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Reminder on Vanilla Interest ratesReminder on Vanilla Interest rates

Didier Faivre

[email protected]

February 2006


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Zero-coupon

Deposit

Libor, Euribor

FRA

Vanilla Swaps

Swaps Forwards

Caplet/Floorlet

Caps/Floors

Swaptions

Volatity Cube

CMS


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Zero-coupon

o Price at date tof one 1 (or any currency) at date T:

o r(t,T) is called the zero-coupon rate at date tfor maturity T

o In the right side of above equation T-tis a year fraction

calculated using ACT/365 convention (or sometimes ACT/ACT

if one wants to take into account the leap years).


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Deposit

o Loan on a period from 1 week to 12 months betweentwo banks

o Interest calculated using monetary interest rate, e.g.

linear interest rates


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Deposit

o Interest Calculated as :If bank lends 1M on a period of 3months, in 3

months banks receives :

Number of days is exact number of days between start

and end of the loan

We speak about Libor1M, Libor3M..

For USD, LiborUSD1M, LiborUSD3M..For EURO, Euribor1M, Euribor3M

360

daysofnumber311 MEuriborM


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Deposit

o For some currencies (GBP, AUD), replace 360 by365


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Deposit

o Example for a 3Month deposito 30 january 2006,offered 3 months rate by BBVA is

2.56%

o 3month loan is from 30 january 2006 + 2 business days

to (30 january +2 business days)+3months businessdays

o So from 2/2/2006 to 2/5/2006, so 89 days period of

interest rates

o For a 100M notional loan, redemption is100*(1+2.56%*89/360)=100.6329M


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LIBOR, EURIBOR

o LIBOR : London Interbank Offered Rate

o Every day, fixing at 11am London Time on mostcurrencies : USD, JPY, GBP

o For EURO, fixing at FRANCFORT Euribor

o Definition :Average of offered rates for a given maturity, on a basketof banks, for deposits operations

Offered rate : means rate at which Bank wants to lendmoney, not to borrow (Bid/Ask spread)

For various maturities from 1 week to 12 months :Using Monetary interest rates as previously explained


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EURIBOR rates : reuter page

List of Euribor interests rate + fixing values +

definition, as of 30/1/2006


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o On the previous slide, we see that the fixing of Euribor

3M on the 30/1/2006 is 2.542%

o The period of interest for a 3M deposit on the 30/1/2006

is from 2/2/2006 to 2/5/2006

30/1/2006 is called the Fixing date2/2/2006 is called the Start date

2/5/2006 is called the End date


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EURIBOR3M : reuter page

Quotations of 3 months offered rate by the official

basket of banks as of 30/1/2006, for calculating thefixing of Euribor3M at this date


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FRA

o Definition of FRA (Forward rate agreement)

A forward Euribor of maturity Tis a forward contract on

Euribor beginning at date T(fixing at date T-2D) and

ending at date T+ .

The maturity Tis calculated taking account business daysconventions, including various end of month rules.

The value of the forward at date tis :


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FRA

o Warning ! : on the the above equation, is a number of

days when added to the date T, for example the numbers

of days for a given standard reference period (3M,

6M) and otherwise on the right side of equation its a

year fraction calculated using the monetary basis

convention of the currency (ACT/360 or ACT/365)

This is usual rule for quants documents notations


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Vanilla Swaps

o In a vanilla swap, two counterparties exchange variable cash

flows based on Euribor (or Libor for other currencies) against

cash-flow based on a fixed rate, in the same currency

o Example : 2 years fixed rate against Euribor6M


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Vanilla Swaps : Schedule

o First step of swap calculation is to set the schedule, we

need :

Total maturity of the swap at initial date : 1Y, 2Y, 3Y

Convention for non business days (holidays)

Fixed leg conventions : period, basisFloating rate conventions : period, basis


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o Basis is a convention to calculate the year fraction

between two cash flows dates for interest calculations

o Cash Flow is : year fraction (calculated according to the

basis ) * interest rate * Notional

o Examples of period : 3M, 6M, 12Mo Examples of basis : ACT/360, ACT/365, 30/360

o For terms linked to stochastic modelling (time value,

convexity adjustment) always use ACT/365


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o Then :

