2.40)
An interesting question is what is the martingale relationship forl/Nt ?Recognizing P ( t , t ) = 1 , we have
E," [l/Nf] = E,N [P ( t , t ) / N t ]= P ( s , t ) / N ,
(05 s I t ) .
(2.41)
There are other commonly used martingales, such as the following.A P-Brownian motion W,' is a P-martingale. The integrationt
o(s)dWr is a P -martingale. Also(2.42)
is a P -martingale. The reader may recognize and will see later on thatthis is the (expected value of) Radon-Nikodfm derivative used in Girsanov's change of measure theorem.The conditional expected value
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101
(OIt IT)
(2.43)
is a P -martingale. In other words,(0 I s I t I T ),
(2.44)
which is the iterated expected value theorem, which, in turn, is the foundation for backward induction.The above are examples of martingales for ideal conditions. In practice, we need to consider effects of the counterparty credit risk and theliquidity, among other issues.An interesting question is what is the martingale relationship of aLIBOR (zero-coupon) bond with a Treasury (zero-coupon) bond as thenumeraire and vice versa? Similarly, how do we, in general, reconcilethe martingale relationships with a LIBOR (zero-coupon) bond as thenumeraire and with a Treasury (zero-coupon) bond as the numeraire? Amore practical question is what are the appropriate numeraires for derivatives modeling and pricing?Even within the Treasury market, there are yield differentials between the on-the-run Treasury issues and the off-the-run Treasury issues.We shall address these issues in some of the later sections.In addition, we shall discuss some interesting martingale relationships needed for pricing cross-currency derivatives in Chapter 3 and forpricing the credit risk in Chapter 8.2.5.6
Martingale Representation and SDE for Derivatives Pricing
The martingale representation theorem provides a convenient way forderiving stochastic processes, or Stochastic Differential Equations (SDE)from martingales. As we shall see, the Harrison-Pliska martingale noarbitrage theorem together with martingale representation and change ofmeasure, which we shall collectively term as martingale arbitrage modeling, essentially are a model factory for creating various arbitrage pricingmodels.The previous section is the first step, i.e., identification of the martingale quantities, towards the martingale modeling. After that, we needto formulate the dynamics of the martingale quantities. This is where themartingale representation theorem comes in. As usual, we start with aperfect market, and then add in various imperfections.
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If MI is a (one-dimensional) P-martingale and satisfies certaintechnical conditions (e.g., continuous and with finite variance), then themartingale representation theorem states that MI can be represented by a
P -Brownian motion yp,Msuch thatdMl = D M (Ml,t)dW,’9M,
(2.45)
where oM(X,t ) is another stochastic process. Essentially, Mt is a diffusion process.In general, for a function f (MI,t ) (which can be the price of a derivatives instrument) that is second order differentiable with respect toMI and fist order differentiable with respect to t, then applying ItG’slemma yields
In general, aside from being a deterministic function of MI ,oM( Mt ,t ) can also be a separate stochastic process driven by a differentBrownian motion. This stochastic volatility parameterization is oftenneeded to model the volatility smile.For a lognormal model or diffusion process,dMl = ~ ( t ) M ~ d ,e ” ~
(2.47)
where o(t ) is lognormal or percentage volatility. Applying It6’s lemmayieldsdlnM, = -$o(t)’ dt + o ( t ) d ~ ; ’ J ’ .
