FINANCIAL E,NGINEERNG

lntroduction

Derivatives can mean fear to some and opportunities for others ...This book. as I explained in the Preface, is meant to open-up and dem.'-stify the'black bor'imaee created by the'Rocket Scientists'and to simplify and expiain the structuring, pricing.trading. hedging. evaluation and risk management oi cieriratives (forwards. futures. swapsand optrons).

There is perhaps no other area ol investment finance where theorv plays as important arole in the practitioner's world as in the field oi derivatives. Sophisticated mathematics havebeen kept to the minimum necessary and in order that the understanding of underlyingconcepts is not compromised. The theory is then applied to the practical pricing, trading,hedging and risk management techniques used in the market. Most importantly, I haveexpiained the different approaches that the market uses wherever there is no one single correctmethodology or solution to, say, hedging or pricing of a complex swap/option.

Arbitrage and exploitation of cashifutures/options relationships are explained. Financiaiengineering methodology and asset,rliability management techniques are covered in detail.Portfolio immunisation and optimisation strategies are based on market practices whileproviding theoretical background.

The understanding. practical uses and'manipulation'of derivatives to create nerv. slnthetic,or structured products is aided by worked-through examples and step-by-step illustrations.The basic models of cash ffo"vs, yield calculations, durations. simulations, projections andrisk management can be used conceptuaily. The diagrams and methodologies can be usedgenerically. Having explained the chess pieces and rules of the game. I would leave you (theReaderl to create erciting trading, hedging and investment strategies as new ideas are notthe monopoly ol any one person.

The phenomenal -qrowth of volume in the derivatives market and the rapid pace o[

innovation have left many sell-proclaimed derivatives specialist houses, with more aspirationsthan technical resources. gasping for breath. And while the lesser securities houses are tryingto muscle-tn with a me-too marketing hype, the corporate treasurers and fund managers arebeine rrpped-off by the big bad wolves ol the city u'ho cleverly ofl'-load a lot of derivatires.which are either unnecessarv or unsuitable. at excessive premiums.

Hence. rvhere options are over-priced (ior new products, off-market structures or wheredemand is sreater than supply'). thrs book shows how the treasurer can replicate the optionat a much reduced cost *'ith dy'namic hedging by buy'ing and selling the underlying cash andfutures contracts. It explains how to modify the original Black-scholes pricing model to suitdifferent t1,pes of options and the related hedging techniques.

Pricing optionsThe most commonly used models lor interest rate options are based on the Black-Scholesmodei which was developed lor short-date equity options. The modified BIack-Scholes modelswork well for simple interest rate options such as caps, floors, collars and European optionson zero-coupon bonds. Howeuer, for more compler options and swaptions, the modifiedBlqck-Scholes models do not price well and, more importantly, they do not hedge well.

Therefore. newer models are needed and the risk management techniques refined. We gothrough these in depth *'ith the associated pros and cons as there is no such thing as a perfectoption prrcing and hedging model universally suitable for all situarions.

Chapter i5, the Financial engineering chapter in the book, deals with applications forderivative techniques. It includes the latest swap innovation, the index differential or.quanto,swaPs, which allow a borrower or an investor to separate currency and interest rate exposures,by paying interest rates based on one currency while taking the currency risk oi another. Itsho*'s how to take advantage of different shaped vield curves to create lower cost financefor the borrower while providing higher returns for the investor, without changing currencyexposure. There is correlation risk for the trader while the risk for the borrower or investoris that the shape of one or both yieid curves will change more rapidly than expected, turninsexoected profits into losses.

Other yreld-curve plavs such as the Libor-in-arrears swap (which is a pla1, on rhe impiieciforward rates by having Libor set say, 6-months in arrears). and the ipread swap (whichenables. lor example, one party to pay the 5-year swap rate and receive the 10-year rite, bothreset semi-annualll', or the counterpartv could even pay the seconci ieg in 5-rnonth Libcr).We examine how the diversity ol new products can be used to tailor very precise risk-rew,aroprofiles anci create acided value.

13. Risk management of complex diffsLet tts now perfornr in-depth anulyric,s ol'inile.r lilJ'ercntiul s*'aps-the pricing, hedging anlrisk ntanagement oJ' inrerest and IX risAs.

\\'e ri'rll pick an exampie rvhere on at least one leq, payments are determined b1'an rnderin one currency but paid in another. Tirat is, instead ol coupon pa)'ments based on a firedrate or a Libor rate rn the base currency. the rate used is an index in another currenct'.6-month Libor or the 5-1'ear swap rate.

ExampleCHF-USS indcrBase currency':Face value:Stirr datc:End date:Frequeno':Receive:Pav:

differential swapUSSUSS200m5 Aug l99l5 Aug i996Semi-annualUSS LrborCHF Lrbor

USS Libor on 3 Aug 199 I was 6ozuCHF Libor on 3 Aug 1991 was 1.9315"/'0

On first coupon payment date (5 Feb 1992), for 6 months and adjusting [or daycount

Net payment : USS200,000,000 x (60/o -

1.93659h)

Pricing the diffA swap is priced by discounting each component cash ffow at the current zero coupon yieldcurve lor that currency. Although luture Libor settings are not known today, they can beimplied irom the current yield curve. Therefore, to price an index differential swap, all cashflows depending on luture rate sets are calculated using implied forward rates in theappropriate currency. Once these cash flows are determined, they can be discounted from thezero coupon curve in the currency the cash flow is denominated in.

Table i-1 shows the component cash ffows for our sample swap. Known cash flows are inbold, implied cash flows are in regular print. Note that the lorward rates implied show USSLibor will be exceeding CHF Libor in a lew years, even though it is currently 300 bp less.

The present value of the cash flons shown, all in US$, is -

2,703,326.

lnterest rate riskAn index differential swap has interesl. rate risk in both the payment and index currencies.In our example, if USS interest rales change, the payments based on USS Libor will change,as will the discounting of cash flows on both sides of the swap. Il CHF interest rates change,the payments based on CHF lonvard rates will change. Note, howeuer, that the CHF/USSexchange rate has no efect on the price of the swap.

As there is risk to both USS and CHF interest rates, there is an interest rate hedge in bothcurrencies. The USS hedee is found in the usual way: the sensitivity ol the swap is founci fora I bp move in each point oi the yield curve, which is hedged by the appropriate amount ola current coupon srvap olthe same maturity. Finding the CHF hedge involves an extra step.The sensitivity is again found for a l bp move in each point of the CHF yield curve, but thisproduces a gain or Ioss rn USS, as the swap payments are denominated in USS. Therefore,this gain or loss needs to be converted at spot to CHF, which is then hedged by the appropriateamount oi a CHF current coupon srvap ol the same maturity as the maturity in the yieldcurve 'blipped'.

Table l6 shows the interest rate hedges (in millions) for our example swap:Note that the CHF hedge is approximately the lace value oI the swap (200m) converted atthe lorri,ard exchange rate at the maturity (1.498).

FX rate riskAs stateci above, the current exchange rate has no effect on the vaiue ol an index differentiaisrvap. FX exposure anses. however, because su'aps in the index currency are put on to heclgeinterest rate risk. Changes in interest rates rvill cause a gain or ioss denominated in the inciexcurrency, rvhich are then converted at spot and netted with the gain or loss on the indexsrvap in the base currency. If the spot rate has not changed irom that used to derermine theinlerest rate heciqe, the gain (loss) on the index swap rvill ecual the loss (gain) lrom the hedee

_converted at spot. If the exchange rate has moveci, horvever, the eain and ioss wiil not cancci

\\^;\et\\o.t \ ,Y\1.r

\rr \ Lv\\-,\v ii- 'iv-9

. \h -,

=r\ \-, \.c \,, \ \

\r \,:..,.! \l\, , \cc \

I^. Itrl=l.!lUILIc. Io. I

EC

,r

,le l?. P/L of inder s$!p + hedge lcorrect hedge)

ingc rn1F IRbpt

PLoirndex sw'rpiUSS '000s)

PLoiindex hedgestCHF '000st

- -< 09'ot.-i66

- 2.0'.;

r.518

P, L in USS'0tl0s for viirlous spot movements-l.0o,o -0.59'o 0.0o,0 0.59i, 1.0901.503 1.495 t..i88 1 .+61 1.J73

l,0o,o ).0"1.459 1.-11

50l0l050

_ I0-20-50

I 1.565 )( r.ll7)

l'l-+l{ 157)