Calculation of theoritical date of swap end

Calculation of theoritical date of cash-flows for both legs

Possible adjustement for taking into account non business

days and convention for non-business days

o Theoritically, all combinations of period and basis are

possible for the two legs

o In practice a standard is set for every-market, used bydefault


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Vanilla Swaps : Standard Conventions

o For the floating leg the basis is always the basis used for

reference rate (Libor or Euribor), same thing for period

o Examples :

Euro marketFor 1 Year maturity swap

1Year period and 30/360 basis and for fixed leg,

Euribor3M for floating leg

For maturities over than 1Year1Year period and 30/360 for fixed leg, Euribor6M for

floating leg


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Vanilla Swaps : Standard Conventions

USD market : two standards, one for New-York working

hours, one for before New-York market open

Before New-York opens

For all maturities, 1Year period and ACT/360 for fixed leg,

LiborUSD3M for floating leg (Money Markets swaps)

After New-York opensFor all maturities, 6 months period and 30/360 for fixed leg,

LiborUSD3M for floating leg (Bond Basis swaps)

Money markets swaps because ACT/360 is the basis for

LiborUSDBond Basis swaps because 30/360 is the basis for USD

corporate bonds


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Vanilla Swaps : Example

o 3Y swap against EURIBOR6M, as of 3/4/2002 ( a

Wednesday)

o Start date = 3/4/2002 + 2 Business days = 5/4/2002

(a Friday)

o Step 1 : theoritical date of swap end : 5/4/2005 (aTuesday)


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o Step 2 : Theoritical dates of cash-flows


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o Step 3 : taking account of non business days :


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o Step 4 : Calculation of interest periods, using true dates of cash

Flows and basis of both legs :

o The are called coverage, calculated as year fractions

between cash-flows dates for both leg, using each basis.

and~


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o From the cash-flow payment dates of the swaps, one

can also calculate the fixing dates, using the -2 Business

days rule.

For example, the fixing of the Euribor6M for the period

6/10/03 to 5/4/04 is the 3/10/03 (4/10/03 is a Saturday)


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Vanilla Swaps : Example2

SWAP 3Y

Fix Leg Frequency 3 M Float Leg Frequency 3 M

Basis ACT360 Basis ACT360

Lib start Lib end coverage pay dates fix dates Lib start Lib end coverage pay dates fix dates

17/02/06 17/05/06 0.247222 17/05/06 15/02/06 17/02/06 17/05/06 0.247222 17/05/06 15/02/06

17/05/06 17/08/06 0.255556 17/08/06 15/05/06 17/05/06 17/08/06 0.255556 17/08/06 15/05/06

17/08/06 17/11/06 0.255556 17/11/06 15/08/06 17/08/06 17/11/06 0.255556 17/11/06 15/08/06

17/11/06 19/02/07 0.261111 19/02/07 15/11/06 17/11/06 19/02/07 0.261111 19/02/07 15/11/06

19/02/07 21/05/07 0.241667 17/05/07 15/02/07 19/02/07 21/05/07 0.241667 17/05/07 15/02/07

17/05/07 17/08/07 0.255556 17/08/07 15/05/07 17/05/07 17/08/07 0.255556 17/08/07 15/05/07

17/08/07 19/11/07 0.261111 19/11/07 15/08/07 17/08/07 19/11/07 0.261111 19/11/07 15/08/07

19/11/07 19/02/08 0.252778 18/02/08 15/11/07 19/11/07 19/02/08 0.252778 18/02/08 15/11/07

18/02/08 19/05/08 0.252778 19/05/08 14/02/08 18/02/08 19/05/08 0.252778 19/05/08 14/02/08

19/05/08 19/08/08 0.252778 18/08/08 15/05/08 19/05/08 19/08/08 0.252778 18/08/08 15/05/08

18/08/08 18/11/08 0.252778 17/11/08 14/08/08 18/08/08 18/11/08 0.252778 17/11/08 14/08/08

17/11/08 17/02/09 0.255556 17/02/09 13/11/08 17/11/08 17/02/09 0.255556 17/02/09 13/11/08


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SWAP 3Y

Fix Leg Frequency 6 M Float Leg Frequency 3 MBasis 30360 Basis ACT360

Lib start Lib end coverage pay dates fix dates Lib start Lib end coverage pay dates fix dates

17/02/06 17/05/06 0.5 17/08/06 15/02/06 17/02/06 17/05/06 0.247222 17/05/06 15/02/06

