40261:16174
$50,000 
 $12:98
N
rnext 1  rY1 rT1 V1  rT1   1  rY1 
4:8%$25,500  $1,000,000 3601:04026
$131:44
5:5%rYn rTn360N $131:51 $1,000,000 TY1:161741  rn   1  rn Finally, we can extract a value for theta:Y!T  $12:98
$131:44  $131:51  $13:05
If we keep now the time constant and change interest rates, we have:34
We assume again for simplicity a 1-day difference between yesterday and today.
CHAPTER 6
YTni  1: rYin1rn1riiirTi  i360360
rYiand:
VALUATION
1  rYi
  1  rTi 
n  1  rYi1 i360
!
rTi
307

n  1  rTi1 i360

!
With rY1  4:85 percent, rY2  5:55 percent, we have:   150  4:8%150  4:85%YT 1 1:040611  r1  1  r1  1 360360
   510  5:5%510  5:45% 1 1:161141  rY2  1  rT2  1 360360
and:X
rYirTi $50,0001  rTi   1  rYi i324:8% 4:85%5:5% 5:45 %150 510 763603607  $20:496541:040611:16114
N  rSW 
N
rnext 1  rY1 rT1 V1  rT1   1  rY1 $25,500  $1,000,000  150 

N
4:8%
4:85%360
1:04061
rYn rTn $1,000,000  510 1  rTn   1  rYn 
5:5%
5:45%3601:16114
 $205:31
 $610:03
and ®nally:Y!T  $20:49  $205:31  $610:03  $835:83YP Y!T  Y!T  $13.05 + $835.83From there we can rebuild VY!T $848.88 compared to a value of $846.56 before, and again the difference is due to the methodology which is only an approximation ofderivative calculus. Note that the de®nition of rho has changed slightlyhere. Because we have two ratesÐr1 and r2 Ðchanging at the sametime, it is not possible anymore to de®ne rho homogeneously by
308
FINANCIAL VALUATION AND RISK ANALYSIS
YPVY!T= rT rY ). Therefore, we consider that rho simply denotes theincremental valuation due to the changes in rates. In contrast to amoney market, it is important to notice that the theta of a swap is ingeneral relatively low because both sides cancel out for the largest part.This property is equivalent to saying that the largest part of a swapvaluation comes from an interest rate variation, which makes swapsideal instruments to hedge interest rate-related risks.
Back Of®ceA back-of®ce valuation for a swap is a strange proposition. On one sidethe back-of®ce requirements are to include only ¯ows known withcertainty, which is why we had to use future market prices or interestexpense on a money market. On swaps, however, variable ¯ows arenot known before they are set, which potentially poses a problem.In practice, the easiest way to address the situation is to adopt ahybrid formulation, in which known ¯ows are accounted for on a costbasis, while unknown ¯ows are evaluated on a front-of®ce methodology. The result is somewhat unpractical, but it has the advantage ofmeeting accounting requirements while keeping all the legs of thetransaction included.
FRAs and EurofuturesFRAsFRAs are extremely simple to evaluate because they are very simpleinstruments to begin with. Their valuation is exclusively mark to market, and there is no distinction between back of®ce and front of®cebecause there is no ¯ow until the FRA begins, at which point itbecomes a regular money market operation.Assume, for example, that we enter an FRA borrowing for anominal N, at a rate rF , and value the position some time later. Theexpected ¯ows are shown in Figure 6-7 where r1 and r2 denoterespectively the zero-coupon mid-market rates to the effective andexpiry dates of the FRA. The valuation VT of this structure is verystraightforward:VT 
N1  r1
N  1  rF 1  r2
Considering that the current market FRA rTF is given by:1  rTF 
1  r21  r1
CHAPTER 6
VALUATION
309
FRA, rFToda y
N
r1N * (1 + rF)
r2
Figure 6-7
FRA
VT can be simpli®ed to:VT  N 
rTF rF1  r2
!
From there we can easily express the time components of VY!T :!
