these martingale relationships in Chapter 7 .One subtlety is that some of the above martingale relationships cannot be used as is to achieve the exact martingale resampling, as themeasure or numeraire depends on the state variable to be resampled.This is because that the numeraire P ( t , q + l ) of the PT+, measure der+l measure changespends on Li ( t ) and when we change Li ( t ), the PT,
accordingly. Since the numeraire P(t,q) of the PT measure is independent of Li( t ), Eq. (4.82) is a better choice for performing the martingale resampling.In general, for the purpose of martingale resampling, we need to explicitly capture the dependency on the state variable S ( t ) so that we canhave a measure or a numeraire that is independent of the state variable.To achieve this goal, we can use the following martingale relationship(with a change of measure),
Sec. 4.5
185
Martingale Resampling
M ( s ) S ( s )=Es [ M ( ' > S ( ' ) ]M ( s ) P ( s , t )= E s [ M ( t ) ]
Y
(4.84)
(s I t )
and choose M ( t ) such that the current measure or numeraire is independent of the state variable.
M ( t ) is related to the RadonEs [ MNikodym derivative for changing the probability measure between thecurrent measure and the martingale measure of the state variable S ( t ) .Commonly, M ( t ) weakly depends on the interest rates (but not on equity spot prices or FX spot rates), which can be seen from, e.g., Eq.The reader may recognize that
(4.82), where M ( t ) =
(43
I
l + q T , T + I ) L i ( t )*Now we are ready to perform the martingale resampling according toEq. (4.84). In other words, we need to resample or transform the statevariable S ( t ) to S'( t ) and M ( t ) to M'( t ) such that the following martingale relationships hold with desired accuracy, i.e.,
M ' ( s ) S ' ( s )= E, [ M ' ( t ) S ' ( t ) ]M ' ( s ) P ' ( s , t )= E s [ M ' ( t ) ] .
(4.85)
(SIt)
If we use a bootstrap procedure such that the state variable S(s) andother quantities at earlier time steps are martingale resampled first, thenthe following boundary condition should also hold, i.e.,S'(s) = S(s)
M'(s)=M(s) .
(4.86)
P'(s,t)=P(s,t)Similar to before, if the state variable S ( t ) is a normal process, thenwe can apply additive resampling, i.e.,
186
Martingale Resampling and Interpolation
Chapter 4
(4.87)(s I t )
If the state variable S ( t ) is a shifted lognormal process, then we canapply multiplicative resampling, i.e.,
(s I t )
It is easy to verify that with the above martingale resampling, thedesired martingale relationship or Eq. (4.85) is satisfied. If M ( t ) and
M ‘ ( t ) do not explicitly depend on the state variable S(t) and S ‘ ( t )(e.g., for equity and FX modeling), then M’(t) is just the normalizedM ( t ) (if needed), i.e.,
(4.89)and Eq. (4.87) or Eq. (4.88) provides explicit solutions for the martingaleresampling. If M ( t ) and M ‘ ( t ) do explicitly (and weakly) depend onthe state variable S ( t ) and S ‘ ( t ) (e.g., for interest rate modeling), thenwe can solve Eq. (4.87) and Eq. (4.88) in several iterations until the desired accuracy and convergence are reached. We can use simple numerical procedures, such as the Newton-Ralphson method, to speed up thecalculations.What we have discussed so far is the first order (or first moment)martingale resampling that is equivalent to repricing the volatility insensitive instruments to their market prices or matching the forward value ofthe state variable. In terms of fixed-income models, this is more or lessthe same as matching or calibrating to the initial yield curve or the initialcredit curve. In terms of other models, such as equity and FX models,
Sec. 4.5
Marringare Resampling
187
this is the same as matching or calibrating to the initial forward equityprices and initial forward FX rates.For short-rate models, matching the initial yield curve is part of thecalibration of the drift parameters. So, in this case, the first order martingale resampling may not be needed. For full yield curve models (andother models, such as equity and FX models), the drift terms are completely determined by the martingale arbitrage relationship discussed inthe previous chapter, and there is no freedom in the model to calibrate tothe initial yield curve (or the initial forward equity prices and initial forward FX rates for equity and FX models).Theoretically, full yield curve models (and other models, such asequity and FX models) should automatically match the initial yield curve(or the initial forward equity prices or the initial forward FX rates), e.g.,the price of an FRA or a swap from the model should automaticallymatch that obtained from the initial yield curve. However, various numerical errors prevent such automatic matching in numerical implementations. The first order martingale resampling discussed here is a simpleand efficient way of achieving such matching.Often times, higher order martingale (or higher order unconditionalmoment) resampling targets are available from the market prices of theliquid volatility hedge or calibration instruments or liquid European options, such as caps/floors or swaptions. For instance, the PV of an ATMEuropean option can be used to imply the (unconditional) variance of theunderlying instrument or state variable.Suppose that the PV of a European call option C ( q,K , ) is related tothe state variable S ( t ) , the expiry q , and the strike K , as follows, i.e.,
or, equivalently,
where P ( s , t ) is the time s price of a zero-coupon bond and, for simplicity and without losing generality, we have assumed that
Martingale Resampling and Interpolation
188
Chapter 4
(4.92)
M(O)=l.