17/08/06 17/11/06 0.505556 19/02/07 15/08/06 17/05/06 17/08/06 0.255556 17/08/06 15/05/06

19/02/07 21/05/07 0.494444 17/08/07 15/02/07 17/08/06 17/11/06 0.255556 17/11/06 15/08/06

17/08/07 19/11/07 0.502778 18/02/08 15/08/07 17/11/06 19/02/07 0.261111 19/02/07 15/11/06

18/02/08 19/05/08 0.5 18/08/08 14/02/08 19/02/07 21/05/07 0.241667 17/05/07 15/02/07

18/08/08 18/11/08 0.497222 17/02/09 14/08/08 17/05/07 17/08/07 0.255556 17/08/07 15/05/07

17/08/07 19/11/07 0.261111 19/11/07 15/08/07

19/11/07 19/02/08 0.252778 18/02/08 15/11/07

18/02/08 19/05/08 0.252778 19/05/08 14/02/08

19/05/08 19/08/08 0.252778 18/08/08 15/05/08

18/08/08 18/11/08 0.252778 17/11/08 14/08/08

17/11/08 17/02/09 0.255556 17/02/09 13/11/08


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SWAP 3Y

Fix Leg Frequency 1 Y Float Leg Frequency 3 M

Basis 30360 Basis ACT360

Lib start Lib end cov pay dates fix dates Lib start Lib end cov pay dates fix dates

17/02/06 17/05/06 1.005556 19/02/07 15/02/06 17/02/06 17/05/06 0.247222 17/05/06 15/02/06

19/02/07 21/05/07 0.997222 18/02/08 15/02/07 17/05/06 17/08/06 0.255556 17/08/06 15/05/06

18/02/08 19/05/08 0.997222 17/02/09 14/02/08 17/08/06 17/11/06 0.255556 17/11/06 15/08/0617/11/06 19/02/07 0.261111 19/02/07 15/11/06

19/02/07 21/05/07 0.241667 17/05/07 15/02/07

17/05/07 17/08/07 0.255556 17/08/07 15/05/07

17/08/07 19/11/07 0.261111 19/11/07 15/08/07

19/11/07 19/02/08 0.252778 18/02/08 15/11/07

18/02/08 19/05/08 0.252778 19/05/08 14/02/08

19/05/08 19/08/08 0.252778 18/08/08 15/05/08

18/08/08 18/11/08 0.252778 17/11/08 14/08/08

17/11/08 17/02/09 0.255556 17/02/09 13/11/08


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Vanilla Swaps : Evaluation of Floating leg

o is the schedule of the floating leg of theswap (for a 4 years with a 6Months period onfloating leg, m = 8, for example).

o The value of the floating leg is :

o It can be shown that it is also :

o is the end date of the swap and theschedule of fixed leg.


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Vanilla Swaps : Evaluation of the fixed rate

o At date 0, the value of the swap is 0, meaning the

value of the fixed leg is the value of the floating leg

o IfS is a fixed rate, the value of a fixed leg using this

rate is :


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Vanilla Swaps : Evaluation of the fixed rate

o The swap rate is the rate such that both legs have

same value at date 0

o We get :

o The term is called the Level of the swap, its value is

close to the sensitivity or duration of a standard bond of same

maturity, period and with coupon rate of .


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Vanilla Swaps : Evaluation of a forward swap rate

o A forward swap is a swap beginning in the future at

date T.o The forward swap rate at date tis the rate

such that the present value at date tof the two legs areequal.

To get the value of a forward swap at date 0, just do t=T= 0 !

i S S f f


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Vanilla Swaps : Schedule of a forward swap

o Example : 2Years in 1Year (fixed against Euribor6M)

as of 3/4/02

o 3/4/02 + 2 Business days is 5/4/02

5/4/03 is a saturday

C l fl l


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Caplet, floorlet

o A caplet is a call option on a Euribor (or Libor..)

forward

o The caplet of strike Kpays at date T+d the difference, if

positive, between Euribor on the period starting T

ending T+d :

pay-off of caplet at date T+d is :

Max(Euribor(T, T+d)-K;0)

The fixing of the Euribor is at T-2D, taking into account

non business days.

o A floorlet is the same thing for a put option.

