Reminder on Vanilla Interest rates

Didier Faivre

didier.faivre2@gmail.com
December 2015
Zero-coupon page 3
Libor, Euribor page 5
Vanilla Swaps page 14
Swaps Forwards page 34
B&S formula page 50
Caplet/Floorlet page 71
Caps/Floors page 73
Swaptions page 77
Volatility Surfaces/Cube page 92
CMS page 108

2
 Zero-coupon

➢ Price at date t of one 1€ (or any currency) at date T:

B(t , T ) =
1
(1 + r (t , T ))T −t

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

➢ T-t is a year fraction calculated using ACT/365 convention
✓ sometimes ACT/ACT in order to take into account the leap years

t is the current market date (in probabilistic terms, all information up to t is

known…)

3
 Zero-coupon

➢ Other equation for zero-coupon:

B(t , T ) = e − r '(t ,T )(T −t ) 1

=
(1 + r (t , T ))T −t
➢ r’(t,T) is the continuous version of the zero-coupon rate:
r ' (t , T ) = Ln(1 + r (t , T ))

r’(t , T )  r (t , T )
➢ r(t,T) also called actuarial version of the zero-coupon rate, by
opposition to the continuous version r’(t,T)

4
 Deposit

➢ Loan on a payment frequency from 1 week to 12 months between two

banks
➢ Interest calculated using monetary interest rate, e.g. linear interest
rates
➢ Interest calculated as:
✓ If bank lends 1M€ on a payment frequency of 3months, in 3 months
banks receives:
 number of days 
1M €  1 + Euribor3M  
 360 

✓ Number of days: exact number of days between start and end of the loan
o Libor1M, Libor3M..for each currency for which Libor are quoted in London

5
 Deposit

o LiborUSD1M, LiborUSD3M, Euribor1M, Euribor3M…

o For some currencies (GBP, AUD…), replace 360 by 365

➢ Example: 3Months deposit

✓ 30 January 2006,offered 3 months rate by BBVA is 2.56%
✓ 3month loan is from 30 January 2006 + 2 business days to (30
January +2 business days)+3months business days
✓ from 2/2/2006 to 2/5/2006, 89 days payment frequency of interest
rates
✓ Redemption for a 100M€ notional loan:
100*(1+2.56%*89/360)=100.6329M€

6
 LIBOR, EURIBOR

➢ LIBOR: London Interbank Offered Rate

✓ Every day, fixing at 11am London Time on most currencies: USD, JPY,
GBP
✓ For EURO, fixing at FRANCFORT
➢ Libor/Euribor definition:
✓ Average of offered deposits rates for a given maturity, on a basket of
banks, for deposit loans
o Offered rate: means rate at which Bank can borrow money, not to lend
(Bid/Ask spread)
o For various maturities from 1 week to 12 months
o Indicative prices: during “subprime crisis”, great doubts about the link
between Libor and real loans transactions

7
 EURIBOR rates: Reuters page

List of Euribor interests rate + fixing values +

definition, as of 30/1/2006

8
 LIBOR, EURIBOR

➢ Fixing of Euribor 3M for the 30/1/2006 is 2.542% (previous slide)

➢ Interest payment period for a 3M deposit on the 30/1/2006 is from
2/2/2006 to 2/5/2006
✓ 30/1/2006 = Fixing date
✓ 2/2/2006 = Start date
✓ 2/5/2006 = End date

9
 EURIBOR3M: Reuters page

Quotations of 3 months offered rate by the official

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

10
 FRA

➢ FRA (Forward rate agreement)

✓ Forward Euribor of maturity T: forward contract on Euribor beginning at
date T (fixing at date T-2D) and ending at date T+δ
✓ Maturity T calculated taking account business days conventions,
including various “end of month” rules (following, modified-
following…)
✓ The value of the forward rate at date t is:

1  B(t , T ) 
FRA(t , T , T + δ ) =  − 1
δ  B(t , T + δ ) 

✓ The settlement at date T for a seller FRA written at date 0:

(FRA(0, T , T + δ ) − Fixing _ Euribor(T , T + δ))δ
(1 + Fixing _ Euribor(T , T + δ)δ)
11
 FRA

➢ Remarks:
✓ Seller (buyer) FRA: expect lower (higher) rates
✓ Settlement in T in accordance to most of clients requests
✓ Discount term is due to the fact that the settlement is in T, not T+d,
o also explains the difference between a FRA and Libor in arrears
➢ Warnings:
✓ In above equation
o δ = number of days when added to the date T, for example the numbers of
days for a given standard reference payment frequency (3M, 6M…)
or…
o δ = year fraction calculated using the monetary Basis convention of the
currency (ACT/360 or ACT/365)
✓ usual rule for quants documents

12
 Pricing of FRA: demonstration

(1+dL(T,T+d))
(1+dF)B(0,T+d) 1
d Replication
of FRA cash
flows

0 T T+d

B(0,T) 1
➢ Self-Financed strategy: (1+dF)
✓ date 0: one borrows (1+d F)B(0,T+d) , lends B(0,T).
✓ date T: one receives 1 and lends 1
✓ date T +d : one receives (1+d L(T,T+d )), pays (1+d F)
➢ Price of a FRA at date 0 = (1+d F)B(0,T+d) -B(0,T)

➢ F = (B(0,T)/ B(0,T+d) -1)/d

13
 Vanilla Swaps

➢ Vanilla swap:
✓ two counterparties exchange variable cash flows based on Euribor (or
Libor for other currencies) against cash-flows based on a fixed rate, in
the same currency
➢ Example: 2 years fixed rate against Euribor6M

14
 Vanilla Swaps: Schedule

➢ First step of swap calculation, the schedule:

✓ Total maturity of the swap at initial date: 1Y, 2Y, 3Y…
✓ Convention for non business days (holidays)
✓ Fixed leg conventions: payment frequency, payment Basis
✓ Floating rate conventions: payment frequency, payment Basis

15
 Vanilla Swaps: Schedule

➢ Payment Basis = convention to calculate the year fraction between

two cash flows dates for interest calculations
➢ Cash Flow = year fraction (calculated according to the Basis ) ×
interest rate × Notional
✓ Examples of payment frequency: 3M, 6M, 12M
✓ Examples of Basis: ACT/360, ACT/365, 30/360
✓ For terms linked to stochastic modeling (time value, convexity
adjustment…) always use ACT/365

16
 Vanilla Swaps: Schedule

➢ Following steps:
✓ Calculation of theoretical swap end date
✓ Calculation of theoretical cash-flows dates for both legs
o Possible adjustment for taking into account non business days and
convention for non-business days (following, modified following…)
o Theoretically, all combinations of payment frequency and payment Basis
possible for the two legs
o In practice a standard is set for every-market for vanilla swaps, used by
default in many OTC (other the counter) exotic swaps

17
 Vanilla Swaps: Standard Conventions

➢ Vanilla floating leg:

✓ Payment Basis = Basis used for reference rate (Libor or Euribor), same
thing for payment frequency

➢ Examples:
✓ Euro market
o 1 Year maturity swap
▪ 1Year payment frequency and 30/360 payment Basis for fixed leg, Euribor3M
for floating leg
o Maturities over than 1Year
▪ 1Year payment frequency and 30/360 payment Basis for fixed leg, Euribor6M
for floating leg

18
 Vanilla Swaps: Standard Conventions

✓ USD market: two standards

o Money Markets swaps
▪ 1Year payment frequency and ACT/360 for fixed leg, LiborUSD3M for floating
leg
▪ Swaps called “Money markets swaps” because ACT/360 is the payment Basis
for Libor USD

o Bond Basis swaps

▪ 6 months payment frequency and ACT/365 payment Basis for fixed leg,
LiborUSD3M for floating leg
▪ Swaps called “Bond Basis swaps” because ACT/365 is the payment Basis for
USD corporate bonds