0-1_i7/ t{

1.128i.57 j

5.t l6l.1l'1.0675ll

0riiit

( r.071)(1.1 50)( 5.,1:0)

1196)(69 trlal( I6)

0l5

55l13

(90 )(16 )(lll{it0,s

81l

;1

(-s,1)( l2)

(1)t-,0I0

(2)(i 3)

( 11)(l)000(l)(3)

( t0)(s l)

( l9)jj20

(3)(7)

( t7)( 69)

(1)l0

,1

0(4)

( l0)(:4)(87)

1?lf

ll60

(6)(1.+)(31)

(r06)

il 15'J I t-lg 1(q at

0{(10) (l :12 I ) (.+.(46) (E,

( r42) (25 :

5le 18. P/L of index swap + 0.5 + hedge (underhedge)

sir,bp)

PLotrndex swaptUSS'000s)

P/L of 0.5index hedges(CHF '000s) -

5.0or'o1.566

-

2.00/or.5r8

P,tL in USS'000s ior various spot movements-

l.Ooh -0.5o/, 0 0% 0'5o/o l 0%r.503 1.495 1.488 1.481 t.1'73

2.00,h1 459

5.0 914t'

50l0r050

-5-10-20-50

r1

4. Minimising interest rate sensitivity directlyLet us assume that the yreid curve is a contrnuous iunction denneci for a finite number o[maturrties and linearly interpolated lor points in bet*'een these. Let us also assume that wehave hedging instruments in each of those maturities. Ideally, *e \r'ant a hedge such that:

AV :0

ior any possible changes in interest rates. This is, of course, impossible to achieve becausewe rvill have a fixed number of hedging instruments. We can, ho*'ever, make the absolutechanqes quite small by requiring that the hedge satisfies:

EV;-: 0cfi ior i:1, ,n

where r,, i : 1,..., n are the interest rates (YTM) ior the n maturities that define the yieldcurve. The hedge could easily be determined by soiving the above system of equations forthe additional amounts of each hedging instrument. This is possible because a system of nequations with n unknowns will always have at least one solution.

These solutions, however, might be very sensitive to changes in the coefficients. This meansthat the hedge might be unstable and frequently lead to very large hedging positions. Thisproblem can be eliminated by the introduction of tolerances in place of the equalities in thesystem oi equations above. The solution will not be unique anymore (in fact we will havean infinite number o[ solutions) and we will have a choice in selecting the hedge. We canIook for some additional properties like low trade volume in the execution of the hedge. Wecan then pose the hedge selection as an LP problem:

minfX,+fY, i:1,...,nsubject to

I : 1,..., n

V':V-rvhere: X, is the additionai long position in the i-th hedging instrumenr,

Y, rs the additronal short position in the i-rh hedging instrument:a, and b' be the limits on the sensitivity o[ the portiolio value to changes in thernterest rate in the i-th maturity;V, V* and V- be the present value of the whole portfolio including all the hedges,of the assets and oi the liabilities respectively; andD* and D- the duration of the assets and liabilities respecrively.

The stability oi the soiution to this LP problem will depend essentially on the magnitudeol the tolerance limits, a' and b,: the larger these limits the more stable the solution. On theother hand, the protection against interest rate risk will be reduced as the absolute size ofthe limits increases. The actual values ol 0YlAr, indicate the magnitude of the change in theportiolio value for a change in the interest rate in the i-th maturity. Examining the set ofthese values the book manager could adapt his positions in order to maximise the benefirsfrom anticipated changes in interest rates.

SummaryThis chapter on hedging theory has defined the concepts of duration and convexity, and usedthem to outline three conditions for portfolio immunisation. These conditions are sufficientto guarantee that the net worth oI the portfolio will be non-negative w'hen subject to parallelshrlts in the yield curve. Because there are many possible hedges to provide this level ofimmunisation, M: is introduced as a means to tighten the cash match between the portfolio'sassets anci liabiiities and thus provide some protection againsr non-parallel rate movernents.This is shown to have some weakness in that it only looks at the variance of cash flows aboutone point in trme-the duration-and it does not take tracing cost into consideratron.Suggesticns for improvement include accepting a trade ofr between iower transaction cosrsand increased M2, and minimising cash mismatch across more than one date. Finailv, ageneral framework for minimising the sensitivity to arbitrary changes in the yieid curve \r,asin r rociuced.

AVa

2. Hedging interest rate risk

Hedging objectives and hedging instrumentsA single swap or a s\\'ap portfolio is hedged in order to protect its value from changes ininterest rares. Iimark-ro-marker accounting is used, these changes in portlolro vaiue are takcninto income direcril. Correct hedging minimises the volatilitl'of earnings and book value atthe same time. Even in the absence of mark-to-market accountins, hedging is extremelyimportant because a change in net market vaiue measurcs the economic gain or loss in presentvalue terms to be reaiised over time.

In managing a swap portfolio, each currency' book should be hedged against interest raterisk, to the extenr efficient hedging instruments exist. Hedge efficiency is determined by trvolactors:

(l) Correlation-how well the value ol the hedge tracks the value of ihe target hedgedposition; and,

(2) Cost-the rransaction and carry'ing cost ol maintaining the hedge position.

Traditional hedge instruments include qovernment bonds, Sovernment interest rate luturesand Eurodollar futures. All of these instruments are imperfect hedges in that their pricebehaviour is not periectly correlated with price changes in the swap markets. Note inparricular that ir is very difficult-if not impossible-to hedge movements in swap spreads(knorvn as spread risk) other than by writing offsetting swaps. For this reason, a srvap bookshould be managed so as to lay ofl-open swap positions as quickly as possible rvith offsettingswaps which are structured as hedges.

For some books such as the CS book, where the bid-offer in the CS bond market can beas high as j point. rhe transaction cost can be a major lactor in the hedging decision. Thebook runner will have to decide on lhe trade-off between interest risk protection and cost.

Of the three main categories of hedge instruments, government bonds are the simplestconceptually. If a neu,swap is entered paying US$100m fixed rate lor 5 years versus Libor,then the hedge will be approximately a USSl00 long position in the 5-year on-the-run USTreasury. There are several practical considerations, however, that limit the use olgovernmentbonds. In DM, for example, it can be very expensive to short the Bunds because there is noactive repo marker. and the pricing of DM swaps should take into account the cost of thenegative carry. In adcjition, the hedge instruments available in Bunds are only in the longerend of the market, i.e. the 7-10 year range. Hedging a 3-year DM sivap with IO-vear Bundstherelore exposes the swap book to greater interest rate risk than il 3-1'ear Bunds wereavailable.

Interest rate futures are advantageous for several reasons. Futures olten are more Iiquidthan government bonds and have lorver bid-ask spreads. Thereiore, the transaction cost oiputting on and laying off futures hedges is lorver than that associated with hedging u'ithtreasuries. Additronaiiy, the transaction cost may be less with futures, depending on theborrowing costs in a given currency. If repo costs ore high, it r+,ill be cheuper to ntaintain thefutures margin account compared to borrov'ing the full price of a treasltry. Futures are alsoadvanrageous in that they allow short positions to be taken in currencies, such as Dlvl andsterling, that do not aliow treasuries to be shorted. There are several disadvantases horvever.Primarily, although changes in futures prices are highly correlated with price changes in theunderlying cash market. the correlation with the swap market is rather poor. Another factorthat diminishes the artraction of lutures as hedges is that there are relatively [eu'maturitiesavailable. This is parricularly true of futures on government bonds. For example, US Treasuryexchange traded lutures are avaiiable only for 3-month, l0-year and l5+ year maturities.Eurodollar futures are useful as short maturity hedges, with quarterly maturities out to 2years and to a certain extent out to 3 years in US dollars, although the liquidity in the farcontracts is very thin.

In currencies such as ECU, DM, Sh and Yen where efficient hedge instruments are notavailable, interest rate risk is unavoidable. In the Yen swap market, for example, althoughrhere are Japanese government securities and futures available, the correlation between themovements in the swap market and the hedge is generally poor. In the past, swap rates havemoved in one direction rvhile the hedge moved in tlre opposite way, due to the spreads. Forihese currencies, the best hedges are other swaps. ln other currencies, particularly ,{S andNZS. iutures either do not exist or are too illiquid to be used for hedging purposes andgovernmenr bonds crnnot practically be shorted. The solurion is that $rap exposure in thesecurrencies should be run short (i.e. net

.fixed ouerborroweil, and hetiged with lona positions ingouerrunent bonds as requiretl. Short hedges are created by writing short swaps (fixed ratePa-YUr /.