C l t fl l t


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Caplet, floorlet

o The market practice to value a caplet at date tis :

o The market pratice to value a floorlet at date tis

BS for Black Scholes, details in next slide

o Of course, the parameter s , volatility ofFRA(t,T,T+d) depends on d, Kand T.

call,Lognormal,,tT,K,T,T,tFRABSpriceT,tB

put,Lognormal,,tT,K,T,T,tFRABSpriceT,tB

C l t fl l t


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Caplet, floorlet

o After having defined the swaptions, we will also explain

what is a volatility cube.

Bl k S h l f L l f d


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Black-Scholes for Lognormal forward

o Let follows a lognormal law with volatility s and

expectation :

o Then is defined as :


TT FKE,FKEMax 0

TT,N

T eFF2

0

0

2

KFE,KFEMax TT 0

call,Lognormal,,T,K,FBSprice 0

put,Lognormal,,T,K,FBSprice 0

0F

TF

Black Scholes formula for a Lognormal forward


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Black-Scholes formula for a Lognormal forward

o If

o then

t

t

t dB

F

dF

T

TK

Fln

d

2

1

2

1

Tdd 12

210 dNKdNFKFE T

102 dNFdNKFKE T

Black Scholes for normal forward


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Black-Scholes for normal forward

o Let follows a normal law :



o Here s is a standard deviation, not a volatility.

TT FKE,FKEMax 0

KFE,KFEMax TT 0

call,normal,',T,K,FBSprice 0

put,normal,',T,K,FBSprice 0

T,N'FFT 00

TF



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tt dB'dF

dnT'dNKFKFE T 0

dnT'dNFKFKE T 0

T'

KFd 0

N= distribution function of normal n(0,1), ndensity



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o Special case : ATM (At the money Option) :

o Call = Put =

o This formula shows that the price of an ATM option

(caplet/swaption) depends in fact only on the standarddeviation, not on and on the volatility

o We remind that a very good approximation of the

relation between s (the standard deviation) and s (the

volatility) for ATM is :

T'.T'

402

0F'

0F

Black-Scholes formulas : a few remarks


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o The above Black-Scholes formulas give the prices of

call and put that would be paid by the buyer of theoption at maturity T

o So at the same date the pay-off of the option would be

paid by the seller of the option to the buyer of the

option.

o To get the price of call and put that would be paid at

date 0, just multiply the above formulas by .T,B 0



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o Call put Parity : the call/put parity for standard

european option is totally independant of the choice ofthe model (lognormal, normal or whatever):

o If the buyer of the options pays the option (and get the

pay-off at maturity) :

o If the buyer of the options pays the option at date 0 (and

get the pay-off at maturity):

KFKFEFKEKFEputcall TTT 0

T,BKFputcall 00

Cap floor


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Cap, floor

o A cap is a sum of caplets, a floor a sum of floorlets

o Value of cap is the value of all the caplets included,same thing for floor

o Example : 1 Year Cap on Euribor3M

Remark : most of the time, the first caplet is notincluded as the value today of the first Euribor isKnown. So most of the times, only 3 caplets in theabove example

Call/put parity for Cap & Floor


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o Using previous definition of Caplet and Floorlet, we get

easily call/put parity for Caplet/floorlet, so we have (using the previous notation) :

o So Cap-Floor = Value of floatLeg Minus Value of

Fixed Leg (strike K)

o The strike Ksuch that Cap = Floor can be seen as a

swap rate corresponding to the schedule of the Cap andfloor, so using frequency and basis of this schedule.

K~

T~

,BT~

,T~

,FRA~

T~

,B ijjji

m

i

j 000 11



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o To give an example, for 5Y Cap/Floor on Euribor3M,

the strike such that Cap = Floor will be :

the 4.75Y rate in 3months, with frequency 3M and basis

ex/360 if the first caplet/floorlet are not included.

the 5Y swap rate, with frequency 3M and basis ACT/360

if the first caplet/floorlet are included.



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o The market practice for caplet/floorlet can be

justified theoritically by introducing the forwardneutral probability tool, but :

its important to understand the practice cannot leadto arbitrage because its consistent with the call/putparity

The traders used this practice long time before quantsused the forward neutral probability tool.

o The ATM rule (p 37) shows that the cheapness ofATM caplet/floorlet depends only of the standard

deviation, so cheapness can be evaluated using theformula :

0F'

Physical settlement Swaptions


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y a S ap

o A physical settlement swaption of strike Kon the

swap is the right to enter into a swap at date Twith fixed rate K.

o Two types swaptions :

Receiver swaptions : receive the fixed rate and pays the

floating rate ; if one buys a receiver swaption, one

believe rates will go down.