19
 Vanilla Swaps: Example

➢ 3Y swap against EURIBOR6M, as of 3/4/2002 (Wednesday)

➢ Start date = 3/4/2002 + 2 Business days = 5/4/2002 (Friday)
➢ Step 1: theoretical swap end date : 5/4/2005 (Tuesday)

20
 Vanilla Swaps: Example

➢ Step 2: Theoretical cash-flows dates

21
 Vanilla Swaps: Example

➢ Step 3: taking account of non business days

22
 Vanilla Swaps: Example

➢ Step 4: Calculation of interest payment frequency, using true dates

of cash Flows and Basis of both legs:

~
➢ d and δ called coverage = year fractions between payment dates for
both leg, using each payment Basis

23
 Vanilla Swaps: Example

➢ Step 5:fixing dates

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

24
 Vanilla Swaps: Example2

SWAP 3Y

Fix Leg Frequency 3M Float Leg Frequency 3M

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

25
 Vanilla Swaps: Example2

SWAP 3Y

Fix Leg Frequency 6 M Float Leg Frequency 3 M

Basis 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

26
 Vanilla Swaps: Example2

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 15/08/06
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
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

27
 Vanilla Swaps: Evaluation of Floating leg

➢ (T~ , d~ )
j = schedule of the floating leg of the Swap (for a 4 years
j 1 j  m
Swap with 6 months payment frequency on floating leg, m = 8, for
example)

~
✓ Tj called payment dates
✓ d~j called payment coverages

➢ Value of the Floating Leg at date 0:

( ) ( )
m
~ ~ ~ ~
 B
j =1
0, T j d j FRA 0, T j −1 , T j −1 + kM ,1  k  12

28
 Vanilla Swaps: Evaluation of Floating leg

➢ Example with a 6M floating leg:

( ) ( )
m
~ ~ ~ ~
✓ PV Floating leg =  B 0, T j d j FRA 0, T j −1 , T j −1 + 6M
j =1

✓ Value of the FRA (cf. pricing of a FRA above):

(
~ ~
) (
FRA T j −1 , T j −1 + 6M = ~ ~
1 

B 0 ,(~
T )
j −1

− 1
~
T j −1 + 6 M = T j
~ ) ( ~
d T j −1 , T j −1 + 6M  B 0, T j −1 + 6M )


➢ Proxy for the value of the floating leg, assuming :

Pv Floating Leg   B(0, T j −1 ) - B(0, T j ) = 1 − B(0, Tn )
m
~ ~
j =1

29
 Vanilla Swaps: Evaluation of Floating leg

➢ Tn = end date of the Swap and (T j , d j )1 j  n = the fixed leg schedule
✓ Proxy formula 1− B(0, Tn ) not linked to the frequency of the floating
leg, only of the maturity of the swap
✓ very good approximation in practice
✓ If floating leg starts at date T , valuation at date t is:

B(t , T ) − B(t , Tn )

30
 Vanilla Swaps: Evaluation of Floating Leg

✓ In fact, even for a spot start Floating Leg, in most currencies there is
a delay (Settlement date = market date + 2Business day)
✓ So the proxy value of a float leg Starting at T0 = 0 + 2 Business days
with first cash flow at T1 will be:

B(0, T0 ) − B(0, Tn )

~
✓ If the first fixing is known (T 0  0), then the value of Floating leg with
first coupon at T1 is:

( (
~ ~ ~
) )( ) ( )
~ ~
1 + Libor 6 M T0 , T1 d1 B 0, T1 − B 0, Tn

31
 Vanilla Swaps: Evaluation of the fixed rate

➢ Value of a fixed leg with fixed rate equal to S:

n n
PV fixed leg(T1 → Tn ) =  Sd i B(0, Ti ) = S   d i B(0, Ti )
i =1 i =1

Physical Level
➢ Fixed Rate Evaluation:
✓ At date 0, swap value is 0: value of fixed leg = value of floating leg

✓ Swap Rate S(0,Tn) = today fixed rate such that both legs have same
value at date 0, for a n years swap

32
 Vanilla Swaps: Evaluation of the fixed rate

➢ Formula for the Swap Rate:

PV Float Leg (T , Tn ) 1 − B(0, Tn )

S (0, Tn ) = n
 n

 d B(0, T )
i =1
i i  d B(0, T )
i =1
i i

o Value of the Physical Level very closed to the sensitivity or duration of a

standard bond of same maturity, payment frequency and with coupon
rate of S (0, Tn )

 d B(0, T ) = Physical Level of a n years Swap fixed leg

i =1
i i

Value of n years fixed leg with rate S = (n years Physical Level) * S

33
 Vanilla Swaps: Evaluation of the fixed rate

➢ Warning: the physical level are in general different (but close) on the
fixed and floating leg:
✓ Application: in order to calculate the number of bp, m, to get a given PV
on a swap:
o If it’s to add/subtract m to the fixed rate, use the fixed leg Physical Level
o If it’s to add/subtract m to the floating leg (as a margin), use the floating leg
Physical Level

34
 Vanilla Swaps: Evaluation of a forward swap rate

➢ Forward Swap = Swap starting in the future at date T

✓ Forward Swap Rate at date t = Rate S (t , T , Tn ) such that present
values at date t of both legs are equal

PV Float Leg (t , T , Tn ) B(t , T ) − B(t , Tn )

S (t , T , Tn ) = n
 n

 d B(t , T )
i =1
i i  d B(t , T )
i =1
i i

✓ To get the value of a Swap Rate at date 0, just do t = T = 0

S (t , T , Tn ) =
Value at t of forward swap rate
for a n years forward swap starting at T

35
 Vanilla Swaps: Schedule of a forward swap

➢ Example: 2Years in 1Year (fixed against Euribor6M) as of 3/4/02

✓ 3/4/02 + 2 Business days is 5/4/02

✓ 5/4/03 is a Saturday

36
 Vanilla Swaps: Cash Level

➢ Cash Level for a n Years Swap

✓ replace the zero-coupon rates by the Swap Rate
✓ take all coverage equal to 1 for Swaps with annual frequency on the
fixed leg:

n
1  1 

i =1 (1 + S )
= 
 1 − / S
n 
 (1 + S ) 
i

✓ to take into account the delay between spot date and Start date, the
Cash Level should be multiplied by B(0, T0 )

37
 Vanilla Swaps: Cash Level

➢ Cash Level General formula for a Swap:

✓ n = number of coupons on the fixed leg
✓ f = payment frequency
o a = 365/360 if Basis for fixed leg is ACT/360, otherwise 1

 
 (1 + aS / f )n  / (aS )
1
1 − 
 

o Example: swap 10Y, S = 5%, a = 1, f = 2 n = 20

Cash Level = 7.794 (7.721 if f = 1)

38
 Vanilla Swaps: Cash Level

Numerical example for various tenors (1Y to 50Y) and swap rates, f = 1

1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 12Y 15Y 20Y 25Y 30Y 40Y 50Y