B Swap options

1. 'Swap derivatives'Swap options are contracts where the underlying asset is a srvap. The most common typesol swaD options are caps and floors and swaptions.

A cap gives the holder the right to pay a predetermined coupon at interest pavment dates.Caps are generally bought by borrowers in order to put a ceiling on interest payments. Afloor, on the other hand, gives the holder the right to receive a predetermined coupon atinterest payment dates. Floors are often purchased by investors in order to guarantee aminimum coupon. Caps and ffoors are essentially a sequence of European options on affoating rate wirh expiry set on the coupon payment dates,

A collar is the combined sale oi a cap and purchase ol a floor, or the combined purchaseol a cap and sale of a ffoor. A corridor is the combined purchase of a low strike cap and saleof a higher strike cap, or the combined purchase of a high strike floor and sale of a lowerstrike ffoor.

A swaption gives the holder the right to enter into a swap at a given date lor a prespecifiedtime paying and receiving predetermined interest rates. We shall see that since the ffoatingrate leg of a swaption has a value close to par, swaptions can be considered as options inthe value ol a fixed rate bond. Examined in this way, the call swaption gives the holder theright to receive fixed, while the put swaption allows the holder to pay fixed.

Other types of swap optrons also include: currency swaptions (an option where theunderlyrng asset is a currency swap); currency options (simply a currency swaption where theunderll'ing currency swap is a zero coupon swap).

rnctng turopean swaptionsDenote:

Then assuming all other notation as in the calculationa European swaption which gives the hoider the right

to be the strike (expressed annually ilcoupons are to be exchanged annually;semi-annualil' coupons are exchanged on a semi-annual basis. etc.):to be ln.n, expressed in the same iorm as X.

of the forward swap rate, the costto receive fixed is given by:

ol

(3)where

I ort-ln-+ "x2 and dz = dr - oJT"A-U1 - oJt"

a2 denotes the volatility of rhe lorward swap rate and N(.) is the cumulative normaldistribution.

Equation (3) has a coupon ol extra terms which merit some discussion.The pay-out in a swaption does not occur at one instant (unless the holder of the swaption

exercises and at the same time enters into a swap to close out the position). Thus thesummation term sums the pay-outs and discounts each by the zero coupon rate relevant lorthat maturity. The division by v ensures that for sw'aptions, where coupon exchanges are notannual, the expected pay-outs are correctly scaled.

The price oi a put swaption is given by:

C= [XN(-dr)-N(-d,)], 1 iv i=il r r,(t,)\"'

_/

laP: UN(dr)- XN(dr)l x - Lv,=n*

c: [2f N(-d.) -

8 ss4 N(-d,)-l * DiscountLr00 r00 'l

(4), (,\'-

notation as above.Consider pricing a DM European call swaption with a slrike ol 9.3% where the maturity

of the option is 2 years and the underlying swap 5 years. We calculated above that the iorward3 year swap rate,2 years forward is 8.554%;therefore the price of the swaption is given by:

r,(ti)\'"

rvhere

Discount : (1 + 0.09091)3 il + 0.08810)' (r + 0.08778)sUsing a volatility oi 10% then:

0.08554ln_ +0.093dr :

-

0.520s0 10J2

dz : -0.5205 - 0.10J2 : -0.6620

Finally, from normal tables N(-d') : 0.698and N(-dz):0.7+S

also Discount : 2. 140

so the cost of the swaption is 2.05% up front.The cost of the put, swaption i.e.. the right to pay fixed. is only 0.46o/o. The put is less

expensive than rhe cail because the iorward rate is below the strike. II the strike was set equaito the forward swap rate then the put and call would be exactl,v' the same price.

r0.10):' (2))

4. CammaWhen the hedging ol currency oprions was studied, it rvas nottceo that in a cielta-hedgedportlolio, ii the trader was long gamma (long on options) and interest rates moved, the valueof the portfolio was always posirive (assuming time decay can be isnored). The concept canbe exrencied drrectly to swap options. Figure 12.1 shows the profit and loss pronle of theEuropean swaption posirion (described above in section 3) under parallel shifts in the yieldcurve. The movement of the hedge (shown as swap on the plot) is also recorded for movementsin the yield curve. The bold continuous line shows the combination of the swaption and thehedge. The profit and loss profile is concave as we would expect from our experience ofhedging currency options.

For a single swaption, the gamma we recall is a measure of how often a hedge should beadjusted in order ro retain delta hedging. A position with a large gamma should be rehedgedfor small moves in interest rates.

5. Hedging time decay and volatility riskTime decay (as noted in section I above) usually only affects long option positions. If thedecay is severe, the option is at-the-money and close to maturity (see Chapter ! and inparticular Figure l.l the trader could sell at-the-money options.

In practice, traders have limits on the time decay of an entire portfolio. Volatility risk, onthe other hand, affects both buyers and sellers of options. If a trader is Iong volatility (ingeneral long at-the-money options) then a fall in volatility will lead to a loss in value.

We discovered empirically (see Chapter !) that the effect of a change volatility ol l.5ohamounted to only a 20 bp change in the value ol a swap option in the worst case. Thus thevolatility risk is small lor a single swaption.

I08.I2TAB.TXT

Table l. Swap positions required to hedgelyear European style DM call swaption withstrike at 9.3% (notional principals shown inmillions of DM, negative indicates shortposition)

Current couponswap maturity

Notional principal(millions of DM)

6-monthl2-month2-year3-year4-year5-year

0.00-

0.02+ 35.00-

0.30-

0.38-35.14

Table 2. Swap positions required to hedgeSyear European style DM call swaption withstrike at l2% (notional principals shown inmillions of DM, negative indicates shortposition)

Current couponswap maturity

Notional principal(millions of DM)

6-mon thl2-month2-year3-year4-yea r5-yea r

0.000.17

49.62-

1.23

-

1.34-

50.91

4. Quanto optionsQuantos are based on the use of a foreign cxchange rate as an index, but with a pavout in USdollars (or some other third currency).

ExampleOn 28 lvlay 1992 rve bought a European-style quanto'put', giving us the right to sell lira iorDeutschmark at 765.4 L/DM on a'notional'ol $100 million but with the payout in USdollars; the option expires 1i December 1992, but delivers the pay-out (if any) 6 months lateron ll June 1993.

Three distinct currencies are involved, so the loreign exchange risks associated rvith aposition in these options can be complex. Hence we need to outline a precise method forcalculating these risks in order to evaluate and hedge them.

How to value a quanto optionThe first step involves a clear statement oi the terms of the option. Rather than using thelanguage of puts and calls, as per the other types of complex options above, it is better tophase the terms of a complex ('non-vanilla') option directly and explicitly in the form of a'contingent claim', i.e. in the form of a'contract'where it is stated that the holder of thesecurity is delivered X on day Y providing condition Z holds, and so on and so forth. In thisway we avoid any risk of ambiguity.

In the case oi a quanto option the ioreign cross rate (L/DM in the exampie above) servesas an index, and the idea rs that the dollar pay-out depends on the value of this index at thematurity ol the quanto.

The terms ol Example A can be stated more explicitly as follows. Let K : 765.4 L/Drv'l(strike), and let S* be the L/DM rate on Il December 1992 (the expiration date). If S* > Kthe holder of the quanto option receives Sl00m U

-

(K/S-)1, payable on 11 June 1993; ifS* < K the holder receives nothing.

As awkwardness arises from the lact that the exchange rate at expiration S* appears in thedenominator in the expression for the contingent pay-out; this results from the convenientbut inessential market convention that has the exchange rate quoted in L/DM rather thanDivI/L. In Example A, however, the structure of the deal is much clearer if the exchange rateis reversed and expressed in DM/L. Then the'strike'rate is K 1.3065 x 10-3 DMiL, whereK: (K)-r. Suppose we write S* for the DNI/L rate at expiration (11 December 1992), thenthe terms of Example A can be rephrased again as follows: If S* < K then the holder receivesC(K

-

S*), where C = USSl00m/K, payable 6 months later; if S* > K the holder receivesnothing.