(2.48)
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Naturally, we can ask what if M iis discontinuous? In that case, Mtcannot be simply represented by a Brownian motion, since any Brownianmotion is continuous. One possibility is to represent M i by a jumpprocess.The most well known example of jump in the real market is probablythe stock market crash on the Black Monday, October 19, 1987. On thatday, the S&P 500 Index lost 20.5% and the Dow Jones Industrial Average (DJIA) fell 22.6%, the largest one-day decline in the stock markethistory. It is interesting for the reader to think that if we were to use alognormal difhsion process with 20% volatility to model the stock market, what would be the probability of one-day 20% market move.Accompanied with this jump was a sharp increase in the impliedstock volatilities. The fear of the crash of the stock market has beenlargely respocsible for the volatility skew whereby the implied volatilityon a stock or an option increases significantly at lower stock price orlower option strike.Other examples of jumps or quasi-jumps in the real market includethe adjustment of the Fed fund rate (an over night interest rate) byFOMC (Federal Open Market Committee), default of an asset or a firm,and the outcome of various events, such as the release of economic indicators (e.g., non-farm payroll numbers, CPI or consumer price index).These events can cause some interest rates and some FX rates to movesharply. They can also cause some implied volatilities to jump.As we pointed out before, some of these jumps can possibly behedged with a paramutuel process. For more information, the reader canrefer to Lange and Economides (200 1) and http://www.longitude,com.An example of dealers for these trades can be found at:http://www2 .goldmansachs.com/econderivs/index.html.The price and the implied volatility of an individual stock and thecredit spread (or CDS spread) of a bond or a firm can respond sharply tothe earnings report of the underlying firm. A bad earnings report thatsurprises the market can cause the stock price to drop sharply, the implied volatility to increase sharply, and the credit spread to widensharply. An FX rate and its implied volatilities can also jump due to thehedging or payoff of the barrier or digital derivatives referencing this FXrate.For a Poisson jump process, we have,
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dM, =kM(M,t)(dNF'M-A(M,t)dt),
Chapter 2
(2.49)
where Np",M is a P -Poisson process with intensity A ( M , t ) and jumpmagnitude kM ( M , t ), and both A ( M , t ) and kM ( M , t ) can be otherstochastic processes.In general, for a function f ( M , , t ) (which can be the price of a derivatives instrument) that is infinitely differentiable with respect to Mtand fist order differentiable with respect to t, then
(2.50)
Since dN:,M is either 0 or 1, so
(dNFYM)" = dNr'M
(n=l,2,3 ...),
(2.51)
then the last term in Eq. (2.50) can be rewritten as
(2.52)
= (f( M , + k,
( M , t ) ,t ) - f (M,,t))dNF'M
Thus, Eq. (2.50) becomes
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More specifically, for a log-Poisson model,
d M , =k(t)M,(dNr’M-A(M,t)dt).
(2.54)
d In M , = In (1+ k (t))dN;,M - k ( t )A( M , t ) d t ,
(2.55)
Then,
ord ~nM , = In (1 + k ( t
1) (~ N F J ”- A ( M ,t ) dt )
+il(M,t)(ln(l+ k ( t ) )- k ( t ) ) d t
(2.56)
’
when k ( t )> -1.While the stochastic volatility diffusion models can handle the volatility smile in general, they tend to have difficulties in handling the volatility smiles in short time horizon (or short option expiry). The typicaldifficulties include: the volatility of volatility being too high and the fitting of the volatility smiles being not accurate enough or being not stable. Intuitively speaking, this is due to that, in a short time horizon, adiffusion stochastic volatility process does not have enough time to diffuse far from its initial value. In other words, how far a diffusion processcan diffuse from its initial value in a short time horizon is approximatelyproportional to the square root of the time horizon. A jump process doesnot have this limitation and can produce high equivalent volatility ofvolatility or fatter tail distributions.So, we can combine a diffusion and a jump process together suchthat M , can be modeled by the following jump-diffusion process (Merton, 1976),d M , =c, (M,t)dW,”M + k , ( M , t ) ( d N F V M-A(M,t)dt).