The first order martingale relationship still needs to hold, i.e.,
M ( s ) S ( s )= E, [ M ( t ) S ( t ) ]
(4.93)
(s S t ) .
Due to various reasons, such as numerical errors, the above equationnormally does not hold in numerical implementations. Similar to before,our goal here is to find a resampled state variable S ' ( t ) , but, this time,by changing its higher order moments such that the following martingalerelationships hold exactly, i.e.,
M ' ( s ) S ( s )= E, [ M ' ( t ) S ' ( t ) ]
(SSt) Y
M'(s)P'(s,t)= E, [ M ' ( t ) ]
(s 5 t )
(4.94)
where C ( q , K l ) is the first order conditional martingale target due to the$6 < S I T I}
factor.
Same as before, the following boundary condition should also hold,i.e.,S'(s) = S(s)
M'(s)=M(s) .
(4.95)
P'(s,t) = P ( s , t )This essentially requires a bootstrap procedure such that the statevariable S(s) and other quantities at earlier time steps are martingaleresampled first.For the simplicity of the notation, we denote
and
Sec. 4.5
189
Martingale Resampling
as well asC"(T,,K,)=C(T,,Ki)+AC(T,,K,).
(4.98)
Both C'(T,,K,) and C"(T,,Kl) are option prices obtained in the numerical procedures (with numerical errors).Combining Eq. (4.94) and Eq. (4.97), we obtain
1= C ( T, , K l ) + AC(
q ,K1)
Combining the above equation and Eq. (4.98), we further obtain
~ o [ ~ ~ ~ r , ~ ~ S ' ~ r , ~ -] ~= C1 "~( T1, IKx, )I. " ; #(4.100)We further denote
which are the first moments or the forward values of the state variableS ( f ) in the numerical procedures (with numerical errors).If the state variable S ( t ) is normal, then we can use the followinglinear transformation or (unconditional) second moment resampling witha deterministic variable k ( s , t > (conditioned on the information availableup to time s) for the resampling of the second order moments while pre-
MartingaIe Resampling and Interpolation
190
Chapter 4
serving the first order martingale relationship, i.e.,
(s I t )
S'( t ) = k (s, t )(S ( t ) - F(~ , t )+) s ( ~ , t ) (S It ) .
(4.103)
Plugging the above equation into Eq. (4.loo), we obtain
It is easy to see that k ( 0 , q ) is positive when Kl is sufficientlyclose to the true ATM strike s(0, I; ) and the ATM strike from the numerical procedure P(0,Ti) (with numerical errors). If the option priceand the forward value of the state variable are matched before the resampling, then, naturally, k (0, T, ) = 1 .The above procedure can be applied to match options prices at different expiries (and at a single strike per expiry), or the term structure ofthe implied volatility by using a bootstrap approach. Linear, piecewiseconstant, or other types of interpolations or extrapolations can be used toobtain k ( s , t ) for t f and O < S .
r,
Regardless of the interpolation and hnctional form of k ( s , t ) , thefirst moment or the forward value of the state variable are matched or thefollowing martingale relationship is satisfied, i.e.,
E ~ [ M ' ( ~ ) S ' ( ~ ) ] = M ' ( S ) S ' ( ~ ) = M ( S ) (S ~( Ss) t ) (4.105).