Note that rvhen the contract is specified in this way the contingent payout is /inear in S*.lvloreover K

- S* is the contingent pay-out lunction ol a single'vanilla' ioreign exchange

option to sell one lira at the rate K, and receive Deutschmark in payment. C is simply aconstant.

The quanto option can therefore be valued according to the lollowing methodology:

(i) First we value an ordinary vanilla foreign exchange option to sellone lira on 11 December1992 at the rate K (:1.3065 x l0-3 DM/L), receiving Deutschmark in payment.(ii) Then we multiply the result by (l + Ro-)' where t is the time leit to expiration; thiseliminates the Deutschmark discount lactor used in calculating the present Deutschmarkvalue of the expected Deutschmark pay-out of the vanilla option, and we are lelt withsimply the expected Deurschmark pay-out of the vanilla option.(iii) This is then multiplied times the'conversion factor'C which derermines the expecreddollar pay-out.

(iv) This finally has to be multiplied times (l + Rp.) -('+o s) in order to discount the expecteddollar payment to the present. Note the extra 0.5 years added to t. which accounts forthe delayed pay-out-this is the only place the delay enters inro the calculation. It is alsoworth noting that the dollar interest rate only appears in this term, and not elsewhere.

Complete formula for quanto valuationSumming up, the Dresent dollar vaiue oi the quanto oprion in Exampie A is given b-v thefollowrng iormula:

e : (l * R"rs) -,'+o.s)C(l + RD"{),p

Equations: hedge positions and delta equivalents

So lar rve have not merje anv snecinc assumptions about thespecialise to the case oi Exampie A, rvhere Q depends oniy'exchange rates, i.e. Q(:, d) = H(zl0), w'here H(x) is a functioncase by the use oi the chain rule rve obtain:

form oi Qtr,0) But nori uson the ratio of the two ioi:ergnof the single variable x ,t lhal

d,Q(r, 0): []-'6q^-

,!-, F

where dq: dHidx is the delta of the quanto option, taken with respect to the underlyingindex rate (i.e. the DMiL cross-rate). Insertion oi these expressions into the relations deriveciearlier than leads to:

Nr_ : 0-'6qNa,Nov : aP-26oNo

These are the required lormulae that state precisely the amount oi Iira and Deutschmarksneeded to hedge Nq quanto optrons at time t. Alternatively, reversing the signs, we can thinkof these figures as representing the delta equiualent currencv spot position corresponding to along position in No quanto options at time t.

In Example A, we have No: l, and do is given explicitly byAa : (1 * Rurr) -,'ro 5)C( 1 + RDM)'ip

where, 6, is the ordinary delta of a vaniila European option entitling the holder to sell onelira in exchange for Deutschmark at the rate K on ll December 1992.

Note that il the DMiL rate increases, the value of the quanto option decreases, so dq isnegative. This implies that the hedge consists of a long position in lira, and short position inDeutschmarks. It should also be observed that when the book is hedged properly the presentdollar value of the lira position aN. should always be precisely opposite the present dollarvalue of the DM position fNo^r.This can be understood directly since if the DM/L rate doesnot change, the book will be immunised against US$iDM fluctuations only if the sensitivitvto USSiL fluctuations is exactly opposite.

Forward currency positionsFor simplicity ol illustration we have assumed that the hedge is achieved by spot positionsin the foreign currencies.ll fortard currency positions are used then the method describedabove remains appiicable with just a few minor modifications.

Let us denote by a, the lorward USS/L exchange rate for time r. That is, z, is the presentcost in US dollar o[ one lira delivered at time r. We shall continue to write a lor ao, rhepresent exchange rate (r : 0). For the number olsuch lorward contracts we shall write N.(r).

Note that r need not be the same as t, the option expiration time. The present US dollarvalue ol the book in this case is given by:

v5 : a,Nt(r) + p,,No"(r') + Q(e, p)Nowhere r' is the DNI lorward delivery date.

To calculate the hedge positions we proceed as be[ore, this time making use olthe relationsd,(r,) : t,fa and ,ls(0,) : 0,10.

The result is:

N-.(r) : -a(t,0)-'dqNo

No*,(r') : a(0,'h- tdoNo

rvhich gives us the number ol lorward positions required in each caseol the forw'ard currencv reiations:

. Alternatively, by use

Rot)- ''0r.: (1 + Rusr)'( i * R1) -'r anci 0, :0 + R,r\sj'il +

15

As the competition among investment banks intensrfies, new products are created by tailor-ing7structuring cash (shares and bonds) and derivative (forwards, lutures, swaps and options)instruments to meet specrfic investment or iunding requirements. The purpose of designingthese strategies is to meet investors'specific hedging or investment objectives wirhout which'new issues'cannot be successfully placed and consequently satisfy borrowers'target fundingrequirements. The secret lies in satis[ying the end objective oi providing borrowers with thelowest cost of funds, while offering investors the highest possible returns.

However, as the gap between the desired cash flows of borrowers and lenders is becomingincreasingly divergent, especially when the borrower has sub-Libor funding target while theinvestor is aiming ior Libor+ yields (not to mention the intermediary arranger,/underwriter'slees), financial engineering becomes the main source of competitive advantage.

Consequently, sv/aps, options, forwards and futures (the basic building blocks of structuringand new products) have become the most important instruments in bridging this gap.However, as the globalisation and complexity oi these derivatives progress, the arbitragesavailable through cross-market differences (e.g. swap windows) are declining, while product-oriented arbitrages are becoming increasingly important. Today, the best arbitrages areachieved by providing tailor-made cash flows targeting specific niches in investor demand,usinq the most efficient static hedges followed with dynamic risk management techniquesutilising sophisticated analytical tools and exhaustive'what-if'simulations.

1. Structuring: techniques and methodologyFinancial engineering involves transforming the risk and exposure profiles as shown in the'CME Futures and Options Strategy Charts'o (see Appendix). The four basic merhodsinvolve:

(l) Eliminating price risk completely by locking into a future price using lorward or swap,while giving up the opportunity of gaining if prices move favourably. For example, usinecurrency swaps which lock into forward currency exchange rates.

(2) Protecting against downside risk while still maintaining the full upside potential by usingoptions. However, lrke any 'insurance' contracts, this costs money and involves thepayment ol an option premium; for example, buying interest rate caps.

(3) Financing the optlon premium in whole or part by selling some or all of the upsidepotential. For example, selling out-oi-the-money currency call options.

(4) Taking on risk by writing options, thereby receiving up-front premiums which are usedto reduce borrowing costs. For example giving put options to bond investors.

It is very important to understand that while derivatives can create risk exposes, they arealso used as a tool ior hedging, managing risks and creating interesting strategies.

Basic building blocks of structured productsUnderlying instrument, plus:

a

a

Swaps (exchange oi fixed lor floating payments).Index or'quanto'swaps/swaptions (fixed payments in one currency exchanged for floatingpayments in the same currency but based on lnterest rate [index] in some other currency).Caps (pay if interest rate rises above 'strike' Ievel).Floors (pay ii interest rate falls below 'strike' level).Forwards/lu t u res.Options.

a

a

a

a

Why invest in structured products?There are tu'o main reasons that structured investments are olten attractive: these arefexibiiity and enitancenient ol re!urns.

FlexibilityAn asset can be created to meet almost any structural reeuirements of an investor: Forexample:

Financial engineering

6. Yen variable redemption/reverse dual currency bonds

The yen variable reciemption,reverse dual currenc\'lssues are e iurther adVancement oI the'Helven and Hell'boncls, but rvhich involve two simultaneous lsstles. One constltutes a'Heaven and Hell'issue denominared in yen terms, where the bond pays the investor a highyen coupon, bul where the redemption is related to the yen/doilar exchange rate at ntaturitr'.The complimenrary reverse dual currency issue has both initial and final principal fixed inyen, bur the bond pays a coupon denominated in doiiars. These bonds are usually issuedtogerher because the redemption lormula in the yen variable bond creates an implicit luturescontract which can be effectively used in the redemption of the reverse dual currency issueto create a lower cost ol fundine ior both issuers.