1% 0.99 1.97 2.94 3.90 4.85 5.80 6.73 7.65 8.57 9.47 11.26 13.87 18.05 22.02 25.81 32.83 39.20
2% 0.98 1.94 2.88 3.81 4.71 5.60 6.47 7.33 8.16 8.98 10.58 12.85 16.35 19.52 22.40 27.36 31.42
3% 0.97 1.91 2.83 3.72 4.58 5.42 6.23 7.02 7.79 8.53 9.95 11.94 14.88 17.41 19.60 23.11 25.73
4% 0.96 1.89 2.78 3.63 4.45 5.24 6.00 6.73 7.44 8.11 9.39 11.12 13.59 15.62 17.29 19.79 21.48
5% 0.95 1.86 2.72 3.55 4.33 5.08 5.79 6.46 7.11 7.72 8.86 10.38 12.46 14.09 15.37 17.16 18.26
6% 0.94 1.83 2.67 3.47 4.21 4.92 5.58 6.21 6.80 7.36 8.38 9.71 11.47 12.78 13.76 15.05 15.76
7% 0.93 1.81 2.62 3.39 4.10 4.77 5.39 5.97 6.52 7.02 7.94 9.11 10.59 11.65 12.41 13.33 13.80
8% 0.93 1.78 2.58 3.31 3.99 4.62 5.21 5.75 6.25 6.71 7.54 8.56 9.82 10.67 11.26 11.92 12.23
9% 0.92 1.76 2.53 3.24 3.89 4.49 5.03 5.53 6.00 6.42 7.16 8.06 9.13 9.82 10.27 10.76 10.96
10% 0.91 1.74 2.49 3.17 3.79 4.36 4.87 5.33 5.76 6.14 6.81 7.61 8.51 9.08 9.43 9.78 9.91
11% 0.90 1.71 2.44 3.10 3.70 4.23 4.71 5.15 5.54 5.89 6.49 7.19 7.96 8.42 8.69 8.95 9.04
12% 0.89 1.69 2.40 3.04 3.60 4.11 4.56 4.97 5.33 5.65 6.19 6.81 7.47 7.84 8.06 8.24 8.30
13% 0.88 1.67 2.36 2.97 3.52 4.00 4.42 4.80 5.13 5.43 5.92 6.46 7.02 7.33 7.50 7.63 7.68
14% 0.88 1.65 2.32 2.91 3.43 3.89 4.29 4.64 4.95 5.22 5.66 6.14 6.62 6.87 7.00 7.11 7.13
15% 0.87 1.63 2.28 2.85 3.35 3.78 4.16 4.49 4.77 5.02 5.42 5.85 6.26 6.46 6.57 6.64 6.66

39
 Vanilla Swaps: Cash Level

Numerical example for various tenors (1Y to 50Y) and swap rates, f = 2

1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 12Y 15Y 20Y 25Y 30Y 40Y 50Y

1% 0.99 1.98 2.95 3.91 4.87 5.81 6.74 7.67 8.59 9.49 11.28 13.90 18.09 22.07 25.86 32.90 39.27
2% 0.99 1.95 2.90 3.83 4.74 5.63 6.50 7.36 8.20 9.02 10.62 12.90 16.42 19.60 22.48 27.44 31.51
3% 0.98 1.93 2.85 3.74 4.61 5.45 6.27 7.07 7.84 8.58 10.02 12.01 14.96 17.50 19.69 23.20 25.81
4% 0.97 1.90 2.80 3.66 4.49 5.29 6.05 6.79 7.50 8.18 9.46 11.20 13.68 15.71 17.38 19.87 21.55
5% 0.96 1.88 2.75 3.59 4.38 5.13 5.85 6.53 7.18 7.79 8.94 10.47 12.55 14.18 15.45 17.23 18.31
6% 0.96 1.86 2.71 3.51 4.27 4.98 5.65 6.28 6.88 7.44 8.47 9.80 11.56 12.86 13.84 15.10 15.80
7% 0.95 1.84 2.66 3.44 4.16 4.83 5.46 6.05 6.59 7.11 8.03 9.20 10.68 11.73 12.47 13.37 13.83
8% 0.94 1.81 2.62 3.37 4.06 4.69 5.28 5.83 6.33 6.80 7.62 8.65 9.90 10.74 11.31 11.96 12.25
9% 0.94 1.79 2.58 3.30 3.96 4.56 5.11 5.62 6.08 6.50 7.25 8.14 9.20 9.88 10.32 10.78 10.97
10% 0.93 1.77 2.54 3.23 3.86 4.43 4.95 5.42 5.84 6.23 6.90 7.69 8.58 9.13 9.46 9.80 9.92
11% 0.92 1.75 2.50 3.17 3.77 4.31 4.79 5.23 5.62 5.98 6.58 7.27 8.02 8.47 8.72 8.97 9.05
12% 0.92 1.73 2.46 3.10 3.68 4.19 4.65 5.05 5.41 5.73 6.28 6.88 7.52 7.88 8.08 8.25 8.31
13% 0.91 1.71 2.42 3.04 3.59 4.08 4.51 4.88 5.22 5.51 6.00 6.53 7.07 7.36 7.52 7.64 7.68
14% 0.90 1.69 2.38 2.99 3.51 3.97 4.37 4.72 5.03 5.30 5.73 6.20 6.67 6.90 7.02 7.11 7.13
15% 0.90 1.67 2.35 2.93 3.43 3.87 4.24 4.57 4.85 5.10 5.49 5.91 6.30 6.49 6.58 6.65 6.66

40
 Physical Level vs. Cash Level

➢ Cash Level and Physical Level at 0 have close values:

n n
  d i B(0, Ti )
1

i =1 (1 + S (0, Tn ))
i
i =1

➢ Dependence on yield curve shape:

✓ Positive Slope : Physical Level > Cash Level
✓ Negative Slope : Physical Level < Cash Level

41
 Physical Level Interpretation

➢ three possible interpretations of n years Physical Level

✓ Present value of 1 bp (0.01%) paid during n years on a 10000€ notional
✓ Present value of 1€ paid during n years*
✓ Approximate profit (resp. loss) in M€ for -1% parallel shift down (resp.
+1% up) of the yield curve, when being receiver of the fixed rate on a n
years market swap (initial PV before shift = 0), notional 100M€
✓ Numerical example:
o 120000€ required P&L on a 10 years swap fixed rate against = Euribor +
margin m, swap notional = 50M€
o Physical Level = 8 (for 10000€ notional)

(interpretation 1 + proportionality rule)

m = 120000/(50*10^6)/8= 0.03% = 3bp

*100% rate on a notional of 1€! 42

 Cash Level of a forward swap

➢ Cash Level for a Forward Swap:

n
 1 
B(0, T )  = B(0, T ) 1 −
1
 / S
n 
i =1 (1 + S )  (1 + S ) 
i

✓ T = Start Date
✓ S = value of the Forward Swap rate at date 0

43
 Vanilla Swaps: link with bonds

➢ Risk equivalence of a Swap (starting at spot date) and a Fixed Rate

Bullet Bond:
✓ 10Y Vanilla Swap:

✓ We add a cash flow of 1 (which is the notional of the swap at maturity:

44
 Vanilla Swaps: link with bonds

➢ Floating Leg value is now 1:

1 − B(0, Tn ) + B(0, Tn ) = 1
✓ Present Value of 1 is B(0, Tn )
✓ Fixed Leg is now a 10Y bullet Fixed rate bond!
✓ For a trader holding a 10Y bond, financing the position at
EURIBOR6M, the analysis is the same
✓ In other words, the float leg of a 10year Vanilla Receiver Swap is the
financing leg of a long position on a 10Y Bond
✓ To Add the notional on the two legs of the swaps, is equivalent to
move all the interest rate risk of the Swap on the Fixed Leg
o When fixing known, a slight risk on the float leg, which is initially
around the index of the float leg (for our example as the float leg is
payer)