(2.57)
More generally, a stochastic process can be modeled by general LCvyprocesses (that contain Brownian motions and Poisson processes as
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special cases). One practical issue is that some jump models, such as theMerton jump-diffusion model, in general are not hedgeable. In otherwords, under these models, while one can price a derivatives instrument,but such price cannot be completely hedged or replicated by a finitenumber of other instruments. This is a significant drawback for marketmaking purposes. Theoretically, the risk neutral argument is derivedbased on the assumption that the un-hedgeable jump risk is completelydiversifiable. Development of hedgeable jump-diffusion models is anon-going research of the authors.It is interesting to mention in passing that the limitation of a diffusion process in short time horizon also occurs in other areas, such as themodeling of a default process with Merton's firm value model (Merton,1974) where the firm value follows a diffusion process and the defaultoccurs when the firm's asset is less than its liability. A practical issue isthat the short term credit spread produced by this model is much smallerthan what is observed in the market. This is due to the same reason that,in a short time horizon, a diffusion process does not have enough time todiffuse far from its initial value. In other words, the short term distanceto default is too high. A promising way to overcome this issue is to introduce a random default barrier in Merton's firm value model to allowjump to default (see, e.g., Finkelstein, Lardy, et al, 2002).The martingale representations that we discussed so far can be applied to various martingale quantities discussed in the previous section,such as forward LIBOR and forward swap rates, to derive their stochasticprocesses.More generally, we can apply the above martingale representation tothe martingale quantity in the Harrison-Pliska no-arbitrage theorem.More specifically, recall that X , / N , is a PN -martingale when t < Twhere X and N are traded assets with no cash flows before time T. Wefurther focus on the situation where X , / N , , X , , and Nr are all continuous with finite variances when t < T . More specifically,d
where
[?) qN= a,"d
(0 I t < T ) ,
qNis a PN -Brownian motion. Or,
(2.58)
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107
(2.59)=O t W 1 N
(0 5 t < T )
Thus,(2.60)
where
As we shall see that Var,( d N , ) and Cov, (al, d N l ) are actually measure invariant.A common case is to model under the spot arbitrage-free measureQ , in which case, Nl =PI. Then
(2.63)
Plugging the above equation to Eq. (2.61), we obtain the familiar
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SDE for X , under the spot arbitrage-free measure Q , i.e.,
dX, = ‘;X,dt + o t d T Q .
(2.64)
In general, N , has an SDE similar to the above under the spot arbitrage-free measure Q . In other words,
dN, = 5 N,dt + o:dW,Q3N .
(2.65)
A more interesting question to ask is what is the SDE of the numeraire Nt under the probability measure PN . To answer this question,we set X , =,B, in Eq. (2.61). Thus,(2.66)Plugging the above equation into Eq. (2.61) yields
d X , = r,X,dt + COY$( d X , , d N , ) d d t + o,dW:Nt
.
(2.67)
This is a generic diffusion model based on the Harrison-Pliska martingale no-arbitrage theorem and the martingale representation theorem.If both X , are N , log-normal, i.e.,
o*= o ( t ) x ,=0, ( t )Nt ’
(2.68)
Then
and
d X , = ( Y, + C0r5 ( d X ,,dN, ) D ( t )O N ( t ) )X,dt
(2.70)
+o(t)X,dW;In some cases, the price of a derivatives instrument can be obtainedanalytically through computing the expectations in the martingale
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relationship. In most cases, the price of a derivatives instrument needsbe solved numerically by solving the SDE of the underling assets or variables with, e.g., Monte Carlo simulation, recombining tree, nonrecombining tree, and explicit finite difference multinomial grid techniques through forward induction and backward induction. These belongto the probabilistic approach.In parallel, there is also the PDE (partial differential equation) approach that directly solves the PDE of the price of derivatives instruments.In general, the PDE approach with implicit finite difference is numerically more robust and has better convergence than the probabilisticapproach. However, unlike some of the probabilistic approach, the PDEapproach suffers from the “dimensionality curse” with its computationtime increasing exponentially as a function of the number of dimensionsor state variables. Typically, it is not practical to solve a PDE with morethan three dimensions. A multi-factor full yield curve interest rate modeland the corresponding PDE for a derivatives price naturally depend onmany state variables. So, such models are typically solved with the probability approach, such as Monte Carlo simulations and non-recombiningtree techniques.As part of the martingale modeling, we often need to perform thechanges of probability measures, which we shall discuss in various futuresections and chapters.
2.5.7
Change of Probability Measure and Importance Sampling
Here we present a brief discussion on the change of probability measure(assuming that the usual technical conditions are satisfied). We shallpresent various applications of the change of probability measure lateron.Let’s first take a look at how to change from (an equivalent martingale) probability measure PNt (with respect to numeraire N ’ ) to (anequivalent martingale) probability measure P,,,(with respect to numeraire N ) . The change of probability measure is also termed aschange of numeraire.This can be achieved by defining a multiplier M i t ( s , t ) for changeof probability measure such that
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Martingale Arbitrage Pricing in Real Market
[
E," [ X ( t ) ]= E,"' X ( t ) M { <( s , t ) ]
(0 I s 5 t ) .