Example issue

Yen uqriable issue Recerse dual currency lssrreAmount Y20,000,000,000 Y20,000,000,000Coupon 870 in yen 7.5% in USSIssue price 101.5% l0l.75o/oCommissions 270 2o/oExpenses 5100,000 5100,000Maturity 10 years 10 yearsRedemption

formulas: P*F* l- (F-S)l Par iiF > 84.5r1+-\{' 'lFo F I Par*(Fi8.1.5) ifF

F^z< u.l7>d, v,>zrul

d.

otr)2,trJFz

ila-

3*>u)r-=

pc-3$> t/)s=

$I

o

I

-oJ

+

J

f-

+*aJO+F,l

_:n??U9>-

f:+>QJ6ifoo

z

tJ

N t-ha!vz

'lli ++a9 xh

I-.1 oa

=fax

t4Ir I's)

o

O

\J

q

>F

x t-X i'rE!lac/,es

o- IZ!z,foX

(Jau1

-

z

xhaJ.B

o.v{z,

Finall-v. one hlts to weieh up the benefit ol possible pro6t compared ro potentiai loss. Forexampie, rn the currency piay oishort seil Iibuy DN{ the upside on the srerling breaki.'- outoi ER\{ rvas far greater than the potential loss on suffering interest rate differential iborr.,vingCBP,4ending DM) between the two currency rates on money-market interest rates, for a shortperiod ol time, until the British Chanceiior let sterling devalue and find the marke: rate onl7 September 1992. Traders then squared their sterling posirion by selling the DI\l andreceiving more f than that needed to pay off the f borrowing, sterling having depreciatedagainst the DM, hence making a huge currency gain. This was a low risk/high gain situationagainst which one has to compare little upside (say possible profit ol 2,h with a 75%probability) against potentially bigge r loss (say possible loss of l0% with a 25o/o probability).In the former case one might profit by 15% against a possible loss of 2.5ol0. However,quantiiying and assigning probability factors to uncertainties on a fast moving trading flooris perhaps the last thing a trader is thinking of while watching the tempo, trend and volatilityoi the exchange rates and digesting new iniormation/data/statistics as they uniold. However,for the longer-term portfolio managers, cross-currency swaps easily provided a quick andliquid method of going short on their sterling portfolios with leverage commensurate withthe firmness of their views and confidence.

Without disciplined analytics and technical research, the thin dividing line between havingmarket views and speculation disappears. Stop-loss limits are vital, as biding time in the hopethat eveitually the market will turn in your favour, costs money and loses other opportunities.

Quanto bond structures and reverse/inverse floatersWe shall now look at some of the latest (March 1993) pieces o[ financial engineering andanalyse how they can be used to isolate desired market views. Note that many are highlyIeveraged and have ernbedded positions in the base currency as well as the indexed portions.These new products enable pension iunds and insurance companies, which might be restrictedin making foreign investments, to create synthetic structures and exploit specific views thatthey have or hedges they require. Quanto trades can be used to create positive carry even inthe negative carry trades, while eliminating loreign exchange risks.

Libor differential noteFor example: Coupon 5 xRedemption: Par in USSOften shown rvith high fixed

Dlvl3m Libor -

FIM 3m Helibor -

3a0 bp), paid on US$

coupon and risk in the principal redemption.

Current market. Market prices widening Libor spread, and base currency is positively sloped.

lnvestment view. Investor believes spreads rvill not rviden much and that base currency rateswiil not rise sharply.

Analytics. The structure is leveraged five times. The 340 bp spread in the DM and FIIvIdrverging curves can be compared to the USS3m Libor ior relative advantage. Furthermore,the payment is in US$-a positive curve currency-to make currency attractive with as highfixed rates as possible.

lndexed inverse floatersFor example: Coupon 10.40%-(AuS6m BBR converted to Act/360), paid on YRedemption: Par in YOften shown with high fixed coupon and risk in the principal redemption.

Market conditions. Index currency and base currency curves are steep.

Rationale. Investor believes that indexed rates (and base rate) will not rise sharply.

Yield curve inverse floaterFor example: Coupon 14.65ok-Flr 5-year offer-side swap rate paid on AuSRedemptron: Par in AUSOlten shorvn with high fixed coupon and risk in the prrncipai redemptron.

Market conditions. Index currency yield curve is inverted and base currencv curve is positive

Rationale. Investor bullish on index currency's bonds (and does not think base cLlrrenci,/."tcc rrrill rise)

Table 12. Intermediarr' (l\1')Yerr Bond's Cesh llcr* F romt(to)rSB Frcm (to) SB Total0 99. 100.000 900,000 t00.000,000I (3.000.000) 3.000,000 (Libor

- 0.10%) ilibor - 0.40%)

, ,r.000.000) 3,000.000 (Libor -

010%) (Libor -

0 40%)I il01,000,000) 3,000,000 rLibor -

010%) ilrbor -

0.10%)

Table 13. Sr+ap brnk (SB)

I nter-Fromi(To) (To)lFrom Fromi(To) (To)/From Froml(To) To7(From) mediaryYear SBZ SBZ INTivlED. INTMED. TPB TPB Fee0 0 (900,000) 0 900,000 0I 7.161,876 (Libor

- 40%) (Libor

-

.t0%) (3,000,000) (1,243,876) 3,000,000 20,0002 7.263,876 llibor -

40%) (Libor -

40%) (3,000.000) (7,243.876\ 3,000,000 20,0003 7.263,876 (Libor -

10%) (Libor -

a0%) (1,000,000) (7,213,876) 3,000,000 20,000Table 14. Third partl' bank

(To)/From From/(To) Option Intermediary Bank's netYear TPB TPB premium spread cash0 (900,000) 0 (10,361,030) (10,478,t24)r (1,000,000) 7,21i,876 0 180,000 4,063,8762 (3,000.000) 1,243,876 0 180,000 4,063,8763 (3,000.000) 7.243,8'16 0 180,000 ,1,063,876

Table 15. Calculation of the reinvestment rate

Treasury note Swap T + Swap spread Present value ReinvestmentYear yield (s.a.) spread (p.a.) of original rate0 ( 10,478,1 24)I 6.4900,',0 1.000% 1.630% 3,775,775 4,063,8762 6.860% 1.000% 8.014% 3,483,187 4,063,8763 6.9209i, 1.000% 8.077% 3,219, t62 4,063,876

10,478, I 24 't .97290A

Table I. Initial exchangeAmro paid DM Amro rec. USS

Wirh Denmark -Dlvll83m rUSSl00mWith Dresdner + DM l83m

- USS l00m

With ltaly None None

Table 2. Principal exchange-Year 2.5

Amro paid Dlt{ Amro rec. US$With Denmark + DM l83m

- USSl0Om

(Final principal exchange)With Dresdner None NoneWith Italy None NoneCitibank

-DMl83m + USSl08.2m(FX forward sale to Citi at DM/USSl.69l3)

20 Risk management

1. lntroductionThe rechnique oi risk mantgen'rent and portfoiio immunisation is one oi the most importantaspects ol profitiloss and cash florv controls for traders, fund managers and borrowers tocreate a risklreward prohle suirable ior their own specific needs. The 6rst step in order toconrrol and risk mlrnage the derivatives ([orwards, [utures swaps and options). is importantto see how they rrade, methodologies by which these products are priced and analyse theirhedging in theory and practice as we have seen in the previous chapters. Next we determineand quantily the risks, present and that expected in the future (as per market indicators,forward prices, forecasrs, etc.), follorved by various simulations ior uncertainties and un-knowns! Finally, we review the behaviour and correlations between the derivatives and theirhedges/underlying cash nrarkets and monrtor their net impact on a portfolio of cash andderivative instruments.

Su,aps are essentially instruments that permit institutions to take interest rate and foreignexchange positions off-balance sheet. They are also the most elhcient instruments availablefor assetiliability portfolio restructuring. As banks and other financial institutions warehouseswaps, i.e. enter into swap agreements without matching counterparties, instead oi hedgingtheir interest rate risk on a deal-by-deal basis which can prove inefficient and uneconomical,they develop portfolio management techniques that permit them to take on mismatches indates and unusual sets o[ cash flows. These risk controi and immunisation techniques havebecome extremely sophisricated with the aim o[ locking in the va]ue oi the trade and enhancingthe net portfolio quantum by at least its cost ol carry.

This chapter demonstrates the management of fixed and floating rate swap exposures andthe risk reporting system lor the ntanagement. It introduces an analytical technique lordecomposing cash flows of either an individual swap or an entire portfolio into a series offixed rate current coupon swaps, known as I + i swaps. These 1 * i swaps provide the meansof marking each book to market, determining the hedges and constructing position reports.They enable the swap book-runners to manage and hedge complex cash flows on a portfoliobasis and simpliiy the task oi reporting risk positions.