Payer swaptions : pays the fixed rate and receives the

floating rate ; if one buys a payer swaption, one believe

rates will go up.



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y p

o Example : payer physical swaption on 2Y (fixed against

Euribor6M) in 1Y as of 3/4/02



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y p

o The market practice to price a receiver physical

settlement swaption at date tis :

o The market practice to price a payer physical

settlement swaption at date tis :

put,Lognormal,,tT,K,T,T,tSBSpriceT,tB ni

n

i

i

1

call,Lognormal,,tT,K,T,T,tSBSpriceT,tBni

n

ii

1

Cash settlement Swaptions


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p

o For a cash settlement swaption, at maturity there is

no settlement of a swap

o For a receiver cash settlement swaption, the buyer of

the option receives at date T:

o For a payer cash settlement swaption, the buyer of

the option receives at date T:

01

1

1

,KT,T,TSMaxT,TS

n

n

ii

n

01

1

1,T,T,TSKMaxT,TS

n

n

ii

n



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o Remarks :

Previous formulae are for swap with a period of1Y on the fixed leg, for a period of 6M, justreplace

By :

The number is called the level cashof the swap ; at maturity, the level cash is just

calculated by replacing the zero-coupon rates bythe swap rate to discount (and also replacing allcoverages by 1).

n

i /inT,TS

/

1 21

21

n

ii

nT,TS1 1

1

n

ii

nT,TS1 1

1



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o The market practice to price a receiver cash settlement

swaption is :

o The market practice to price a payer cash settlementswaption is :

o Volatility depends on features of forward swap andstrike

put,Lognormal,,T,K,T,T,tSBSpriceT,T,tS

T,tB n

n

ii

n1 1

1

call,Lognormal,,T,K,T,T,tSBSpriceT,T,tS

T,tBn

n

ii

n1 1

1



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o For a same features (same swap forward and same

strike), the volatility is the same for a physical or cashsettlement swaption.

Call/Put Parity for swaptions


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o Physical Payer Swaptions-Physical Receiver swaption =

o Cash settlement Payer Swaptions-Cash settlement

Receiver swaption =

KT,T,tST,tB ni

n

i

i

1

KT,T,tST,T,tS

T,tB n

n

ii

n1 1

1

Swaptions : a few remarks


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o The market practice can be justified by introducing

the Q Level probability tool, but as for caps/floors :Its important to understand the market practice cannot

lead to arbitrage because its consistent with above

call/put parity formulas for both kind of swaptions.

The traders used the market practice long time beforequants created the Q Level probability tool.

o The ATM rule (p 37) shows that the cheapness of ATM

swaption (if we forget the level) depends only on the standarddeviation, and so can be seen using the formula

0F'

Volatility cubes


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o First lets define a volatility surface for a given

reference rate, let say the Euribor3M

o We need to be able to price a caplet on Euribor6M for

any strike and any maturity

o A volatility surface for this reference rate will be as

following :

Volatility surface for Euribor3M, as of 31/1/06


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3M 0 .50% 1 .00% 2 .00% 2 .50% 3 .00% 3.50% 4 .00% 4 .50% 5 .00% 6 .00% 7 .00% 8 .00% 9 .00% 10 .00% 11 .00% 12 .00% 13.00% 14 .00%

1M 7.44% 5.41% 4.22% 4.35% 4.58% 4.82% 5.03% 5.22% 5.39% 5.66% 5.88% 6.07% 6.22% 6.34% 6.46% 6.55% 6.64% 6.71%

3M 10.60% 8.16% 6.76% 6.75% 6.86% 7.02% 7.18% 7.33% 7.46% 7.70% 7.90% 8.07% 8.21% 8.32% 8.43% 8.51% 8.59% 8.66%

6M 17.42% 13.62% 11.33% 11.20% 11.28% 11.44% 11.63% 11.82% 11.99% 12.31% 12.57% 12.80% 12.98% 13.15% 13.29% 13.41% 13.52% 13.62%

1Y 33.20% 25.22% 19.94% 19.68% 20.01% 20.52% 21.06% 21.59% 22.08% 22.92% 23.63% 24.21% 24.71% 25.13% 25.49% 25.81% 26.10% 26.35%