Chapter 2
(2.7 1)
The multiplier M{l (s, t ) for change of probability measure is relatedto the Radon-Nikodqm derivative dPN/dPW through
and it is also easy to see thatE,N'[M;- ( s , t ) ]= 1
(0 I s It) .
(2.73)
The easiest way to understand the above change of probability measure intuitively is to think the stochastic processes X ( t ), N ( t ) , andN ' ( t ) as the prices of simple traded assets with no intermediate cashflows. In this case, we have the following martingale relationship^,^^
(2.74)(OIsIt)
Thus,
Consequently, by comparing the above equation with Eq. (2.71), we obtain(2.76)Now, let's discuss a simplified version of Girsanov's theorem ofHere we have ignored some subtleties in the real market, such as different fundingcosts, which we shall discuss more later on.
65
Sec. 2.5
111
Martingale Arbitrage Modeling
change of probability measure that is quite practically useful. More specifically, if
(OIsIt)
where W, is a one-dimensional PNt-Brownian motion and o(t ) is a demotion y Nis a
terministic function of time t, then, a P,-Brownian
PNt-Brownian motion W," plus a drift term. In other words,dW,N = -o(t )p ( t )dt + dW,"
(0 It ) ,
(2.78)
p(t)=E,"[dW,"dW,
(OSt).
(2.79)
where]/dt
In approving the above Girsanov's theorem of change of probabilitymeasure, we restrict ourselves to the task of finding the drift for y Nunder the probability measure PNf.Let's assume thatdW,N = p ( t)dt + dW,"
(2.80)
(OIt).
The key is to recognize that
E,"
[d (W,NM", ( O J ) ) ] = dE," [
W,NM"r
=dE"W,N]=O
(041(OIt)
The left-hand side of the above equation can be rewritten as
.
(2.81)
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Martingale Arbitrage Pricing in Real Markei
Chapter 2
=E,N'[dW,NM,N(O,t)+W~dM~~(O,t)]
+ EON' [d Y NdM" (0,t )]
[d Y " ] + E," [MpM;, (0,t )a(t ) d W , ] .+ E,"' [dW,NM;# (0,t)a(t)dW, ]= p ( t )dt + O (t )E," [dW,"dW, ]= E,"'
= p ( t )dt
+ a(t ) p ( t) dt
(2.82)
(Olt)
Thus, combining the above equations, we obtain
P ( t )= - O ( t ) P ( t )
(2.83)
*
In deriving the above equation, we have useddM,$ ( 0 , t )= M" (0,t)dW,
E,N' [W,NM$<(0,t)a(t )dW, ] = 0
(2.84)
(0 I t )
We can also approve that
(OIt)
which is very useful derivatives pricing.Another interesting application of Girsanov's theorem of change ofprobability measure in numerical implementations, such as Monte Carlosimulation, is the importance sampling. For instance, rather than simulating dW, , one can equivalently simulate dW, - p ( t) dt and apply thefollowing multiplierexP(-iP2 ( t ) + P ( t ) d W , )
(2.86)
to the probability of the simulation path.This allows more simulation paths reaching the tail states of the
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distribution and typically helps in improving the numerical accuracy andcomputation speed, especially for pricing OTM (Out of The Money) options and default modeling.2.5.8
PDE for Derivatives Pricing and P&L Decomposition
In addition to the SDEs that we discussed earlier, we can also derive thePDE for the price of a derivatives instrument from Harrison-Pliska martingale no-arbitrage theorem. Once this is achieved, we can leveragevarious existing techniques for solving PDEs analytically and nurnerically (e.g., through finite difference techniques) with proper boundaryconditions.Here we provide a few examples to illustrate how to derive the PDEfor the price of a derivatives instrument.Let’s start with a simple case where the price of an underlying assetU (t) and the price of a derivatives instrument V ( t ) follow the martin-
gale relationships with the same numeraire asset N ( t ). In other words,similar to what we discussed earlier,(2.87)
and
(2.88)where the expectation E“,
...I is taken under the probability measure
PN conditioned on the information available up to time s.As before, we assume that the numeraire asset N ( t ) does not payintermediate cashflows or income prior to time T, and the underlyingasset U ( t ) and the derivatives instrument V ( t ) either do not have intermediate cashflows or income prior to time T, or have included all theirintermediate cashflows or income.We define the state variable F, as the numeraire-normalized underlying asset price, i.e.,
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(2.89)oru ( t )= N ( t ) 4
Y
(2.90)
The intuition of the state variable is as follows. If U ( t ) is the priceof the floating leg of a forward starting swap and N ( t ) is the price of thecorresponding annuity, then 5 is the corresponding forward swap rate.This also includes the forward LIBOR as a special case (with U ( t ) being the price of the floating leg of a forward rate agreement (FRA)). IfU ( t ) is the price of a non-dividend-paying stock and N ( t ) is the priceof a LIBOR zero-coupon bond (or rep0 zero-coupon bond), then F, isthe corresponding forward price of the stock. This case covers, e.g.,simple stock options (on a non-dividend paying stock) and simplecaps/floors and swaptions.Let’s start with diffusion processes without jumps. Recall from theMartingale Representation Theorem, if 4 is continuous with finite variance, then 4 can be represented by a (one-dimensional) PN -BrownianMotion Wl asdF, =u,FdW,.