45
 Swaps: Building a zero-coupon curve without taking account of
Basis swaps

➢ Basics: DEPOSIT
DEPOSIT
O/N
1W
0.12
0.886
DEPOSIT 2W 0.91

✓ List of Markets Rates : Tenors DEPOSIT

DEPOSIT
1M
2M
0.953
1.094
DEPOSIT 3M 1.281
o (deposits, Libor futures , swaps FUTURE
FUTURE
JUN09
SEP09
98.6925
98.6725
rates) FUTURE
FUTURE
DEC09
MAR10
98.4675
98.2875

✓ A zero-coupon rate for each

FUTURE JUN10 98.0175
FUTURE SEP10 97.7425
FUTURE DEC10 97.4375

tenor FUTURE
SWAP
SWAP
MAR11
3Y
4Y
97.2075
2.51
2.852
swaps définition for this example :

✓ One interpolation method

SWAP 5Y 3.112 Fixed leg :
SWAP 6Y 3.324 payment frequency : annual,
SWAP 7Y 3.49 payment basis : 30/360
SWAP 8Y 3.621

✓ Find Zero Coupon Rates SWAP

SWAP
SWAP
9Y
10Y
12Y
3.73
3.822
3.985
Floating leg
payment frequency : quarterly
payment basis : ACT/360
consistent with Markets Rates SWAP
SWAP
15Y
20Y
4.154
4.221
o Zero-Coupon curve used both for SWAP
SWAP
25Y
30Y
4.137
4.037
discounting and calculate Forwards SWAP
SWAP
40Y
50Y
3.837
3.731
Rates
 B(T0 , Ti )d i'  3.822
10
o Constraints on Swaps:
i =1
▪ Fixed Leg PV = Floating Leg PV for all
( )
Tenors 40

▪ Example
N.B: données du 08/06/09
for 10Y tenor
=  B T0 , Ti d i  EuriborForward3M (Ti )
i =1

46
 Swaps: Building discount and fixing curve: taking account of Basis
swaps

➢ Additional contracts and Type Maturity value

constraints BASISSWAP 6M/12M
BASISSWAP 6M/12M
1Y
2Y
4.8bp
3.8bp

✓ Contracts: Basis swaps

BASISSWAP 6M/12M 3Y 3bp
BASISSWAP 6M/12M 4Y 2.6bp
BASISSWAP 6M/12M 5Y 2.3bp
o Paying Euribor 3M +6bp and receiving BASISSWAP 6M/12M 6Y 2bp
Euribor6M during 7Y BASISSWAP 6M/12M 7Y 1.8bp
BASISSWAP 6M/12M 8Y 1.7bp
o Paying Euribor 6M+3bp and receiving BASISSWAP 6M/12M 9Y 1.5bp
BASISSWAP 6M/12M 10Y 1.4bp
Euribor12M during 3Y BASISSWAP 6M/12M 11Y 1.3bp

✓ Additional constraints: Pv of BASISSWAP 6M/12M

BASISSWAP 6M/12M
12Y
15Y
1.3bp
1.1bp
BASISSWAP 6M/12M 20Y 0.9bp
Basis swap = 0 BASISSWAP 6M/12M 30Y 0.8bp

o Example Basis swap 6M/12M at 4Y:

 B 6 M (0, T ) 
EuriborForward 6 M (Ti ) =
360
 6 M − 1
(Ti + 6M − Ti )  B (0, T + 6M ) 
 B12 M (0, T ) 
EuriborForward12 M (Ti ) =
360
 12 M − 1
(Ti + 12M − Ti )  B (0, T + 12M ) 

 B (T , T ) d ( )
 (EuriborForward 6 M (Ti ) + 0.026) =  B d T0 , T j  d 12j M  EuriborForward12 M (T j )
8 4
d 6M
0 i i
i =1 j =1
N.B: data as of 08/06/09

47
 Basis swap: numerical applications
Courbe Zéro-coupons Fixing

➢ Zero-Coupon Rates 5.00%

4.50%
4.00%
3.50%
fixing 1M
3.00%
fixing
0.05 1M fixing
0.39 3M fixing
0.53 6M fixing
0.60 12M fixing 3M
2.50%
1M 0.60% 0.85% 0.95% 1.00% fixing 6M
2.00%
3M 0.95% 1.19% 1.29% 1.34% fixing 12M
1.50%
6M 1.15% 1.39% 1.49% 1.54%
1.00%
9M 1.17% 1.41% 1.52% 1.56%
0.50%
1Y 1.29% 1.53% 1.64% 1.68%
0.00%
18M 1.49% 1.74% 1.84% 1.88%
2Y 1.73% 1.97% 2.07% 2.11% 0Y 10Y 20Y 30Y 40Y 50Y
3Y 2.24% 2.42% 2.52% 2.55%
4Y 2.64% 2.79% 2.88% 2.90% Type Maturity value
5Y 2.95% 3.07% 3.15% 3.17% BASISSWAP 3M/6M 3Y 9.9bp
6Y 3.20% 3.31% 3.38% 3.40% BASISSWAP 3M/6M 4Y 8.5bp
7Y 3.40% 3.50% 3.55% 3.57% BASISSWAP 3M/6M 5Y 7.5bp
8Y 3.56% 3.65% 3.70% 3.71% BASISSWAP 3M/6M 6Y 6.6bp
9Y 3.70% 3.77% 3.82% 3.83% BASISSWAP 3M/6M 7Y 6bp
BASISSWAP 3M/6M 8Y 5.4bp
10Y 3.81% 3.88% 3.92% 3.94%
BASISSWAP 3M/6M 9Y 5bp
12Y 4.02% 4.07% 4.11% 4.12% BASISSWAP 3M/6M 10Y 4.7bp
15Y 4.24% 4.28% 4.31% 4.32% BASISSWAP 3M/6M 11Y 4.4bp
20Y 4.31% 4.34% 4.37% 4.37% BASISSWAP 3M/6M 12Y 4.1bp
25Y 4.15% 4.17% 4.20% 4.20% BASISSWAP 3M/6M 15Y 3.6bp
30Y 3.97% 3.99% 4.01% 4.01% BASISSWAP 3M/6M 20Y 3.1bp
40Y 3.61% 3.62% 3.64% 3.65% BASISSWAP 3M/6M 30Y 2.8bp
50Y 3.43% 3.44% 3.46% 3.46% Type Maturity value Type Maturity value
BASISSWAP 1M/12M 2Y 33.4bp BASISSWAP 6M/12M 1Y 4.8bp
BASISSWAP 1M/12M 3Y 27.2bp BASISSWAP 6M/12M 2Y 3.8bp
BASISSWAP 1M/12M 4Y 23bp BASISSWAP 6M/12M 3Y 3bp
BASISSWAP 1M/12M 5Y 19.7bp BASISSWAP 6M/12M 4Y 2.6bp
BASISSWAP 1M/12M 6Y 17.1bp BASISSWAP 6M/12M 5Y 2.3bp
BASISSWAP 1M/12M 7Y 15.2bp BASISSWAP 6M/12M 6Y 2bp
BASISSWAP 1M/12M 8Y 13.7bp BASISSWAP 6M/12M 7Y 1.8bp
BASISSWAP 1M/12M 9Y 12.6bp BASISSWAP 6M/12M 8Y 1.7bp
BASISSWAP 1M/12M 10Y 11.7bp BASISSWAP 6M/12M 9Y 1.5bp
BASISSWAP 1M/12M 11Y 10.9bp BASISSWAP 6M/12M 10Y 1.4bp
BASISSWAP 1M/12M 12Y 10.2bp BASISSWAP 6M/12M 11Y 1.3bp
BASISSWAP 1M/12M 15Y 8.7bp BASISSWAP 6M/12M 12Y 1.3bp
BASISSWAP 1M/12M 20Y 7.3bp BASISSWAP 6M/12M 15Y 1.1bp
BASISSWAP 1M/12M 30Y 6.5bp BASISSWAP 6M/12M 20Y 0.9bp 48
BASISSWAP 6M/12M 30Y 0.8bp
 Basis swap: numerical applications