YPNVY!T
rTF rF1  rT2
V TD  N 
rTF rF1  rT2
N
rYF rF1  rY2
!
!
Naturally these expressions are valid only if the FRA has not startedyet, but if it has, it can be valued as a regular money market anyway. Itis then very straightforward to extract from these equations the valuesof theta and rho:!rY1360#!Y!T  N  "n1  1  rY1n1  rY11 1360360
1  rF  N"
#n2  1  rY21360
rY2360
!
n  rY1 2 2360
!
310
FINANCIAL VALUATION AND RISK ANALYSIS
n1  1  rY1
rT1

#360"#n1  1  rY1n1  1  rT1 11360360n2  1  rY2 rT21  rF  # " 360#N"Yn2  1  r2n2  1  rT2 11360360
Y!T  N  "
Consider, for example, a 4  7 FRA at 5 percent. If we buy the FRA, weborrow at 5 percent 4 months from now, for a period of 3 months.Assuming we entered this transaction 1 month ago (30 days), the rateshave changed and we have now r1  5 percent (90 days) and r2  5:5percent (180 days). The ¯ow diagram for this situation is shown inFigure 6-8. The new market FRA is therefore:180  5:5%1180 90 T360 rF  )rTF  5:93%190 5%3601360and the valuation of the transaction can be calculated from there:VT  N 
rTF rF1  r2
!
90  5:93% 5%360 $2,286:24 $1,000,000 90  5%1360
This ®gure is positive, consistent with the fact that we originally borrowed and the rate increased. Suppose now that the next day r1  5:1percent and r2  5:4 percent. The new market FRA valuation fromyesterday to today is:
Today
FRA, rF = 5%, 90 daysr1 = 5%, 90 days
30 days
r2 = 5.5%, 180 daysFigure 6-8
FRA
CHAPTER 6
179  5:4%1180 90 T360 rF  189  5:1%3601360290  5:63% 5%6YP360VY!T  $1,000,000 6489  5:1%1360 $741:38
VALUATION
311

)
rTF  5:63%
390  5:93% 5%7360790  5% 51360
rMF went down so the valuation is negative as expected. If we decompose it into theta and rho: 25%6360 Y!T  $1; 000; 000  6490  5%89  5%1 1360360 390  5%5:5%17360360 7 $11:04180  5:5%179  5:5% 51 1360360290  5% 5:1%6360Y!T  $1,000,000  6490  5%90  5:1% 11360360318090  5% 1 5:5% 5:4%7360360 7 $723:55180  5:5%180  5:4% 51 1360360The total is then $11:04 $723.55 = $734.59, an approximation ofthe accurate ®gure of $741.38. We ®nd again that the theta is muchsmaller than rho, indicating again that FRAs are essentially dependenton interest rates.EurofuturesIn contrast with FRAs, eurofutures are listed, and there can be a markto-market valuation based on closing prices. There are therefore twoways to value them: theoretical, which essentially comes back to aFRA, and mark to market, like a regular futures contract. Not surprisingly the mark to market is predominantly used as a back-of®ce ®gure.
312
FINANCIAL VALUATION AND RISK ANALYSIS
Theoretical ValuationsIn a theoretical valuation, the theoretical price of the futures is thecurrent market FRA, but the ¯ow resulting from the liquidationappears on expiry, not at the end of the FRA. Consider, for example,that we bought a euro contract at a historical price PH . If rTF is thecurrent market FRA, PTF = 100 ± rTF is the current theoretical price forthe futures contract, and the liquidation ¯ow is exactly PV  PTF PH ),occurring at the expiry of the contractÐthat is, at the beginning of theFRA.35 Figure 6-9 represents this situation. The valuation of the futuresis then:!!PTF PHrH rTF QT  PV VT  QT  PV 1  rT11  rT1And the components are easy to express:PTF PHYP Q Y  PV VY!T1  rT1
V
TD
QT

PYF PH1  rY1
QY   PV  PTF 1  rT1
P
!