(2.91)
We further assume that the volatility 0;is a deterministic h c t i o n of4 and not a separate stochastic process. We shall discuss stochasticmodels in the next section.Now, we are ready to derive the PDE for the price of derivatives instruments that satisfy the condition that the numeraire-normalized ornumeraire-discounted derivatives price is a deterministic function of tand 4 , i.e.,
(2.92)that is differentiable respect to t and twice differentiable respect to 4 .
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The above equation actually covers a wide range of derivatives instruments. However, this does not cover some of the path-dependent ornon-Markovian instruments, such as an average instrument whose payoffdepends on the past average of the state variable. Such average needs tobe an additional argument in the above function f ( t , F , ) . While interestrate derivatives, in general, depend on the entire yield curve with manystate variables, some simples, such as caps/floors and European swaptions, can actually be capture by the above equation.With the probabilistic approach, we need to compute the followingexpectation (obtained from Eq. (2.88) and (2.92)),f(t,F,)=E,”f(t,F,)]
(OlsltlT).
(2.93)
For the PDE approach, we first need to obtain the SDE of the numeraire-normalized derivatives price f ( t ,F, ) ,using Taylor’s expansion,
df(t,F,) =-dtaf
+--dF,af .+?(dF,)*,a2f
(2.94)
a684where we have assumed that 4 is the only stochastic variable in ourat
model.It is interesting to mention in passing that the above equation is aspecial case for the P&L explanation or decomposition, a critical component of the derivatives modeling and risk management for helping oneunderstand why, how, and where one is making or losing money. Morespecifically, the left hand side of the above equation is the realized (say,daily) P&L of the numeraire-normalized derivatives price, and the righthand side is decomposed into Theta
(z), [E),Delta
and Gamma
[$)
P&L.
It is important to point out that the SDE of the numeraire-normalizedderivatives price only contains P&L contributions from the stochasticvariables being modeled, but not the P&L contributions due to thechange of any deterministic model parameters. For instance, since thevolatility is not modeled or stochastic in our model, so the Vega P&L ismissing from the above equation. The Vega P&L is very important in
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Martingale Arbitrage Pricing in Real Market
Chapter 2
practice and we shall discuss it more in the next section. Another example is that in an equity derivatives model, the interest rate is typically notmodeled or stochastic, so the P&L due to the interest rate change is missing from the above equation.Since both 5 and f ( t , F , ) are IF',, -martingales, by considering allthe drift terms, we further obtain
af
a2f
1-dt+-_(dF,)at2a6
=o
(2.95)
Plugging in Eq. (2.91), we obtain the following PDE(2.96)While the above PDE appears to be different from the originalBlack-Scholes PDE, it is a more general PDE and through transformations of variables, the original Black-Scholes PDE can be reduced to theabove PDE as a special case.Eq. (2.95) reveals an interesting fact that the Theta P&L and theGamma P&L exactly cancel each other under infinitesimal marketmovement. This actually provides a way to effectively hedge the ThetaP&L.As we know, Delta hedging, the most fundamental hedging for market markers or dealers, is aimed at hedging the Delta P&L caused by thestochastic movement of the state variable 5 . It is interesting to notethat, in contrast, the Theta P&L is with respect to the deterministic passage of time. Nonetheless, the Theta P&L is a significant component ofthe overall P&L that needs to be analyzed and handled properly.In general, Theta has two parts. One is due to the time value of adeterministic future cashflow, such as a (riskless) zero-coupon bond, orthe time value of money. The second part is due to the additional timevalue coming from the stochastic nature of cashflows from a derivativesinstrument, which is exactly the Theta of the numeraire-normalized derivatives price.The time value of money is priced in by discounting (or numerairediscounting or numeraire-normalization). It is hedged by taking proper(long or short) positions in the numeraire by investing the excess cash inthe numeraire and borrowing cash by shorting the numeraire. In this