Sec. 4.5
191
Martingale Resampling
Similar to before, if'{K,S.S'(T)} = '{KI<S(7i)} '
(4.106)
and M ( t ) and M ' ( t ) do not depend on S ( t ) and S ' ( t ) , then Eq.(4.104) provides the solution for the resampling coefficient k (O,T, ) .However, if Eq. (4.106) is not satisfied andor M ( t ) (weakly) depends on S ( t ) , which is often the case for interest rate derivatives, thenEq. (4.104) provides an equation (rather than a solution) for the resampling coefficient k (0, ) . We can use simple numerical procedures,such as the Newton-Ralphson method, to find the solution iteratively.Similarly, if the state variable S ( t ) is lognormal or shifted lognormal as before, then we can use the following linear transformation or(unconditional) second moment resampling on In(S( t ) + As ( t ) ) for theresampling while preserving the first moment, i.e.,
--As
(4
Plugging the above equation into Eq. (4.100), we obtain an equationfor k (0,T, ) , which can be easily solved by using simple numerical procedures, such as the Newton-Ralphson method. For any deterministicvariable k (s, t ) (conditioned on the information available up to time s),the first moment or the forward value of the state variable are matched orthe following martingale relationship is satisfied, i.e.,
E, [ M ' ( t ) S ' ( t ) ]= M ' ( s ) S ' ( s )= M ( s ) S ( X )
(S It).
(4.108)
The European put option price P(T,,K,) is matched based on theput-call parity because we match the call option price and the forward (orthe unconditional first moment). In other words,
192
4.5.2
Martingale Resampling and Interpolation
Chapter 4
Conditional Martingale Resampling at the State Variable Level
The first and second order unconditional martingale resampling discussed in the previous section can be used for incremental model calibration that is usually much faster than the full model calibration. It canalso be used to calibrate to the hedge instruments that are not in the calibration instruments set. Particularly, it can be applied to match optionsprices at different expiries (and at a single strike per expiry), or the termstructure of the implied volatility by using a bootstrap approach.The resampling of additional higher order (unconditional) moments,in principle, can even be used to calibrate to options at many differentstrikes and a single expiry (so as to match the volatility skews andsmiles).However, recognizing that an option is essentially a conditional firstmoment of the corresponding state variable, thus it is easier to calibrateto options at many different strikes using the conditional first momentmartingale resampling on the underlying state variables than using unconditional higher order moment martingale resampling presented in theprevious section.This procedure essentially creates non-parametric local volatility orstochastic volatility models, for matching the volatility skews and smilesthrough transformations on the state variables. Preferably, this procedureshould be applied to or combined with a good prior solution or approximation.More specifically, if the state variable S ( t ) is normal, then we canemploy the following conditional first moment martingale resampling
Sec. 4.5
Martingale Resampling
193
after the unconditional first (and possibly second) moment martingaleresampling at all time steps. In other words,
where K; is given by solving the following equation, i.e.,
or, simply,(4.112)and the resampling coefficient a(s,t,K , ) is a deterministic variable(conditioned on the information available up to time s).Since, as we just indicated, the unconditional first moment martingale resampling has been performed on the state variable S ( t ) at all timesteps, we have the following martingale relationship, i.e.,
E,[M'(t)S(t)]=M'(s)S(s)
(sI~).
(4.113)
It is easy to verify that the conditional martingale resampling or Eq.(4.110) preserves the first moment martingale relationship or the forward, if M ' ( t ) does not depend on S ( t ) and S ' ( t ) , i.e.,E, [ M ' ( t ) S ' ( t ) ]= E, [ M ' ( t ) S ( t ) ] =M ' ( s ) S ( s )(s I t )
Plugging Eq. (4.1 10) into Eq. (4.100), we obtain
(4.1 14)
194
MartingaIe Resampling and Interpolation
Chapter 4
ora(O,T,,K,)=C”(T,,K,)-C’(~,K,).
(4.116)
After performing the above resampling, we can proceed with the following bootstrap process (by using S ’ ( t ) to replace S ( t ) in the equations below) to further match the call option price C ( T , , K 2 ) with a different strike while preserving the previously matched call option priceC ( & , K , ) and the forward, i.e.,S’( t ) = s( t)
( s s t , K ; IS(t))
where K; is given by solving the following equation, i.e.,
or, simply,(4.1 19)
Solving the above equations yields
Sec. 4.5
195
Martingale Resampling
ora( 0, q ,K2 ) = C"(T 3
2
) - C'( q x 2 ) .