CFT
The term  CFT re¯ects the fact that we can aggregate all trades duringa single trading day because listed contracts of the same maturity arefungible. From there all other components of the valuation, notablytheta and rho, are easily accessible and similar to those of a regularFRA.FRA, rF
Toda y
T
H
PV.(PF - P )
r1r2
Figure 6-9
35
Eurofutures
Note that PTF is a theoretical ``market'' price: It is derived from actual market rates, but itis theoretical in nature because the futures does have a market price that may bedifferent.
CHAPTER 6
VALUATION
313
Mark-to-Market ValuationsAs we indicated, eurofutures are listed products, so they have a marketprice possibly different from their theoretical one. It is possible to perform a mark-to-market valuation, which is actually similar to that of anindex future:!!PTF PHrH rTF QT  PV VT  QT  PV 1  rT1  rTwhere PTF is now the futures market price, QT the position held, PV thefutures point value, and r is the market rate from the eurofuturessettlement date to the relevant discount date for the valuation.36 rHand rTF represent respectively the FRAs associated with the currentmarket price of the futures and the price at which it was traded inthe pastÐthat is, PTF  100 rTF and PH  100 rH .YPand V TD are then equal to:VY!T!PTF PH PYF PHYPVY!T  QY  PV 1  rT1  rY
V TD 
QT
QY   PV  PTF 1  rT
P
CFT
If we were to adopt the simplifying assumption rT  rY , we would ®ndagain that the valuation of the position to be very close to the margincall:PQT  PTF QY  PYF  PV  CFTVY!T 1  rTThis expression is, not surprisingly, similar to that of an index future.On the treasury side, because of the existence of margin calls, wealso have a situation similar to that of a regular futures contract:where MCT  QT  PTF QY  PYF   PV  CFT is the maintenance margin, and ICT  ICY  QT QY   IMis the initial margin.. IT still given by IT  IY  CY  IY   rOY.. CT  CY  MCT  ICT
36
In the United States, for example, index and eurofutures are settled at D  1; therefore, r  0 and VT  QT  PV  PTF PH ) if the relevant discount date is the futuressettlement date.
314
FINANCIAL VALUATION AND RISK ANALYSIS
And the complete valuation is therefore:PQT  PTF QY  PYF  PV  CFT CY  IY   rOVY!T Y1  rTwhere rOY is the relevant overnight rate from yesterday to today.
Foreign Exchange Forwards and SwapsSwapsConsider the swap ¥=$ shown in Figure 6-10. Assuming we receive theyen payments, the valuation of this swap is in principle exactly equivalent to that of a transaction with only one currencyÐthat is, we discount all expected ¯ows at the prevalent zero-coupon mid-market rate:!!44¥$XX1CFCFii CF¥1 CF$1VT$  T ¥$f¥=$1r1riii2i2If we take the speci®c case of a variable-variable structure, we canoperate as we did before a transformation of the variable sides, asshown in Figure 6-11. The valuation becomes:!!1CF¥2  N¥N¥CF$2  N$N$$¥$ CF1 CF1VT  T :f¥=$1  r¥21  r¥41  r$21  r$4which can be generalized to any number of ¯ows:"!#¥Xr111V;nextVT$  T  N  N¥ CF¥past¥;T¥;Tf¥=$1  r¥;T1r1rn11"!#$X $rV;next11CFpast N$ :N$ 1  r$;T1  r$;T1  r$;Tn11r3$, r3¥
Toda yCF1$
CF2$
CF3$
CF4$
CF1¥
CF2¥
CF3¥
CF4¥
r1$, r1¥
r2$, r2¥
Figure 6-10
Forex Swap
CHAPTER 6
CF1$
N$,CF2$
N¥
CF1¥
N¥,CF2¥
N$
Figure 6-11
where:. N¥ 
r¥V;next
1  r¥;T1
VALUATION
and
N$ 
315
Forex Swap
r$V;next1  r$;T1
represent the present value of the next variable ¯ows.. N¥ :
1
1
1  r¥;T1
1  r¥;Tn
!and
N$ 
1
1
1  r$;T1
1  r$;Tn
!
represent the present value of the remaining unknown variable ¯ows..