➢ Swaps calculations
✓ Example 1: Swap 3Y

swap 3Y (EUR) 3y basis swap datas:

fixed leg with annual payment basis swap 1M/12M 27.2bp
basis swap 6M/12M 3.8 bp
index for floating leg : EURIBOR 1M EURIBOR 3M EURIBOR 6M EURIBOR 12M basis swap 3M/6M 9.9bp
fixed rate 2.26 2.41 2.51 2.54

✓ Example 2: Swap 2Y/1Y

swap 2Y/1Y
jambe fixe coupon annuel

index jambe variable : EURIBOR 1M EURIBOR 3M EURIBOR 6M EURIBOR 12M

taux fixe : 2.83 2.94 3.05 3.07

49
 Black-Scholes for Lognormal forward

➢ Practical use of B&S formulas in dealing rooms:

✓ To put in a nutshell, the B&S volatility can be seen as the wrong
number to put in the wrong model to get the right price
✓ Let’s note: Ft,T the price at date t of a Forward of maturity T on
underlying S
✓ F can be a forward on any kind of underlying: equity, interest
rate, bond, FX rate…
➢ Simplified notations:

F0 = F0,T
Ft = Ft ,T
FT = FT ,T

50
 Black-Scholes for Lognormal forward

➢ Let FT follows a lognormal law with annual volatility σ and

expectation F0 :
2
N (0,T )− T
FT = F0e 2

➢ Price (in a lognormal model) of a Call on FT :

BSprice (F0 , K ,T ,, Lognormal,call) = E (Max (FT − K ,0 )) = E (FT − K )
+

T = maturity
K = strike

+ +
𝐸 𝑆𝑇 − 𝐾 = 𝐸 𝐹𝑇 − 𝐾
+ +
𝐸 𝐾 − 𝑆𝑇 = 𝐸 𝐾 − 𝐹𝑇
51
 Black-Scholes for Lognormal forward

➢ Price of a put on FT :

BSprice (F0 , K ,T ,, Lognormal, put ) = E (Max (K − FT ,0 )) = E (K − FT )

+

➢ In practice, a dedicated model for each underlying, especially:

✓ Choice of probability (risk neutral, forward neutral…)
✓ Calculation of the Forwards (dividend in % or in value for an equity, repo
rate for bond or equity, stripping of the curve for interest rate…)
✓ Volatility (interpolation rule, volatility surface (equity), volatility cube
(interest rate)

52
 Black-Scholes formula for a Lognormal forward

➢ If dFt
= dBt
Ft
u2
➢ then x
1 −2
N (x ) =  e du
E (FT − K ) = F0  N (d1 ) − K  N (d 2 ) 2
+ −

E (K − FT ) = K  N (− d 2 ) − F0  N (− d1 )
+

T = year fraction in ACT/365 Basis between 0

and settlement of forward T

F  1 F 
Ln 0  +  2T Ln 0 
d1 = K 2 =  K  + 1 T
 T  T 2
Better use the second
d 2 = d1 −  T way of writing
d1 d 1

53
 Black-Scholes for Gaussian forward

➢ Let FT follows a Gaussian law:

✓ FT = F0 +  ' N (0, T )
✓ E (Max (FT − K ,0)) = E (FT − K )+ = BSprice (F0 , K ,T ,' , normal,call )

✓ E (Max (K − FT ,0)) = E (K − FT ) = BSprice (F0 , K ,T ,' ,normal, put )

+

✓ σ’ is a annual standard deviation, not a volatility

54
 Black-Scholes for Gaussian forward

dFt = ' dBt

E (FT − K ) = (F0 − K )N (d ) +  ' T n(d )

+

E (K − FT ) = (K − F0 )N (− d ) + ' T n(d )
+

F0 − K
d=
' T −
x2 x
, N (x ) =  n(u )du
1
n( x ) = e 2
2 −

N = distribution function of Gaussian law mean 0 variance 1 n

density function of Gaussian law mean 0 and variance 1 (N(0,1))

55
 Black-Scholes for Gaussian forward

➢ For Vanilla Caps and Swaptions, traders use both models,

Gaussian or lognormal
➢ Gaussian model also very useful for getting analytical
formulas/proxy/control variable for:
✓ Spreads options (interest rates, index…)
✓ Asian options
✓ Exchange options on index with non zero strike

Special case: ATM (At the money Option):

' T
Call = Put =  0.4 ' T
2

56
 Black-Scholes for Gaussian forward

➢ Very good approximation of the relation between σ’ (the standard

deviation) and σ (the volatility) for ATM strike:
 '    F0
✓ relative cheapness of European ATM options:
o just compare the exact or approximated standard deviations
o Standard deviations often called « Gaussian volatilities » by many traders or
quants
▪ confusing but seems to be the market practice
o this formula shows that the price of an ATM European rate option (Caplet/swaption
before multiplication by level or call/put on CMS) depends in fact only on the
standard deviation, not on the forward rate and on the volatility

57
 Black-Scholes for Gaussian forward
➢ ATM options in Practice:

Interest Rates ATM Options Equity ATM Options

ATM Caplets/Floorlets +
 S − F0,T 
Pay − off ATM Ca plet = (Euribor (T , T + d ) − F (0, T + T + d )) d  N
+ Pay − off ATM Eq uity Option =  T  N

 F0,T 
Pay − off ATM F loorlet = (F (0, T + T + d ) − Euribor (T , T + d )) d  N
+

0.4 ' T
Price ATM Equity Option  N
Price ATM Caplet / Floorlet  0.4 ' T  d  N F0
 '    F0 
ATM Cash Settlement Swaptions

Pay − of f ATM Payer Swaption = CashLevel(S (T , Tn )) (S (T , Tn ) − S (0, T , Tn ))

+

Pay − off ATM Receiver Swaption = CashLevel S ( (0,T ,Tn ) − S (T ,Tn )) (S (T ,Tn ))+ Price ATM Equity Option  0.4 T  N

Price ATM swaption  0.4 ' T  CashLevel(S (0, T , Tn )) N

➢ Conclusion : ’ (standard deviation) for interest rate options,  (volatility) for equity options
✓ All prices are calculated as if options paid at maturity (non-discounted)
✓ (N = Notional of the option)

58
 Black-Scholes formulas: a few remarks

➢ Black-Scholes formulas above = prices of call and put that would be

paid by the buyer of the option at maturity T
➢ At the same date the pay-off of the option will be paid by the seller of
the option to the buyer of the option
➢ To get the price of call and put that would be paid at date 0, just
multiply the above formulas by B(0,T )

59
 Black-Scholes formulas: a few remarks

➢ Call put Parity:

✓ call/put parity for standard European option is totally independent of the
choice of the model (lognormal, Gaussian or whatever):
✓ If the buyer of the options pays the option at maturity (when he gets the
pay-off or 0):
call − put = E (FT − K ) − E (K − FT ) = E (FT − K ) = F0 − K
+ +

✓ If the buyer of the options pays the option at date 0 (and gets the pay-off
or 0 at maturity):
call − put = (F0 − K )B(0,T )

60
 Main Greeks of B&S formulas on forwards

➢ Black-Scholes for Lognormal forward

✓ Delta Call-Delta Put = 1

dC
deltaCall = = N (d1 )
dF0
✓ Gamma Call = Gamma Put d 2C d 2 P 1
= = n(d1 )
d F0 d F0 F0 T
2 2

