that determine the Bermudan option price. Oneway to appreciate the complexity of Bermudan options as compared toEuropean options is to realize that a Bermudan option is a highly compound option on a basket European options whereby the exercise of oneof these options cancels all the options at later dates. In other words, aBermudan option contains a switch option for choosing between itsshortest expiry European option and the rest of the options.
Sec. 2.2
Subtleties in Arbitrage Pricing in Real Market
53
In the U.S. interest rate market, there is more natural supply thandemand for the Bermudan swaptions (or Bermudan volatilities) due tothe callable debt issuance and the debt issuers swapping with the dealers.In other words, by issuing a callable debt (which has an embedded Bermudan call option), the debt issuer buys Bermudan volatility from thedebt holder (and, as a return, the debt holder gets an enhanced coupon).The debt issuer, then, in order to cover the cost of the option or volatilitybought, sells the volatility to the dealer by selling the Bermudan swaption or entering into a cancelable swap. Thus, the Bermudan swaption orvolatility market tends to be one-sided with the dealers having long positions of the option or volatility. Consequently, the Bermudan swaptionstend to have lower implied volatilities as compared to the Europeanswaptions.2.2.5
Primary Model Calibration and Secondary Model Calibration
As we discussed earlier, in order to achieve arbitrage pricing, (implicit orexplicit) model calibration is needed. In other words, the model parameters need to be adjusted so that the model prices the calibration instruments or hedge instruments correctly to the market.The primary model calibration calibrates to the liquid underlyinginstruments and the liquid European options on the underlying instruments by adjusting various model parameters by using, e.g., LM (Levenberge-Marquardt) optimi~ation~~.For an interest rate market, calibratingto the liquid undcrlying instruments is the same as (implicitly or explicitly) calibrating to the initial yield curve, which is typically simple andstraight forward, and calibrating to the liquid European options on theunderlying instruments is the same as calibrating the volatility of theyield curve, which can be much more difficult. For an interest rate swapmarket, examples of the liquid underlying instruments include cash deposits, Euro futures, and swaps, and examples of the liquid Europeanoptions on the underlying instrument include capdfloors and Europeanswaptions.46 A popular and free Levenberge-Marquardt solver software was developed by ArgonneNational Laboratory and can be downloaded from http://www.netlib.org/minpac!d.Thef2c utility for converting FORTRAN to C is often useful and can be downloaded fromhttp://www.netlib,orglflcl. Netlib (http://www.netlib.org) is a useful repository of freemathematical/numerical software.Another useful resource is Quantlib(http://www.quantlib.org/) that provides a useful free/open-source library for QuantitativeFinance.
54
Martingale Arbitrage Pricing in Real Market
Chapter 2
For a simple IR model, the calibration instruments depend on thederivatives instrument to be priced. It typically calibrates to all the liquidunderlying instruments or the initial yield curve that the derivatives instrument depends on, and selected Vega points (or liquid caps/floorsand/or European swaptions that the derivatives instrument depends on).This is also referred to as local model calibration.For an advanced IR model, not only can it calibrate to the entire initial yield curve, it can also calibrate to the entire ATM volatility matrix(of liquid capdfloors and/or European swaptions) and possibly best fit tothe volatility skews and smiles globally. This is also referred to as globalmodel calibration.Various martingale control variate techniques presented in Chapter 4can be used to improve the calibration accuracy.Essentially, the primary calibration calibrates to the hedge instruments so that the model produces market compatible prices for the hedgeinstruments. However, as we pointed out, after the primary model calibration, the model still may not be able to produce market compatibleprices for certain derivatives instruments, particularly, exotic and hybridderivatives instruments, due to their dependencies on the un-hedgeablevariables. This is where the secondary model calibration comes in.The secondary model calibration attempts to calibrate to similar orthe same derivatives instruments or the secondary calibration instruments(which normally cannot be used as hedge instruments due to their largebidask spreads) by adjusting additional model parameters while maintaining good primary calibration. This is essentially to handle or estimate the un-hedgeable variables.The first step is to identify the model parameters that correspond toun-hedgeable variables and are suitable for the secondary model calibration based on the sensitivities the PV of the derivatives instrument withrespect to these parameters, These are typically parameters that (explicitly or implicitly) determine the correlation and fiture correlation (e.g.,among various interest rates), such as the correlation parameters in amulti-factor model, and parameters that determine the volatility skewsand smiles and future volatility skews and smiles, such as the parametersfor a stochastic volatility model or a jump model.The second step is to estimate some of these model parameters fromhistorical data, if feasible. For instance, one can estimate various historical correlation, e.g., among various interest rates, and then estimate themodel correlation parameters.