P
P $CF¥past andCFpast represent the algebraic sum of all ¯owspaid to date on each leg.
. N¥ and N$ denote the notional of the swap, with N¥  f H ; N$ ; f H¥=$¥=$
as the spot exchange rate at the time the swap was traded.
In terms of treasury, a foreign exchange swap is identical to a regularswap in that the ¯ows accumulate in CT when they are paidÐthat is,on each payment date. From there the overnight ®nancing is straightforward to compute, with the small subtlety that it requires keepingtrack of dollars and yen separately.As it appears, the expression of VT$ hardly ®ts in a full page and isfar from being user friendly in any way. Considering that we havealready developed all the pieces of the formalism earlier, in the interestof time and space, we will not develop the valuation further but ratherconcentrate on a numerical application.Consider a structure similar to the one above, in whichN$  $1,000,000, N¥  ¥107,000,000, CF$1  $13,750 (5.5 percent on90 days), CF$2  $15,000 (6 percent on 90 days), CF¥1  ¥40,125 (0.15percent on 90 days), CF¥2  ¥42,800 (0.16 percent on 90 days), r$1  5:8percent, r$3  6:5 percent, r¥1  0:15 percent, r¥3  0:20 percent, andT ¥108, as shown in Figure 6.12. The dollar leg is valued at:f¥=$
316
FINANCIAL VALUATION AND RISK ANALYSIS
210 days, r3$ = 6.5%, r3¥ = 0.20%CF1$
N$, CF2$
N¥
CF1¥
N¥, CF2¥
N$
30 days, r1$ = 5.8%, r1¥ = 0.15%Figure 6-12
Forex Swap
0$VT;1
1
B$15,0001  $1,000,000  B@210  6:5%30  5:8%11360360$13,750 
C1C30  5:8%A1360
$60,399:28
The yen leg is valued at:21 6¥42,800 ¥107,000,0006430  0:15%108136001
$VT;2
B1B@30  0:15%1360
3
C71C  ¥40,1257  $1,798:46A5210  0:20%1360
Which leads to the valuation of the entire swap:$$VT$  VT;1 VT;2
$60,399:28  $1,798:46 
$58,600:82
This valuation is negative because we are paying interest rates on thedollar notional and receiving yen interest in exchange, at a much lowerrate. If parameters were to change tomorrow to r$1  5:9 percent, r$3 T ¥107:5, the6:6 percent, r¥1  0:15 percent, r¥3  0:25 percent, and f¥=$valuation would become:01B$15,0001$ $1,000,000  B VT;1@209  6:6%29  5:9%11360360$13,750 
$60,851:43
C1C29  5:9%A1360
CHAPTER 6
and:
2
$VT;2
VALUATION
317
0
6 ¥42,800B11 ¥107,000,000  B64@29  0:15%29  0:15%107:51136036013C71C  ¥40; 1257  $2; 093:65209  0:25%A1360
Finally:$$VT$  VT;1 VT;2
$60,851:43  $2,093:64 
$58,757:79
The difference between the two dates is $156.97, and if we want tounderstand the source of this difference, we need to dig one more layerin the valuation. Indeed, if we compute theta, delta, and rho, wehave37:2Y!T  $4:284 Y!T  $169:21Y!T  $8:36which gives a value of $156.57 for VY!T , close to the actual ®gure of$157.97. The largest piece is rho, negative because rates are going upby a higher increment on the dollar side.38
COSTS AND FEESIntroduction
The examination of costs and fees related to establishing and maintaining a trading position may appear somehow misplaced here. It is fair tosay that in general trading costs associated to ®nancial instruments arerelatively low, especially in a high-volume professional environment.From the perspective of a rigorous valuation, however, it is necessary to revisit these issues, for three reasons. First, because we considerthe liquidation of the entire position as a starting point, expenses negligible for an individual transaction may become signi®cant and should3738
TDelta denotes the variation with the spot exchange rate, f¥=$:$¥$¥The impact of r1 and r1 is relatively minor; only r2 and r2 make a difference: r$2 goesup by 0.10 percent from 6.5 percent to 6.6 percent, r¥2 by 0.05 percent only.