In general, the system should analyse pure cash flows which represent actual payments toor from counterparries. Actuai cash ffows will be sufficient to value and hedge properlypositions u,here no notional principal is involved, i.e. all fixed rate currency swaps, Iong dateforeign erchange and ioreign currency zero coupon swaps. For swaps with notional principaiamounts (fixed interest rate srvaps), the notional principal must be included in the cash florvsanalysed. This is because inreresr rare swaps have the same volatility as current coupon bondswhere principal is returned at maturity, and this relationship must be preserved. Omissionol rhe notional principal amount will also prohibit the I + i hedging swaps from eliminatingthe associated floating exposure.

1+ i swaps rvill be used in examples throughout this chapter and are of fundamentalimportance to swap exposure management. By definition, the / + i s*,aps for a series of fredcash J7ot,s contprise tt series of current ('ouporl bullet svraps x'hose aggregale fxed cash Jlowsexocth.defeose the originul pa)tments. Since the cash flows olthe portfolio are identical to thecash floivs created by rhe I + i swaps, the present value, duration, and all other riskcharacteristics of the original cash tlows are preserved. In particular, if these 1 + i srvaps arecalculated and written ior the fixed rate swap book, there rvili be no interest rate exposureremaining in the book.

Unlike the origrnal swaps that create our portiolio, the I + iswaps have recognisable cashflorvs and are easy to analyse. These swaps are the basis on which the books rvill bemarked-t.o-market and position reports constructed. In the process, it will also be shown thatin managing the risk ol the cash and fixed rate books with I * i swaps, Libor exposure inthe floating book will be eliminated.

The requirements ol rhe swap management system rvill be outlined in the next section. Thefollowing two sections wrll then examrne the exposures created by cash events in the fixedrate book and their impact in the cash account. In the process, it will be shown that bymanaging the fixed rate book, the Libor exposure in the floating book rvill be eliminated.Sections 5 and 6 rvill then expiain the implementation of the risk management reportingsysrem, in theory and practice. The appenciices to this chapter include sampie reports to show'the layout anci methodology ol being arvare anci managing the risks anci exposures.

21 Risk management reports

1. lntroductionThis chapter deals with the risk management o[ an integrated portiolio, by currency books,containing: Trading Book-swaps; options; swaptionslcaps/floors; and Hedges--cash andfu t u res.

Separate risk and evaluation reporting oi each or any singie product type, I believe. defeatsthe whole concept oi portfolio management, especially where the derivatives are hedged ona net exposure basis by the traders/book-runners and where splitting hairs over profit andloss allocation is quite futile.

Besides the corporate treasuries and iund managers, the banks and security houses dealingin derivatives manage colossal portfolios of billions of dollars equivalent in a multitude ofcurrencies and prociucts. Often these books are inadequately reported to the senior directorswho may or may not be capable of understanding the risks involved and the potential lossesto the company in case of unexpected or adverse market movements.

Unscrupulous traders with or without the co-operation of those directly in charge of them,are capable of 'couering-up'certain trades and exposures and manipulating the profit and lossresults. Furthermore, the back-office (settlement and accounts) clerks are easily overawed bythe complexity of the structures, and at times too timid to approach the traders with'silly'questions. It is also the case that so many back-offices are far behind the sophistication ofthe front-office due to:

r inadequate training; andloro outdated accounting/recording/ssltlement systems; and/oro lack ol integration between diverse front-office pricing tools; and/oro incompatible software programs handling the front- and back-office operations.

Hence, it is not always possible to enter the transaction into the firm's systems in the lormit was struck. Consequently, the deal may need to be replicated as a series of simplertransactions, which have not, in reality, been made: or suspense accounts are set up to'dump'the differences between the accounting and the trading P&L, and reconciliations withunidentified errors and omissions simply hidden among the tons of paperwork. Unfortunately,automatic processing lacks integration of some systems.

This chapter also shows how risk exposures can be quantified and reported in a practicaland concise way so that sen;or management are aware of what their book-runners are up to,realise what risk each individual trader is taking, and the total net exposure to the firm:thereby directors are enabled to participate actiuely in the decision-making process of riskmanagement.

The reporting formats and information contained should also cater ior traders' needs andaid traders in managing their positions with technical efficiency and within their limits. Themark-to-market should be properly carried out with the correct market data and the resultantP&L analysis followed-up with post-mortem meetings, and whether a profit or a loss wasmade-as both are educative.

Portfolios should meet target returns, after charging cost-of-carry for the capital utilised,and the up-front trading profits/premiums conservatively accrued over the life of the deal.Reserves (for potential losses due to market adversities or deficiencies in the pricingmethodologies) and provisions (for bid-offer spreads, future hedging and rollover costs) ihouldbe regularly reviewed for adequacy. Stop-loss controls, position limits (overnight andintra-day) and the quality and type of deals struck by individual traders should be monitoredsystematically, without overlooking the credit and liquidity factors.

The rest of this chapter illustrates specimen reports and explains the importance ol various'greeks'and exposure analytics. The reports are by no *.un, perfect, but they cover all thecriticalrisks and sensitivities and identify the characreristics of individualbookslby currencies)and the aggregate exposure to the company as a whole. Mark-to-market (m-t-ml evaiuationand P&L are also shown.

We start by checking the adequacy olprovisions (future hedging and other costs lor specificdeals) and reserYes (contingency for uncertainties). It should U. not.a that this distincrion isalso important for tax reasons, as the former artract tax reliel while the latter cio not. Theaccounting treatment also differs and makes a difference to the bottom-line p&L as well asmanagement and traders' perlormance evaluation.

ilr positions and P&L (mark-to-ntarket rcvalualion) Drtc: l0-June-1992\otes ri)

Positionat l0-6-92

r1\Positionat 9-6-92

{3) ({)Daily CCY Darlv (USS)

P&L Equrv MVT(5) i,l

\l-T-D CCY \l-T-D tUSchange movement

rs) (9)Nlonthly Tooal'sP-tareet FX-rate

(s)Positron

ar 3l-5-92

*apsis lJook:: T-bond

Futures

S5

ps Bookrs Books-Gilts

Futures

rwapSns Book's-Bunds

Futures

)!ISwapsnsB35-t_-11-' FuturesI Futuresr:CU

5.806.700(19,372)

(314,923)lt0

i r?1 ?)s

I,223,400451,860

0(r s,0s5)

I,660,205

4,639.000.

r 2,6300

( 14,915)4,636,715

(382,300)00

(57,28 7)0

(439,s 87)

5,8r2,r00(18,3i l)

(315, I 96)4,610

5,461,201

r,235,000447,013

0( r 3,1 80)

r,668,833

4,652,20013,590

0(20,645)

4,615, I 45

(388,700)00

(59,r87)0

(41't,987)

5,69 5,600(?0,5 r 6)

(287,285)4,430

5,392.229

(s,036,800)49t,347

6,234,996( 1,867)

t,687,67 6

4,773,600r 4,209

0(99,e26\

4,687,883

(276,000)n

0(53,465)

(270)(329.135)

l I 1,100l,l+{

(27,638)(4, l l0)

80,,{96

6,260,200(39,487)

(6,234,996\(r3,r88)(27,471)

( 1 34,600)( l,s79)

085,011

(5 1,1 68)

( 1 06,300)00

(3.822)132

(109,900)

USS/Il.8300

DIVT/USS1.5963

US/ECUL2813

Yen/U$127.1250

(5.400) (5,100)(1,052) (i.051)ln 1?': ln l?1(4,300) (1,i00)o

5 Portfolio immunisation

1. Risks and hedging of risks of a single swapHavinr: investrgated the mark-to-nrarket rnethodology to value swaps, we shall in this chapteridentrly the rrsks ol entcring into u swap, and then go. on to explore how these risks can behedged. One such method, involving constructing a portfoiio of the swap together wirh aseries of current coupon swaps, is described in detarl. Although the suggested metnod is quireinefficient to hedge a single swap, the extension of hedge portfolio of swaps (and indeed swapoptions) is trivial.

The risks of a swapThe holder of a swap laces three types oi risk. Credit risk, the first of the three, is the riskthat one counterparty in a swap defaults on payments and has an exposure to the othercounterparty. This subject is discussed exhaustively in Chapter l.