(4.121)
The above bootstrap procedures can be repeated for other strikes andexpiries, if needed. Linear, piecewise constant, or other types of interpolations or extrapolations can be used on the resampling coefficients.Similar to before, if(4.122)
and M ( t ) and M ' ( t ) do not depend on S ( t ) or S'(t ) , then Eq. (4.1 16)and (4.121) provide the solution for the resampling coefficientsU(O,T,,K,) and u(O,T,,K2).However, if Eq. (4.122) is not satisfied and/or M ( t ) (weakly) depends on S ( t ) , which is often the case for interest rate derivatives, thenEq. (4.116) and (4.121) provide an equation for the resampling coefficients u(O,q , K 1 ) and u(O,T,,K,). We can use simple numerical procedures, such as the Newton-Ralphson method, to find the solution iteratively.Similarly, if the state variable S ( t ) is lognormal or shifted lognormal as before, then we can employ the following conditional firstmoment martingale resampling on In(S( t) + As ( t)) after the unconditional first (and possibly second) moment martingale resampling at alltime steps. In other words,
Martingale Resampling and Interpolation
196
Chapter 4
sr(t ) = k ( 3 yt’ K1 E, [M‘( t )( S ( t )+ A, ( t))] ( S ( t )+ A, (I))k’ ( s, K l )t7
(4.123)
-As ( t )( 3 I t,s(t)<K;)
where K; is given by Eq. (4.1 11) or (4.112) and k(s,t,K,) is a deterministic variable (conditioned on the information available up to time s),and
k’(s,t,K,) =k(s,t,Kl
+
lEs
[M ’ ( t ) ( S ( t ) As( t ) ) 1 { ~ , q 4 ] ]+
(4.124)
[
4 M ’ ( t ) ( W + A s (f))1JS(I)<K;)]
is a normalization divider.It is easy to verify that the conditional martingale resampling preserves the first moment martingale relationship or the forward, if M ‘ ( t )does not depend on S ( t ) or S ‘ ( t ) and the unconditional first momentmartingale resampling has been performed on the state variable S ( t ) atall time steps. In other words,
E , [ M ’ ( t ) S ‘ ( t ) ]= E , [ M ‘ ( t ) S ( t ) ]= M ’ ( s ) S ( s )
( sIt )Plugging Eq. (4.124) into Eq. (4. loo), we obtain
(4.125)
Sec. 4.5 Martingale Resampling
197
Similar to before, ifl{K,Sf(?)} = I { K 1 < S ( Q }'
(4.127)
and M ( t ) and M ' ( t ) do not depend on S ( t ) or S ' ( t ) , then Eq. (4.126)provides the solution for the resampling coefficient k (O,T, ,Kl ) .However, if Eq. (4.127) is not satisfied andor M ( t ) (weakly) depends on S ( t ), which is often the case for interest rate derivatives, thenEq. (4.126) provides an equation (rather than a solution) for the resampling coefficient k (O,T,, K , ) . We can use simple numerical procedures,such as the Newton-Ralphson method, to find the solution iteratively.Bootstrap procedures can be repeated for other strikes and expiries, ifneeded. Linear, piecewise constant, or other types of interpolations orextrapolations can be used on the resampling coefficients.Similar to before, the put option price is matched based on the putcall parity and the fact that both the call option price and the forwardvalue are matched.
4.5.3
Brownian Bridge Resampling at the State Variable Level
This is the next step after the Brownian bridge interpolation that we discussed previously. The Brownian bridge interpolation is at the driving
198
Martingale Resampling and Intelpolation
Chapter 4
factor level and is a good starting point. However, it does not automatically ensure the exact martingale relationships on the state variables insome cases.Similar to a vanilla floater that is priced (almost) at par as of thecoupon reset dates, a vanilla zero-coupon floater should also be priced(almost) at par as of the coupon reset dates (both without including theaccrued interests).Intuitively speaking, some of the martingale resampling effectivelyensures that to a plain vanilla floater is priced correctly at all of its resetdates. Similarly, a Brownian bridge resampling ensures that to a plainvanilla zero floater is priced correctly at all of its reset dates.