318
FINANCIAL VALUATION AND RISK ANALYSIS
be accounted for. Second, we need to explore the proper accounting oftransaction costs in general, and from there the notion of provisionedcosts. Finally, and this is clearly the most important aspect, expensesmay be ®xed or variable, in which case they can and probably shouldbe hedged appropriately because they represent in effect an additionalrisk.
Realized versus Provisioned
Consider that we build today an index arbitrage position on the S&P500 for a nominal of $100,000,000. At an assumed average price of$43.88, we buy an approximate market of 2,300,000 shares, and sell286 futures if the index is around $1,400. If we consider that the totaltrading cost is 3 cents per share and $5 per contract, the total establishment cost is $70,430. This amount is due as a result of today's activity;therefore, it is a realized cost and should be accounted for as a pureout¯ow in the valuation.There are basically two ways to incorporate this information. Onone hand, we can add 3 cents to the price of each share bought, andsubtract 2 cents from the price of each futures sold. In so doing, transaction costs are embedded in the transactions' market prices and do notneed to be kept separate. On the other hand, it is possible to establish aseparate account, dedicated to trading costs that would show a loss of$70,430, partitioned into $69,000 and $1,430 to re¯ect cash and derivative activity.If, technically speaking, both approaches are valid, the second hasmuch more appeal because it allows an evaluation of the tradingexpenses for what they areÐperipheral costs not necessarily relatedto the fundamental business. In particular, a distinct accountingkeeps the P&L statement clean and gives access to the intrinsic pro®tability of the activity. If this pro®tability is mediocre, trading should beadjusted accordingly. This type of ®ne-tuned analysis is not possiblewithout an explicit and distinct tracking mechanism.Now consider the evolution of the position. At some point it willhave to be reversed, even if we do not have any idea about when thiswill happen. The bottom line is, trading costs will accrue again, as faras we can tell to an equal amount. Therefore, there should be $70,430provisioned somewhere. When the position is actually unwound, thisprovision is reincorporated into the P&L, and actual costs are deductedas regular expenses. Figure 6-13 illustrates the process. In essence, wetake a double loss today and ``save'' transaction costs when the position is decreased or closed.
CHAPTER 6
VALUATION
319
$70,430, provisioned
+-
Toda y
$70,430, realizedFigure 6-13
Fees
This operating mode is not a requirement but has the advantage ofbeing conservative and accurate. However, it should be noted that itdoes not apply to all instruments. Money market loans, for example,cannot be described this way because the transaction cost is included inthe bid-ask spread, not captured in a commission as for stocks andfutures.
Fixed versus Variable
In the above example, the trading costs associated with stocks andfutures are variable because their magnitude depends on the size ofthe position, but in the absence of trading, they remain ®xed. Considernow the case of a brokerage cost proportional to the nominal traded,say, 0.10 percent. The total amount of transaction costs varies with thesize of the position and the level of the market. If we do keep the sameaccounting mode of provisioning future expenses known today, wepay $100,000 for a $100 million position, and we should charge$100,000 more in anticipation of the unwinding. This simple calculation, however, does not solve the problem entirely, because if the market is 50 percent higher at the time we eventually sell, the total chargewill be $150,000, not $100,000.This trivial example shows that the 0.10 percent cost is more than atrading cost; it is truly a risk that we should track daily and hedge bytaking a long position of 0.29 future (= 0.10 percent  $100,000,000/($1,400  $250)). If the original amount of $100 million is increased to$150 million, the gain on the future is ($2,100 $1,400)  0.29  $250 $50,000Ðthat is, exactly equal to the incremental trading expenses.In reality, the situation may be