2Y 39.79% 30.01% 22.29% 21.31% 21.31% 21.76% 22.35% 22.98% 23.57% 24.64% 25.54% 26.30% 26.94% 27.50% 27.98% 28.40% 28.78% 29.11%

3Y 43.64% 32.97% 23.56% 21.65% 20.96% 21.02% 21.44% 21.99% 22.58% 23.69% 24.67% 25.50% 26.22% 26.84% 27.38% 27.86% 28.29% 28.67%

4Y 44.28% 33.60% 23.76% 21.42% 20.27% 19.99% 20.21% 20.66% 21.20% 22.28% 23.27% 24.13% 24.87% 25.52% 26.08% 26.59% 27.03% 27.43%

5Y 43.71% 33.36% 23.61% 21.08% 19.63% 19.05% 19.04% 19.35% 19.80% 20.79% 21.74% 22.58% 23.31% 23.96% 24.52% 25.03% 25.47% 25.88%

6Y 43.01% 33.04% 23.50% 20.90% 19.26% 18.42% 18.18% 18.32% 18.65% 19.51% 20.39% 21.18% 21.89% 22.51% 23.06% 23.55% 23.99% 24.38%

7Y 42.18% 32.58% 23.33% 20.72% 18.98% 17.97% 17.54% 17.51% 17.71% 18.42% 19.21% 19.95% 20.61% 21.21% 21.73% 22.20% 22.63% 23.01%

8Y 41.28% 31.99% 22.99% 20.41% 18.61% 17.50% 16.94% 16.80% 16.91% 17.51% 18.23% 18.93% 19.57% 20.14% 20.65% 21.11% 21.52% 21.90%

9Y 40.33% 31.33% 22.60% 20.06% 18.28% 17.12% 16.49% 16.27% 16.32% 16.81% 17.47% 18.12% 18.73% 19.27% 19.76% 20.20% 20.60% 20.96%

10Y 39.32% 30.62% 22.16% 19.69% 17.93% 16.75% 16.08% 15.80% 15.79% 16.19% 16.78% 17.39% 17.95% 18.47% 18.94% 19.35% 19.73% 20.08%

15Y 36.16% 28.20% 20.40% 18.07% 16.35% 15.15% 14.40% 14.02% 13.91% 14.17% 14.67% 15.21% 15.72% 16.20% 16.62% 17.01% 17.36% 17.68%

20Y 34.90% 27.11% 19.41% 17.09% 15.36% 14.14% 13.37% 12.98% 12.88% 13.16% 13.68% 14.22% 14.74% 15.21% 15.64% 16.02% 16.37% 16.68%

25Y 34.47% 26.63% 18.91% 16.60% 14.92% 13.77% 13.09% 12.80% 12.79% 13.17% 13.72% 14.29% 14.81% 15.28% 15.70% 16.08% 16.43% 16.74%

30Y 34.62% 26.71% 18.96% 16.65% 14.97% 13.84% 13.19% 12.92% 12.91% 13.30% 13.85% 14.41% 14.92% 15.39% 15.81% 16.18% 16.52% 16.83%

Vertical axis : maturity of

caplets/floorlets

Horizontal axis : strikes ofcaplets/floorlets

Volatility cubes


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o If we define these surfaces for all reference rates

(Euribor1M, Euribor2M, 3M, 12M) and all swap(1Y, 2Y, 10Y, 30Y) we can price any vanilla cap or

swaption

o The set of vol surfaces is called a volatility cube

o In practice, more complex models are used and

calibrated on previous sufaces and used to get the

volatilies for any caplet/floorlet/swaption

o The purpose of these models is to avoid numerical

problem due to non proper interpolation methods.

CMS, CMS options


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o Everyday, fixing of fixed rate swap for every maturities

(1Y, 2Y, 3Y, 10Y, 30Y)o Buying at date 0 a CMS n years at date Tis buying the

right to get the fixing of the swap rate n years at date T,

ending at .

o The price at date 0 of this operation will be called :

o A call on this CMS gives at date T:

nT,T,CMS 0

00 ,KT,T,TSmax,KT,T,TCMSmax nn

nT

CMS, CMS options


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o A put on this CMS gives at date T:

o The pricing of CMS and CMS options is notstraightforward derivation of swap forward and

swaptions

o but follows the call/put parity formula (p 39)

00 ,T,T,TSKmax,T,T,TCMSKmax nn

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