4.5.4
Martingale Control Variafe at fhe Underlying lnsfrurnenf Level
After the above martingale resamplings, the martingale relationships forvolatility insensitive underlying instruments and volatility sensitive underlying instruments in the calibration instrument set should be satisfied.However, the martingale relationships for volatility sensitive underlying instruments that are not in the calibration instrument set can still beoff. Examples of this situation include compound options, especiallythose with digital or barrier option features, such as callable range accrual swaps or notes, particularly when the volatility skews and smilesare significant.In this case, we can resample the underlying instrument prices to enforce the required martingale relationship. The numerical procedure issimilar to what has been discussed. More specifically, we can use theunderlying instrument prices in the place of the value of the state variableS ( t ) in Eq. (4.72) or (4.73). This is also similar to changing the notional, the strike, or the barrier level, etc., to match the market prices ofthe underlying instruments.Particularly, the following is a simple martingale resampling technique for resampling the underlying instrument prices.Suppose that U is the price of the underlying instrument (or betteryet the price of a leg of the underlying instrument) at a given state in anumerical procedure and U’ the true price of the underlying instrumentat the same state. Normally, we do not know the value of U’ per se,rather, often times, we know some of the moments of U’.Suppose we only know the first moment of U’ or E[U’]. Then, recognizing that the inaccuracy mainly comes from the volatility being not
Sec. 4.5
199
Martingale Resampling
modeled accurately, we can perform the following resampling (whichdoes not change the sign of U ) , 7 3i.e.,l+k)U O I U1-k)U U<O’
(4.128)
where k is a deterministic variable. Thus,
E [U’]- E [u](4.129)
Normally, we need to require k << 1 .A better situation is when we know some conditional moments of
[
]
U‘ , such as E U’1{ K<O,J and EIU’l{uf<,,], which are the Europeanoptions on the underlying instrument. The treatment of this situation isvery similar to that in the previous section.This is effective in addressing the inaccuracy and imprecision forcredit charges resulting from the market process modeling or the scenariogeneration engine and the pricing engines not being fully consistent. Forinstance, the scenario generation can be implemented with the BGM/Jmodel (with one or multiple factors), whereas the pricing engines can beimplemented with various short rate models (with one or multiple factors). This is very helphl in the implementation of a credit charge system that needs to bring together the scenario generation engine and thepricing engines from different trading systems.This is also very useful in using BGM models as control variate tothe short rate models.An important situation where the underlying instrument is modeledaccurately is that the underlying instrument has a non-zero Vega ( V )(from, e.g., caps/floors) and its volatility is not modeled or calibratedaccurately. This is often the case for pricing a callable trade on an underlying instrument containing caps/floors (such as a callable capped floater,a callable inverse floater, a callable range accrual, and a callable CMS orCMS spread floater) using a short rate model. In this case, Vega adjustment is more appropriate.73
In some cases, this type of resampling is better performed on each of the legs of theunderlying instruments, such as the fixed and the floating legs of a swap, which requiresthe sign of the value of the leg to remain the same before and after the resampling.
200
Martingale Resampling and Interpolation
Chapter 4
More specifically, the resampling is achieved by
U’=U + k V ,
(4.130)
E[U’]- E [ U ]
(4.131)
or
k=
EPI
4.5.5
Martingale Resampling at the Derivatives Price Level
Unlike the situation in the previous sections, for the derivatives (especially exotic derivatives) instrument that we try to price, we normally donot have the martingale target readily available. As a mater of fact, themartingale target is what we try to find as the derivatives price to beginwith. Nonetheless, that does not mean that we cannot perform a martingale resampling. As we can often find necessary conditions for achieving accurate derivatives prices, even though we may not know the sufficient conditions for achieving accurate derivatives prices.A typical situation is that a model cannot price the hedge instrumentsof a derivatives instrument to the market, which is quite common in interest rate derivates, particularly, in the presence of volatility skews andsmiles.Since the model price or value V of the derivatives instrument is afunction of the model price of its hedge instruments, by knowing howmuch the martingale relationship or price is off for its hedge instruments,we can estimate how much the martingale relationship is off for the derivatives instrument as follows, i.e.,n
av
AV = C-AHi,i=l aHi
(4.132)
where AHi = Hi - H i M , and Hi and H i M are the model price and themarket price (or the martingale target) of the hedge instrument i, respectively.In this case, the martingale resampled or adjusted price, V - AV , is abetter estimate of the derivatives price. Different models can give very
See. 4.5
201
Martingale Resampling