✓ Vega Call = Vega Put (variation of price for a variation of 1% of

volatility)
dC dP
/ 100 = / 100 = F0 T n(d1 ) / 100
d d

61
 Main Greeks of B&S formulas on forwards

➢ Black-Scholes for Gaussian forward

✓ Delta Call-Delta Put = 1

dC
deltaCall = = N (d )
dF0
✓ Gamma Call = Gamma Put
d 2C d 2 P 1
= 2 = n( d )
d F0 d F0  ' T
2

✓ Vega Call = Vega Put (variation for a variation of  ' equal to 1 )

dC dP
= = T n( d )
d ' d '

62
 Interpretation of N(d)

➢ Black-Scholes for Lognormal Forward:

✓ N (d 2 ) is the probability that the spot (forward = spot) at expiry) is above
the strike at expiry

F 
ln  0 
d1 = 
K  1 d 2 = d1 −  T
+  T
 T 2

➢ Black-Scholes for Gaussian Forward:

N (d ) is the probability that the spot is above the strike at expiry
F0 − K
d=
' T

N(d) also gives the delta when using the B&S

Gaussian formula

63
 Interpretation of N(d)

➢ If K = F0, ATM option call = put

➢ Then, whatever the value of  and ’

1
d2 = −  T  0
2 N (d 2 )  0.5
1 N (d1 )  0.5
d1 =  T = −d 2
2 N (d ) = 0.5
d1  0
d =0

64
 Interpretation of N(d)

➢ Probability of being above the strike for ATM call:

✓ less 0.5 when one uses the Lognormal formula
✓ 0.5 when one uses the Gaussian formula
✓ Premium paid at maturity

➢ Delta:
✓ above 0.5 when uses the Lognormal formula
✓ 0.5 when uses the Gaussian formula
o Result true whatever  and ’
o especially if for a given  (resp. ’) one chooses ’ (resp.  ) so that the
Gaussian price of ATM (resp. Lognormal) is equal to the Lognormal price
(resp. Gaussian price) given by  (resp. ’)

65
 Interpretation of N(d)

➢ Interpretation valid only in the B&S world, not in the real world,
because of the smiles

➢ Digital cannot be priced using the previous results but with the « call
spread method »

66
 B&S formulas: Variance of Lognormal forward

➢ X is a Gaussian variable of mean m and variance  2:

( )  2 

Ee X
= exp m + 
 2 
( ) ( )( ( ) )
V e X = exp 2m +  2  exp  2 − 1

( ) 2 T using the previous results we have:

2
N 0, T −
➢ FT = e
F0
E(FT ) = F0
, as expected (expectation of forward at T is F0)

➢ Variance of FT:

( ( ) )
V (FT ) = F0 2 exp  2T − 1

67
 B&S formulas: Variance of lognormal forward

➢ Standard deviations:
✓ Standard deviation of FT = F0 exp ( 2T ) − 1

✓ annual standard deviation of

FT =
( )
F0 exp  2T − 1
T
➢ First order Taylor development:

✓ Standard deviation of F  F  T
T 0

✓ Annual standard deviation of F T =  '    F0

o Explanation for  '    F0

o approximation, not too bad for ATM strike (K = F0 )
We remind that it’s the annual standard deviation (resp. annual volatility)
that has to be used in the Gaussian (resp. lognormal) version of bsprice
68
 B&S formulas: Variance of Lognormal forward

➢ “Conversion Lognormal/Gaussian volatilities”:

✓ For a given K, the exact  ' (resp  ) to use in the B&S Gaussian (resp.
lognormal) formula to recover the lognormal (resp. Gaussian) price, has to
be computed numerically

✓ For a given  ' (and also (F0 , T )), if for any strike the implied volatilities to
recover the Gaussian prices are computed, the smile is in fact a skew, as we
see on the following graph:

69
 B&S formulas: Variance of Lognormal forward
Implied Volatility skew from the normal model

14.00%

13.00%

 ' = 10
12.00% F0 = 100
T =1
Implied Volatility

11.00% implied volatility

10.00%

9.00%

8.00%
60 70 80 90 100 110 120 130 140
Strike

70
 Caplet, Floorlet

➢ Caplet = call option on a Euribor (or Libor..) forward

✓ Caplet of strike K pays at date T+δ the difference (if positive) between
Euribor on the payment frequency starting T ending T+δ and K
o pay-off of Caplet at date T+δ is : Max(Euribor(T, T+δ)-K;0) *δ
o fixing of the Euribor is at T-2D, taking into account non business days

➢ Floorlet = same thing for a put option

71
 Caplet, Floorlet

➢ Market practice for Caplet/Floorlet pricing

✓ Price of a Caplet at date t:

B(t , T + δ) δ  BSprice(FRA(t , T , T + δ), K , T − t , σ, Lognormal, call )

✓ Price of a Floorlet at date t:

B(t , T + δ) δ  BSprice(FRA(t , T , T + δ), K , T − t , σ, Lognormal, put )

✓ σ = volatility of FRA(t,T,T+δ) depends on δ, K and T

✓ B & S for Black Scholes, details in next slide

72
 Cap, Floor
➢ Cap = sum of Caplets, Floor = sum of Floorlets
➢ Value of cap is the value of all the Caplets included, same thing for
floor
➢ Example: 1 Year Cap on Euribor3M

✓ first Caplet :
o not included most of the time as the value today of the first Euribor is
Known.
o most of the times, only 3 Caplets in the above example

73
 Call/Put Parity for Cap & Floor

➢ Call/put parity for Cap/Floor :

( ) ( ) ( )
m
~ ~ ~ ~ ~ ~
Cap − Floor =  B 0, T j d j FRA 0, T j −1 , T j − B 0, T j d j K
j =1

➢ Cap-Floor = Value of Floating Leg minus Value of Fixed Leg (strike

K)
➢ ATM strike:
✓ Strike K such that Cap = Floor
✓ ATM strike = swap rate corresponding to the schedule of the Cap and
floor, using payment frequency and payment Basis of this schedule

74
 Call/Put Parity for Cap & Floor

➢ Example: for 5Y Cap/Floor on Euribor3M

✓ ATM strike = Strike such that Cap = Floor:
o 4.75Y rate in 3months, with payment frequency 3M and payment Basis
ex/360 if the first Caplet/Floorlet are not included
o 5Y swap rate, with payment frequency 3M and payment Basis ACT/360 if
the first Caplet/Floorlet are included

75
 Call/Put Parity for Cap & Floor

➢ Comments on market practice for Caplet/Floorlet:

✓ can be justified theoretically by introducing the forward neutral
probability tool
✓ more important to understand the practice cannot lead to arbitrage
because it’s consistent with the call/put parity
✓ Traders used this practice long time before quants used the forward
neutral probability tool
➢ The ATM rule (page 55) shows that the cheapness of ATM
Caplet/Floorlet depends only of the standard deviation, so
cheapness can be evaluated using the formula:

' =   F0

76
 Physical settlement vs. Cash Settlement Swaptions

➢ Physical settlement Swaptions:

➢ A physical settlement Swaptions of strike K on the swap T →Tn is the
right to enter into a swap at date T with fixed rate K
➢
➢ Cash settlement Swaptions:
➢ No settlement of a swap at maturity but a cash pay-off

➢ Receiver vs. Payer Swaptions:

✓ Receiver Swaptions: to receive the fixed rate and to pay the floating
rate; if one buys a receiver Swaptions, one believe rates will go down.
✓ Payer Swaptions: to pay the fixed rate and to receive the floating rate;
if one buys a payer Swaptions, one believe rates will go up