Sec. 2.2
Subtleties in Arbitrage Pricing in Real Market
55
The third step is to find the market prices of the similar or the samederivatives instruments (or the secondary calibration instruments) andthen adjust some of the model parameters estimated from historical datasuch that the model produces market consistent prices for the secondarycalibration instruments.Even though exotic and hybrid derivatives instruments are usuallynot liquid, their prices can often be discovered from the market if there isa competitive market where dealers or dealers need to bid in order to winthe trades. Brokers are also a source of market price discovery. Thereare also other ways for dealers or dealers to exchange information onpricing, e.g., through private communications on inter-dealer trades.Pricing information providers, such as Totem Market valuation^,^^ provide extensive pricing information contributed by dealers on a widerange of derivatives instruments across product areas.The reader may recognize the subtlety that the secondary model calibration is’ almost the same as calibrating to the derivatives instrumentthat we are trying to price. The reader may want to spend a few minutesto think about this before reading on.In practice, this is handled in an iterative way that the secondarymodel calibration performed in the past is used for the current trade. Thehope is to find a model such that the parameters for the secondary modelcalibration do not change much over time.One needs to have a process to monitor the performance of the secondary model calibration. For instance, if one keeps losing trades toother dealers, then it is an indication that something may be wrong. Thismay be because one’s secondary model calibration may be not good orone’s model may be very different from the models of other dealers.One also needs to be careful, if one keeps wining trades against otherdealers. Unless one can justify that, e.g., by risk reduction or hedgingcost reduction from portfolio benefit (such as the situation whereby anew trade significantly offsets the risks in the existing portfolio), it mayalso be an indication that something is wrong. The portfolio benefit canbe significant in reducing the counterparty credit risk and price.A major model risk is that the parameters for the secondary modelcalibration can indeed change much over time, especially when there is aregime change in the market.Thus, it is important for one to take a market or model reserve on thePV of the derivatives instrument. Consequently, part of the P&L from47
See www.TotemValuations.com or www.Markit.com for more information.
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Martingale Arbitrage Pricing in Real Market
Chapter 2
this derivatives instrument, rather than being considered realized rightaway, is put into a reserve, which will be released over time as a realizedprofit. The magnitude of the risk reserve can be measured by and linkedto certain number of units of the sensitivities or hedge ratios of the PV ofthe derivatives instrument with respect to the un-hedgeable variables.2.2.6
Models for Pricing, Models for Hedging, and Hedging Calibration
The primary objective of a model for pricing (or pricing model) including the primary and the secondary model calibration is to produce marketconsistent prices (that are typically expected values of certain dynamicprocesses), whereas the primary objective of a model for hedging (orhedging model) is to produce market consistent hedge ratios or hedgedynamics (that are the dynamic processes themselves).Ideally, one would hope that the pricing model and the hedgingmodel should be identical.48 But, in practice, especially for exotic derivatives instruments, it is often not the case due to various issues, suchas the dynamics implied by the model being inconsistent with the market,the existence of un-hedgeable variables, numerical issues, and computation speed issues.In general, even if a model can calibrate to all the liquid hedge instruments, it by no means guarantees to give correct hedge ratios orhedge dynamics. The intuition is simple, as calibrating to the liquidhedge instruments or the expected values cannot uniquely determine themodel dynamics. In other words, models with different dynamics cangive the same expected values, but different hedge ratios or hedge dynamics.The local volatility model is such an example. The model can produce market-consistent volatility smiles by calibrating to all the volatilitysmile points of liquid European stock (or stock index) options, yet it stillproduces a volatility smile dynamics that is inconsistent with the market(See, e.g., Hagan, Kumar, Lesniewski, and Woodward, 2002).More specifically, in the real market, when the underlying variablemoves, the associated volatility smile moves along, whereas in the localvolatility model, the associated volatility smile appears to move in theopposite direction. But, this does not necessarily mean that the localvolatility model is wrong. Rather, it means that one needs to use hedging48 There also may be a need to use different pricing models for booking (for end of day or
EOD) P&L and for structuring (or for live trade pricing).
See. 2.2
Subtleties in Arbitrage Pricing in Real Market
57
corrections or a different model for hedging.In addition, in order to produce numerically stable hedge ratios, thehedging model needs to be parsimonious (i.e., with fewer model parameters) and thus may not fit the market prices well enough for pricing purposes.A similar example is related to part of the volatility skew dynamicsthat the Black or percentage volatilities of liquid European options increase when the underlying variable (such as the stock price or the interest rate) decreases,49 and vice versa. For instance, in an IR market, theBlack volatilities of liquid European options tend to rally (in an expectedvalue sense) when the market rallies (or the interest rates decrease).If the pricing model is a lognormal model on the underlying variables (such as the stock price or the interest rate), then the model predictsthat when the underlying variables change, the Black volatilities of theEuropean options on the underlying variables do not change.” This obviously is inconsistent with the market skew dynamics. A good hedgingmodel needs to produce different and skew adjusted hedge ratios, such asDelta and Vega.For instance, if one is short in volatility (or short in Vega, i.e., has anegative Vega) in an IR market, then, when the market rallies (or the interest rates decrease), one would lose money on one’s Vega position, asthe volatility tends to rally (in an expected value sense). Intuitivelyspeaking, one is shorter in the market than one would have been withoutthe volatility skew or as compared with the Black Delta (or the Deltaproduced by a lognormal Black model). In other words, part of the skewadjusted Delta is from (Black) Vega, which we shall term as Delta due toVega and quantify later on.If one uses a lognormal model for pricing, then one needs to properly49 The reader may want to think more about volatility skew dynamics in the FX market
when the exchange rates (or FX rates) are the underlying variables. This is one of theinteresting situations related to the exchange rate symmetry in that if X is an exchangerate, 1/X is also an exchange rate with the same Black or percentage volatility. So, itseems that we have a paradox that if the FX Black volatility is a decreasing function of X,then, by symmetry, the same FX volatility should also be a decreasing function of 1/X,but they cannot be true at the same time. In the real market, it turns out that typically theFX Black volatility increases when the underlying FX rate moves in the direction of thedevaluation of the (market perceived) weaker currency. In other words, the market typically implies more probability for the weaker currency to devaluate than the other wayaround.50This statement is an approximation for IR caps/floors and European swaptions if theunderlying variable is the short rate.
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Martingale Arbitrage Pricing in Real Market
Chapter 2
incorporate the Delta due to Vega in one’s Delta hedging, otherwise oneis not truly Delta neutral. Similarly, one needs to properly handle one’svol