different prices for exotic derivatives. But after the above martingaleresampling or adjustment, the prices tend to get much closer.The typical reason for the hedge instrument price being off is thatexotic derivatives, especially compound derivatives, often require morehedge instruments than what the model can calibrate to. For instance,callable range accruals, callable capped floaters, callable inverse floaters,and ARM (adjustable rate mortgage) derivatives, should be hedged withboth caps/floors and swaptions. However, short-rate models or reducedfull yield curve models with few state variables, often cannot calibratewell to both caps/floors and swaptions sirn~ltaneously~~especially whenthe volatility skews and smiles are significant. General full yield curvemodels, such as general HJM models and BGM models, can calibratewell to many ATM caps/floors and swaptions simultaneously. However,general HJM models and BGM models can only calibrate well to limitedvolatility skew and smile points, even with state-dependent or stochasticvolatilities.To put it in another way, the reason for the hedge instrument pricebeing off is that the model parameters ( Aj), such as the local or instantaneous model volatilities, are not calibrated accurately. Thus, Eq. (4.132)can be rewritten as(4.133)Let
aHi
A . . =-,dAj
(4.134)
74 This statement has two meanings. Firstly, if too many caps/floors and swaptions areincluded as the calibration instruments, then the calibration of short-rate models or reduced full yield curve models with few state variables may not have a solution, e.g., themodel prices of the calibration caps/floors and swaptions may not match those from themarket within the required accuracy. Secondly, as the number of caps/floors and swaptions included as the calibration instruments is reduced, the calibration can start to havesolutions. But, sometimes the solutions have highly time-dependent model parametersand thus are not stable and cannot produce stable hedge ratios and good P&L explanations. A good calibration not only requires good calibration accuracy to the calibrationinstruments, but also requires parsimonious (stationary or slightly time-dependent) modelparameters so as to produce stable hedge ratios and good P&L explanations.
202
Chapter 4
Martingale Resampling and Interpolation
which can be easily obtained by repricing the hedge instrument after perturbing the model parameter (without recalibration). This is also true for
av-
aaj
*
If A-' ,or the inverse of A ,exists, then
(4.135)which is often termed as the Jacobian approach.
4.5.6
Application to Secondary Model Calibration
There are situations where all the plain vanilla hedge instruments arepriced accurately to the market in a model, but an exotic derivatives tradecan still be priced off the market. Part of the reason for this is that theexotic derivatives trade can depend on parameters or variables (or unhedgeable) that the plain vanilla hedge instruments do not depend on orare not sensitive to, such as correlation, forward volatilities, and forwardvolatility skews and smiles, and forward correlation. Another reason forexotic derivatives trades being priced off the market is that trade-level ormicro models do not price in unexpected risk and liquidity premium, etc.This is essentially the secondary model calibration that we mentioned earlier.Even though exotic derivatives instruments are usually not liquid,there are often market observable prices if there is a competitive marketwhere dealers need to bid in order to win the trades. There are also otherways for dealers to exchange information on pricing, e.g., through private communications on inter-dealer trades or through pricing information providers, such as Totem Market valuation^.'^Thus, adjustment similar to Eq. (4.135) is needed in all these situations to compensate the additional trade premium not priced in. Unlikethe previous case, in this case, AV or the additional trade premium isknown from the market. The goal is to break down the additional tradepremium into additional units of hedge instruments (A-') . AHi (or,
I.'
hedge ratios) to capture the commonalities of the additional premium
75
See http://www.totemvaluations.com for more information.
Sec. 4.6
Other Applications of Martingale Resampling
203
among the same types of the exotic derivatives trades.Similarly, market risk reserves and model risk reserves can becharged based on certain units of appropriate hedge ratios, typicallyhedge ratios of the parameters that cannot be easily observed or hedgedin the market and the hedge ratios of the parameters that cannot be calibrated or modeled accurately, commonly Vega or Vega on volatilityskews or smiles or forward volatilities, or sensitivities on correlations.The supply and demand bias (e.g., in Bermudan swaptions) may also beestimated by certain units of appropriate hedge ratios, typically Vega.Derivatives modeling often times is more about finding the correctand optimal hedge ratios or hedging strategies than finding the price, asthe price can often be discovered from the market, but hedges ratios aremuch less transparent. Using wrong hedges ratios or hedging strategies,a dealer is bound to lose money. There two objectives for finding thecorrect and optimal hedge ratios. One is to lock in the profit, which is abasic objective of a dealer. The other is to make more profit, which islike proprietary trading within market making. The profit of dealercomes from “buy low sell high” or, more practically, “buy low hedgehigh” or “h