The second risk, which is perhaps the most important is that of In(erest rate risk. As interestrates change the mark-to-market olthe cash flows alters and thus the value of a swap changes.

The third risk faced is Currency risk. Swap dealers manage swaps in single currency books.By doing so currency swaps are simply treated as two sets ol cash flow and hedged withother cash ffows ofthe same currency in different books. Foreign exchange hedging ofa swapbook is discussed in grearer detarl in Chapter t.

Hedging the risksThe perfect hedge lor a single swap, as mentioned before, is to enter into an equal and opposireswap. As the cash flows are exactly equal and opposite, the only risk which remains is creditrisk.

In a large portfolio, however, this method of hedging is extremeiy inefficient and it wouldbe vtrtually impossible to execute as a result of bid-ask spreads. Clearly an alternativeapproach is required.

Prevlouslv, we saw that swaps can simply be considered as a strip oi cash ffows and thatthe value ol an interest rate swap is given by:

n

v : f C(ri)D(ti)

where there are n cash ffows and ti, ...., tn represents the time at which these cash florvs aremade. The cash flows at time t, is C(t,) and the relevant discounting factor at time t, is D(t,).

It is possible to hedge the interest rate risk oi a swap by creating a portfolio consisting ofthe swap plus a set of new current coupon swaps. Let the value ol the portlolio ...ut.J b.denoted by v. Then:

v: f X,v,j=o

(1)

(2)

where vo is the original swap, Xo is l, v1,..., v.swaps with notional principal I and X,,..., X-required to neutralise interest rate risk. Notice thatCOUpOn Swaps vr : V2 : ...V- : 0 SO the presentequal to that ol the original swap vo.

To be certain that the value oi (2) is preservedlact ensure that:

are the value of the added current couponare the number of current coupon swapssince v,,..., v- are the value ofthe currentvalue of the created portfolio V is simply

for changes in the yield curve we shall in

dV m dv,\ v r :^ar,(t,)

,?o 'Jr.(t,) lor 1 : 1, 2..... L (3)

T.,irlcs I rnti .l ,riler ilte lu.irk-(o-ntltrkets\\.tLr rirte rs blrpoetl. Thus tlre estrmate ol

is crlculated b1

citlctrlrttit,rr rt't elujr ,ri tirc ctrrcs rritcrc thc j-rrlu

,/ r'n

rlr.{ j )

_1S7.S"15 -

196,+01-

{1i,800 pe r i o.o:(0.0 I i

Tirble.l shorvs tlte elTect on the present value oi the D\l s\vap ol blipping each of the currentcoupon swap rates in turn.

To find the hedees we must also calculate the eflect of a blip in cuch of the srvap rates tothe value ol each ol the current coupon s\\'aps. AII but the diagonal elements are zero. Table5 gives the effect on the value of each oi the current coupon sw'aps lor a l0% move of eachswap rate.

The hedge is calculated by direct implementation of equatron (6). The srvap positions whichshould be entered into are shown in millions. The siens assume that the trader is long thefixed side oi aj DM swap. (See Table 6.)

The trader should therelore pay fixed on a 5-year currenr coupon swap *'ith a principalof D)vlilm and pa1, fixed on a 4-1'ear current coupon swap rr,ith a principal oi DNI39m. Inprilctice the trader rvould probabl-v- not rvorrv ab0ut the other positions as these are small.

2. Delta hedgingThe method descrrbed and illustrated above is called delta hedging. The idea, rve recall, is tohedge using an approximation of the first derivative with respect to the underlying variable.We shall discuss this method, in detail, together rvith a couple olother hedge possibilities, inthe context ol currency options, in the next chapter. The discussron will set the scene lormuch ol the marke t terminoloey of srvaptions.

It should be noted that the delta hedge is onll'a good local hedge (snrall changes); iiinterestrates move 100 bp the hedge is not perfect. Let us now investigate how effective the hedgeis, in the example in the previous section. to a (non-parallell move tn the yieid curve. Weshall consider t\+'o portfolios. The first consists of the naked posrtion (r.e. only,rhe 4j yearDIVI srvap unhedged). The second port[olio consists oi the inrtial swap rogerher w'ith a shorrposition o[ tlte 5-year current coupon swap to the tune oi DM llrn and a shorr position olDN{i9m ol the 4-year current coupon swap.

Let us superimpose a couple olnon-parailel shiits to the yreld curve;rnd record rhe chaneesto the value oi each portlolio under these shi[ts. The first shilt represents a ffattening oi thecurve by an increase in the long-term swap rates, while the second shilt also represents affattening but by a lall in short term interest rates. The shiits are shown in Table 7.

The change in value or P&L (Profit & Loss) ior both port[olios under the nvo scenariosis given in Table 8.

First consider the yield curve shift A. Thc hedged portiolio fairs lar better, the loss is limiteclto only DM13,000. If we had entered into all six oi the suggested hedges this figure rvouldbe even closer to zero.

The second yield curve shiit did not afTect the 4- or 5-year rare and rhus the unhedgedportiolio is relatively unchanged in value. The hedged portiolio has recorded a profit mainlybecause we are receiving ffoating at a relati!el1'high rate (remember the l2-month Libor ratelell l5 bp in this case).

As inrerest rares move. rhe hedge ol a swap position itself alters slightly. The delta hedgingexcrcise abovc rvus repcated for a llat 1"1, yield curve representing a 200 bp move in mostmarurities. Table 9 compares the hedges under the two yield-curve scenarios.

The 4- and 5-year hedge positions have hardly altered, the 6-month swap position has,however. changed significantly. This change is because we are paying floating. and a floatingcoupon o1"9.833% is due in 3 months'time, although the market rate is 7%.

The concepr of a change in a hedge caused by movement oi the underiving is reiatei torhe gamma oia porriolio Swaps have practicaily zero qamma, while options have, in generai,non-zero samma. We shall study this concept in depth by looking at currencv options inChapter I and apply this to swaption hedging in Chapter l.

5. The Black-Scholes pricing formula for European-style optionsWe shall initially develop the Black-Scholes lormula in order to price European-style optioitson srocks which pay no dividend. The lormula will then be extended to consider European-style options on indices and currency options. We will then progress on to interest rrte andbond options and finally explode the complexities oi exotrc options.

In order to be able to pnce options on stocks we shall assume:

(l) The risk free rate is deterministic (usually constant).(2) The underlying stock price follows a log-normal random walk*.

' It is oflen questioned why a (normal) random walk rs not used. II we were to use a normal random walk it wouidbe possible to obtain negarive srock pricesl The log-normal distribution ensures that the stock price is al*'aysnon-negative.

European-style options on stock (no dividends)Let the stock price S be some function of time and a random component:

S : f (time, random component)

(We shall assume the random component is a Weiner process.)The cost of the optron is some other function of the stock price and time:

g:g(time,S)Now construct a portfolio ol the stock and option in such a way as to eliminate the randomcomponent from the formula of the portfolio's return. Since the portfolio's return is in-dependent of a random component it should be equal to the risk-free rate. Then by lto'sLemma:

Equation (l) is the heat equation rvhich models heat ffow.We know that ior a European call when t : 0, C : max(S

-

K, 0) Thus the solution ro(l) ior a call is:C: SN(d,)

-

Ke-"N(d:)where:

CC

-+rSAt^ o-LS'- : rC

AS'

('.5)

(l)aL o-as2

r" (!) *\K/

/')\

d::dr-oJl

Nrxr: I ls-,":6tJ _, J2"

For a European put P = max(K -

S,0) and so the solution to (1) for a put is:

dr:oJl

P: Ke-"N(-dr) -

SN(-dr)

An approximation for N(x)To calculate N(x) by evaluating the integral numerically woulda number of approximarions exist. One such approximation is

(3)

take a long time. Fortunarelygiven belorv.

Remember, S, K must be quoted in American stvle; r, I nrust bi in continuous comnoundedform.

Lf!) *(,-r*a)'\K/ \ 2/d,= = 0.188oJtdz=dr

-oJr=0.025Using the approximation referred to above:

N(dr) : 0.5745 N(d2) : 0.5100Thus by equation (4):

c = SN(dr)e-r'- Ke-"N1dri= 0.2855 - 0.2492

= 0.0364

The cost of the option is 3.64 cents per DM or 6.l9oh up-front (3.641K).