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117
case, the Theta hedging is the same as interest rate hedging (for all interest rate terms and buckets), which, for interest rate derivatives, is part ofthe Delta hedging. An example of this is the valuation and hedging of a(riskless) zero-coupon bond.In the case of stochastic cashflows, the Theta P&L and the GammaP&L of the numeraire-normalized derivatives price exactly cancel eachother under the SFTS for hedging and replication associated with theprobability measure P,,,(if the volatility is constant). So Theta andGamma naturally hedge against each other.For instance, if one is long a simple European option, then one islong Gamma (i.e,, has a positive Gamma) and short Theta (i.e., has anegative Theta). So, as time goes on, the value of the option decreases,but the value of the hedge portfolio increases due to the cash incomebrought in by the long Gamma position. Since these two values offseteach other, the dealer can lock in the initial profit (or the bidask spread).The reason that a long Gamma position can bring in cash income in thehedge portfolio is quite intuitive. For instance, with a Delta neutralhedging strategy and a long Gamma position, one needs to sell progressively more underlying asset when the underlying asset price is higher,and buy progressively more underlying asset when the underlying assetprice is lower.It is important to point out that even though Eq. (2.95) and (2.96)indicate the Theta P&L and the Gamma P&L offset each other, it doesnot mean that one does not need to hedge the Gamma. One interestingquestion is that what if the market does not move (as much)? In thiscase, how can the Theta P&L and the Gamma P&L offset each other?This is actually a typical situation that the conditions based on whichthese equations and derived can be violated in the real market, e.g., whenthe volatility of the state variable is stochastic (or when there are jumpsin the state variable), in which case these equations do not hold anymore. More specifically, if the market does not move (as much), it typically means that volatility has decreased, so one needs to add the VegaP&L into the equation.One of the trader’s headaches is to long Gamma and short Vega, asthe former benefits from a more volatile market and the latter benefitsfrom a less volatile market. This situation requires the trader to reducehidher Gamma and Vega positions.Also, large and rapid movements in the state variable can impairDelta hedging accuracy and incur large hedging cost in the real market.
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So, one typically needs to reduce one's Gamma exposure to within certain risk limit.In the next section, we shall discuss more about the stochastic volatility.2.5.9
SABR Stochastic Volatility Model
We consider the volatility of the following dynamic process0; =o,&(t,F,),
(2.97)
where & ( 4 )is a deterministic function of 4 and a, is a separate stochastic process.There are many possibilities to specify the stochastic process forvolatility or including jump processes. Since a, is not traded, its process, particularly, its drift, may not be preference-free.If or follows a diffusion process, then
af
af
at
a4
df (t,F,)=-dt+-dF,
1 a2f+--(dF,)22 a42
af1 a2fdF,do,+--do, + -y(do,) + 30,
2 ao,
.
(2.98)
a4ao,
In this case, the P&L decomposition further includes the Vega P&L[$do,),
volatility Gamma P&L
The P&L decomposition is actually not unique. For instance, part ofTheta can be exchanged with Delta and part of Delta can come fromVega, which we shall term as Delta due to Vega. Delta due to Vega is aneffect of the volatility skew and smile on hedging, in addition to theireffect on valuation. Proper management of Delta due to Vega is a critical part of the hedging strategies, which we shall discuss in more detailsin the next chapter. As we discussed before, Theta itself can also be decomposed into several parts, including the effect of intermediate cashflows (e.g., coupons received or paid), which is termed as the carry in the
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