Example 2Assuming the following price a put on Ucall on US$ with a strike of 1.60 and maturity of 3monthsl

Spot : 1.67 SILUS$ risk-free rate : 9.7% (annual)I risk-free rate : l4.5oh (annual)Volatility :9Yo

S : 1.67K : 1.60t : 0.25 (3 months, not 3 years)r : 0.0926l' : 0.1354o = 0.09dr : 0.73614 d2 : 0.69114N(-d,) : 0.23083 N(-dz) = 0.24475hence replacing values in equation (5) the cost of the put is I cent per I or 0.624" up front.

Note. This option is quite inexpensive because the strike, K, is much less than the spot S andthe implied forward for the 3-month period.

European-style options on stock indices

Equations (2)and (3) can be adjusted to price European-style options on stock indices:C : Se-q'N(d,)- Ke-"N(d2) (6)

The formula lor a European style put is:

P: Ke-"N(-dr)- Se-q'N(-d,) (7)where:

S is equal ro the value of the index, o is equal to the volatility ol the index, q to the averageannualised yield on the index during the life of the option. In calculating q, oniy dividendsrvhere the ex-dividend date is during the life of the option should be included.

Note. Equations (6) and (7) are the same as (3) and (4) but with f replaced by q.

FII\AI{CIAL ENGINEERING&

RISK MANAGEMEI{T

INDEX

coMpr.-rrER prsKETrE (. .DOC) / CHAPTER TITLE PagesFII.{-ENGN.DOCCOVER

DEDICATIONCONTENTSCONTENTS (I-8)PREFACE (9-lo)INTRODUC(1) INTRODUCTTON TO DERTVATTVES (10/A-F)SWAPS(2\ SWAPS (11-42)

i. Swap Mechanics2. Why Use Swap Products?3. Benefits of Swaps versus Bonds and Futures4. Pricing Swaps5. Swap Market Rates6. Mark-to-Market of Swap Positions7. Zero Coupon Swap Valuation8. 1+i Analysis9. Basis Swaps

10. Arrears Swaps1 1. Cross-Currency Basis Swaps12. Yield-Curve Swaps13. Index/Quanto Swaps14. Risk Management of Complex Diffs

(1)

S\\'APACCT

1. P&L-and Balance-Sheet Considerations2. Mant-to-:l,rturket -.methodologies3. Discounting method4. Evaluation

TAXATION(17) TAXATTON1. Forwards & Futures2. Options3. Swaps4. Tax Planning

(xxx-xxx)

SWAPCRDT(18) SWAP CREDTT POLTCY (37s-383)1. Credit Risks on Swaps2. Netting of Counterparty Exposures3. Credit Policy for Swaps4. Counterparty Credit Limits5. Active Credit Management

REGULATS(19) REGULATORY TMPACT1. Securities and Banking Regulations2. Capital Adequacy

Afterword -mentiort of the Strategic Report "Financial lingtneering - Risk Management in Practice"

Appendix - Formulae and mathematical models- The'Greeks'

435

(s)

Euromoney Strategic Report

Financial Engineering- Risk Management in Practice

DRAFT INDEX *:extra/newmatenal

FIN.ENGN.DOC

CO\IER

DEDICATIONCONTENTS

CONTENTS

PREFACEINTRODUC(l) TNTRODUCTTON TO DERTVATTVESFOREX(2)* FORETGN EXCHANGE

1. Forward Foreign Exchange Contracts2. Currency Discounts and Premiums3. Trading & Opportunities4. Hard and Soft Currencies5. Currency Blocs and Snake6. Deposits and Loans

(1-8)

(e-10)

(10/A-r)

SWAPS(3) SWAPS (ll-42)1. Swap Mechanics2. Why Use Swap Products?3. Benefits of Swaps versus Bonds and Futures4. Pricing Swaps5. Swap Market Rates6. Mark-to-Market of Swap Positions7. Zero Coupon Srvap Valuation8. i+i Analysis9. Basis Swaps

10. Arrears Swaps1 1. Cross-Currency Basis Swaps12. Yield-Curve Swaps13. Index/Quanto Swaps14. Risk Management of Complex Diffs

(D

OPT]ONS(e) oPTroNS (89-133)1. Notation & Basic Definitions2. PayoffDiagrams3. Variables which affect the price of a stock option4. Assumptions made in the Black-Scholes and Binomial models5. The Black-Scholes Pricing Formula for European-Sryle Options6. Plotting the value of European-options prior to maturity7. A Diffusion Model for American - Style Options8. Volatility9. Interest Rate Options

10. Derivatives of Cost (the Greeks)I i. Hedgrng and the importance of the delta and gamma12. Options Portfolios13 Trading Volatility

SWAPTION(10) swAP oPTroNS (lsl-1s8/A-r)1. 'Swap Derivatives'2. Capsffloors3. Yield-Curve Caps/Floors - strucruring & Pricing4. Complex/QuantoCapslFloors5. Swaptions6. Risks7. Option Valuation Issues

cT.TRRENCY (134-150)(11)CURRENCY OPTION PRICING,asan Introduction to Interest Rate Options

1. Factors Affecting the Price of a Currency OptionInterest rate differentialsVolatilityIntrinsic value and Time value

2. A Brief Derivation of the Black-Scholes Model3. American Options and the Binomial Tree Approximation4. Interest-Rate Options

INTOPTS(r2) PRTCING OPTTONS ON TNTEREST RATES (1s9-175)1. Caps and Floors2. European Swaptions3. Modelling the Term Structure4. Pricing European Swaptions Using An Interest Rate Tree5. American Style Swaptions6. In-The-Money and Out-Of-The-Money Srvaptions

(Ir)

SWAPACCT(r8) ACCOUNTING (361-371)i P&L and Balance-Sheet Crinsiderations2. Mark-tolMarket methodologies3. Discodnting method4. Evaluation

RISKMANG(19) RrSK MAr.{AGEMENT (315-347)1. Introduction2. Srvap Risk Management System Objectives3. The Cash Book4. Managing Fixed and Floating Rate Exposures5. Implementation - In Theory & Practice

Appendix l. Zero CouPon SwaPCurveAppendix II. Sample of Risk Management Report

to do...delta sensitivity and graph !t!t?l???/lll

NEWRISK(20)* RISK MANAGEMEI.IT REPORTS (348-360)

1. Introduction2. Reserves Policy3. fusk Profiles

- Yield Curve fusks- Spread Risks and Mismatches/Gaps- Basis risks- Delta (Portfolio Equivalents)- Theta (Time Decay)- Gamma (Bond Convexiry,;- Vega (Volatility Changes)

4. Exposure RePorts5. P&L'. Mark-To-Market6. Sensitivities: derivatives

hedges7. Management RePorts8. Trader Limits & Performance Measurementsg. Pricing and Quality of market data used for Evaluation & M-T-M

10. Short-term Book Management1 l. Checks & Controls - Monitor & Authorise

- Procedures & ComPliance- Deal Confirmations and Settlements

(v)

TAXATION(25)* TAX TREATMENT OF DERfVATIYES - Coopers & Lybrand

Tax relief for Reserves & Provisions -Taking advantage of timing tax cash-flows

Deferrin g i ncome, bringtn g forward costs/expensesTax avoidance (not evasion) schemes (involving Capital Gainstax treatment instead of taxation of profits at less favourableCorporation Tax levels e.g. creating culrency gains/losses vs'.wer interest rate currency on New Issues, etc.)

TAXANON(26) TAX PLANNING (xxx-xxx)1. Forwards & Futures2. Options3. Swaps4. Tax Planning

SWAPCRDT(27) SWAP CREDTT POLTCY (37s-383)1. Credit Risks on Swaps2. Netting of Counterparty Exposures3. Credit Policy for Swaps4. Counterparty Credit Limits5. Active Credit Management

REGI'LATS(28) REGULATORY TMPACT1. Securities and Banking Regulations2. Capital Adequacy

LEGAI(29)* LEGAL - Rogers & Wells

Global Trading and Booking TransactionsUsing legal and regulatory jurisdiction of different countriesObligation on parties to ensure counterparties transactgd-wrthin therr powers

D(rcUMENT(30)* STANDARD DOCUMENTATION

Master Agreements & ConfirmationsISDA

Appendix - Formulae and mathematical models-

*Futures & Options Strategy Chart- The'Greeks'-

*ISDA Report on Swaps Accounting and Reporting ...prmission ?!?!...

*Glossary !?(!T)