77
 Schedule: Swaptions vs. Swap forwards

➢ Reminder: Standard swap (10Y)

Theoretical End = Spot date + 10Y,

Today Spot date = Today + Spot Lag,
non adjusted
Adjusted

➢ Spot Lag = 2 days for most currencies (EUR, USD, JPY…), 0 for GBP
➢ Spot date = Start date for a Standard Swap
➢ Schedule of fixed leg and float leg first calculated using Theoretical End to
calculate cash-flows dates, then all cash flows are adjusted (Using business
days conventions: following, modified-following…)

78
 Schedule: Swaptions vs. Swap forwards

➢ Swap Forward: 7Y in 3Y
Theoretical End = Spot Date + 3Y + 7Y,
Not adjusted

Fixing Date =
Start Date – Spot Lag,
Adjusted

Spot Fixing Start Theoretical End

Today
Date Date Date

Start Date = Spot Date + 3Y,

Adjusted

➢ Swaption: 7Y in 3Y
Exercise date =
Expiry + Spot Lag, Start Date + 7Y,
Today + 3Y,
Adjusted Not adjusted
Adjusted

Exercise Start
Today Theoretical End
Date Date

79
 Schedule: Swaptions vs. Swap forwards

➢ Swaption vs. swap forward example:

Swaption 2Y/1Y au 3/04/02 Swap forward 2Y/1Y au 3/04/02

today 03/04/02 Wednesday today 03/04/02 Wednesday

exercice date 03/04/03 Thursday fixing date 05/04/02 Friday
start date 07/04/03 Monday start date 07/04/03 Monday
theoretical end 07/04/05 Thursday theoretical end 05/04/05 Tuesday

Schedule of underlying swap swap forward Schedule of swap forward

fixed leg (30360) fixed leg (30360)

Pay. Dates Pay. Cvge Pay. Dates Pay. Cvge
07/04/04 1.0000 05/04/04 0.9944
07/04/05 1.0000 05/04/05 1.0000

floating leg (ACT360) floating leg (ACT360)

Pay. Dates Pay. Cvge Lib Start Lib End Fix. Dates Pay. Dates Pay. Cvge Lib Start Lib End Fix. Dates
07/10/03 0.5083 07/04/03 07/10/03 03/04/03 06/10/03 0.5056 07/04/03 06/10/03 03/04/03
07/04/04 0.5083 07/10/03 07/04/04 03/10/03 05/04/04 0.5056 06/10/03 06/04/04 02/10/03
07/10/04 0.5083 07/04/04 07/10/04 05/04/04 05/10/04 0.5083 05/04/04 05/10/04 01/04/04
07/04/05 0.5056 07/10/04 07/04/05 05/10/04 05/04/05 0.5056 05/10/04 05/04/05 01/10/04

80
 Schedule: Swaptions vs. Swap forwards

➢ Swaption schedule example:

✓ Swaptions on 2Y (fixed against Euribor6M) in 1Y as of 03/4/02, EUR, strike 4.48% (on
03/4/02: swap 2Y = 4.48%, swap 2Y/1Y = 5.13%)
Exercise date = Today + Expiry + Spot Lag, Start Date + 2Y,
1Y, Adjusted Not adjusted
Adjusted

→ Today
Exercise Start
Theoretical End
Date Date
03/04/02
03/04/03 07/04/03 05/04/05

If on 3/4/03 exercise date, the swap 2Y against Euribor6M is above 4.48%, the buyer of the option
enters in a 2Y swap payer at 4.48%, receiver of Euribor6M (physical swaption)
Payer Swaption gain at maturity
45%
pay-off as percentage of notional

40%

35%

30%

25%

20%
Strike
15% 4.48%
10%

5%

0%
0% 2% 4% 6% 8% 10% 12% 14%

swap rate at maturity

81
 Physical settlement Swaptions

➢ market practice to price a receiver physical settlement swaption

at date t:

 n 
  d i B(t , Ti )  BSprice(S (t , T , Tn ), K , T − t ,  , Lognormal, put )
 i =1 
Physical Level
➢ market practice to price a payer physical settlement swaption at
date t:

 n 
  d i B(t , Ti )  BSprice(S (t , T , Tn ), K , T − t ,  , Lognormal, call )
 i =1 

( )
S t , T , Tn =
= value a t of n years forward swap rate
(swap starting at T)
82
 Cash settlement Swaptions

➢ Cash settlement Swaptions in practice:

✓ no settlement of a swap at maturity
✓ receiver cash settlement swaption pay-off at T:

n
 Max(K − S (T , Tn ),0)
1
 (1 + S (T , T ))i
i =1 n
Cash Level (at T)
✓ payer cash settlement swaption pay-off at T:

n
 Max(S (T , Tn ) − K ,0)
1
 (1 + S (T , T ))i
i =1 n

S (T , Tn ) = n years market swap rate at swaption maturity T

83
 Cash settlement Swaptions

➢ Swap with semi-annual payment on fixed leg:

✓ If payment frequency of fixed leg = 6M, just replace
n  
 / S (T , Tn )
1 1

i =1 (1 + S (T , Tn ))
= 
 1 − n 
 (1 + S (T , Tn )) 
i

By:
×
2n  
 / S (T , Tn )
1/ 2 1

i =1 (1 + S (T , Tn ) / 2 )

= 1 − 2n 
 (1 + S (T , Tn ) / 2) 
i

✓ In both cases, n = tenor of underlying swap rate (e.g. n = 10 for

10Y)

84
 Cash settlement Swaptions

➢ market practice to price a receiver cash settlement swaption:

n
B(t , T )   BSprice(S (t , T , Tn ), K , T − t ,  , Lognormal, put )
1
i =1 (1 + S (t , T , Tn ))i

➢ market practice to price a payer cash settlement swaption:

n
B(t , T )   BSprice(S (t , T , Tn ), K , T − t ,  , Lognormal, call )
1
i =1 (1 + S (t , T , Tn )) i

Cash Level (at t)

➢ Volatility in practice:
✓ depends on features of forward swap and strike
✓ For same features (same swap forward and same strike), the volatility is
the same for a physical or cash settlement Swaptions

85
 Call/Put Parity for Swaptions

➢ Call put parity formulae for Swaptions:

✓ Physical Payer Swaption-Physical Receiver Swaption =

 n 
  d i B(t , Ti )  (S (t , T , Tn ) − K )
 i =1 

✓ Cash settlement Payer Swaption-Cash settlement Receiver Swaption* =

n
B(t , T )   (S (t , T , Tn ) − K ) =
1
i =1 (1 + S (t , T , Tn ))i

  
B(t , T ) 1 −  / S (t , T , Tn )  (S (t , T , Tn ) − K )
 1
 ( + ( ))n  
 1 S t , T , Tn  

*For cash level formula, Annual payment on the underlying swap fixed leg, see before for semi-annual frequency

86
 Physical Level vs. Cash Level for Swaptions

➢ Thanks to the discount factor B(t , T ), Cash Level and Physical Level at t are
homogenous:

n n
B(t , T )    d i B(t , Ti )
1
i =1 (1 + S (t , T , Tn ))
i
i =1

➢ Dependence on yield curve shape:

✓ Positive Slope: Physical Swaption Level > Cash settlement Swaption Level
✓ Negative Slope: Physical Swaption Level < Cash settlement Swaption
Level

87
 Swaptions: a few remarks

➢ Market practice justifications:

✓ “Q Level probability” tool, but as for caps/floors:
✓ market practice consistent with above call/put parity formulas for both
kind of Swaptions no arbitrage possibility
✓ Traders used B & S formulae long time before quants created the “Q
Level probability” tool

➢ ATM rule (page 57)

✓ cheapness of ATM Swaptions (if we forget the level) depends only on the
standard deviation
✓ can be seen using the formula
' =   F0

88
Cash Discount formula for Cash Settlement Swaption

▪ A slightly different formula for Cash Settlement Swaption:

n n
1 1
➢ Replace  (1 + S (T , T ))i by  (1 + S (T , T ))i in the Level Cash
i =1 n i =1 i i

✓ New formula using for zero-coupon rate of maturity i the Swap

of maturity i
✓ Exotic product, not vanilla

89
Cash Settlement Swaptions: summary for pay-off calculation

Default formula

For level of cash Data used Main market

settlement
swaptions

EUR, GBP, CHF, Cash level ISDA FIXING Cash settlement swaptions
SEK,..
USD, JPY Physical level Counterparty Physical swaptions
agreement

90
 Swaptions: a few remarks

➢ Market practice justifications:

✓ “Q Level probability” tool, but as for caps/floors:
✓ market practice consistent with above call/put parity formulas for both kind of
Swaptions →cannot lead to arbitrage
✓ Traders used B & S formulae long time before quants created the “Q Level
probability” tool

➢ ATM rule
✓ cheapness of ATM Swaptions (if we forget the level) depends only on the standard
deviation
✓ can be seen with the formula ' =   F 0

91
Swaptions volatility surfaces at three dates: 13/02/06 -15/10/08 - 22/10/09 (EUR)

Swaptions B&S Volatility Surface, Tenor 10Y,

13/02/06

60% 50.00%-60.00%
40.00%-50.00%
50% 30.00%-40.00%
20.00%-30.00%
40%
10.00%-20.00%

30% 0.00%-10.00%

20%
350%
10% 200%
100%
0% 50% Strike in %
0.1Y
0.5Y 2Y 4Y 6Y Y 10% of Forward
8 10Y 0Y Y
2 30 Swap Rate
Option Maturity
Swaptions B&S Volatility Surface, Tenor 10Y, Swaptions B&S Volatility Surface, Tenor 10Y,
15/10/08 22/10/09
90.00%-100.00%
80.00%-90.00% 80.00%-90.00%
90% 70.00%-80.00% 100% 70.00%-80.00%

80% 60.00%-70.00% 90% 60.00%-70.00%

50.00%-60.00% 80% 50.00%-60.00%
70% 40.00%-50.00% 40.00%-50.00%
70%
60% 30.00%-40.00% 60% 30.00%-40.00%
20.00%-30.00% 20.00%-30.00%
50% 50%
10.00%-20.00% 10.00%-20.00%
40% 40%
0.00%-10.00% 0.00%-10.00%
30% 30%
20% 350%
20% 350% 200%
200% 10% 100% Strike in %
10% 100% Strike in % 0% of Forward
50%
0% 50% of Forward Swap Rate
0.1Y
0.5Y 2Y 4Y 6Y 8Y 10%
0.1Y0.5Y 2Y 4Y 6Y 10% Swap Rate 10Y20Y30Y
8Y10Y 0Y
2 30Y Option Maturity
Option Maturity

92
 Volatility Smiles 10Y/5Y, 10Y/1Y at three dates:
13/02/06 -15/10/08 - 22/10/09 (EUR)

Smile Swaptions 10Y/1Y

90.0%
80.0%
70.0%
60.0%
Volatility

13/02/06
50.0%
15/10/08
40.0% 22/10/09
30.0%

20.0%
10.0%
0.0%
0% 50% 100% 150% 200% 250% 300% 350%
Strike in % of Forward Swap Rate

Smile Swaptions 10Y/5Y

70.0%

60.0%

50.0%
Volatility

40.0% 13/02/06
15/10/08
30.0% 22/10/09

20.0%

10.0%

0.0%
0% 50% 100% 150% 200% 250% 300% 350%
Strike in % of Forward Swap Rate

93
 Swaptions standard deviation surfaces at three dates:
13/02/06 -15/10/08 - 22/10/09 (EUR)

Swaptions B&S Standard Deviation Surface,

Tenor 10Y, 13/02/06

2.50%-3.00%
3.0%
2.00%-2.50%
1.50%-2.00%
2.5%
1.00%-1.50%
0.50%-1.00%
2.0%
0.00%-0.50%

1.5%

1.0%
350%
0.5% 200%
100% Strike in %
0.0% 50% of Forward
0.1Y
0.5Y 2Y 4Y 6Y Y 10% Swap Rate
8 10Y 0Y
2 30Y
Option Maturity

Swaptions B&S Standard Deviation Surface, Swaptions B&S Standard Deviation Surface,
Tenor 10Y, 13/02/06 Tenor 10Y, 22/10/09

2.50%-3.00% 3.0%
3.0% 2.50%-3.00%
2.00%-2.50%
2.00%-2.50%
1.50%-2.00% 2.5%
2.5% 1.50%-2.00%
1.00%-1.50%
1.00%-1.50%
2.0% 0.50%-1.00% 2.0%
0.50%-1.00%
0.00%-0.50%
1.5% 0.00%-0.50%
1.5%
1.0%
1.0%
350%
350% 0.5% 200%
0.5% 200% Strike in %
Strike in % 100%
100% 0.0% of Forward
50%
0.0% 50% of Forward
0.1Y0.5Y 2Y 4Y 10% Swap Rate
0.1Y0.5Y 2Y 4Y 6Y 10% Swap Rate 6Y 8Y 0Y
8Y10Y 0Y 1 20Y30Y
2 30Y
Option Maturity
Option Maturity

94
 Standard Deviations Smiles 10Y/5Y, 10Y/1Y at three dates:
13/02/06 - 15/10/08 - 22/10/09 (EUR)

Smile Swaptions 10Y/1Y

2.50%

2.00%
Standard deviation

1.50% 13/02/06
15/10/08
1.00% 22/10/09

0.50%

0.00%
0% 50% 100% 150% 200% 250% 300% 350%
Strike in % of Forward Swap Rate

Smile Swaptions 10Y/5Y

2.50%

2.00%
Standard deviation

1.50% 13/02/06
15/10/08
1.00% 22/10/09

0.50%

0.00%
0% 50% 100% 150% 200% 250% 300% 350%
Strike in % of Forward Swap Rate

95
 Tenor 10Y swaptions standard deviation and volatility surfaces
13/02/06
0.1Y 0.3Y 0.5Y 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 15Y 20Y 25Y 30Y
10% 0.67% 0.67% 0.77% 0.66% 0.66% 0.65% 0.65% 0.64% 0.64% 0.64% 0.63% 0.62% 0.61% 0.56% 0.52% 0.48% 0.45%
20% 0.74% 0.81% 0.77% 0.66% 0.66% 0.66% 0.66% 0.66% 0.66% 0.66% 0.65% 0.64% 0.63% 0.58% 0.54% 0.50% 0.47%
25% 0.74% 0.80% 0.76% 0.66% 0.66% 0.66% 0.66% 0.66% 0.66% 0.66% 0.65% 0.64% 0.63% 0.59% 0.54% 0.50% 0.48%
30% 0.74% 0.78% 0.74% 0.65% 0.65% 0.66% 0.66% 0.66% 0.66% 0.66% 0.65% 0.64% 0.63% 0.58% 0.54% 0.51% 0.48%
40% 0.72% 0.74% 0.71% 0.63% 0.64% 0.64% 0.65% 0.65% 0.65% 0.65% 0.64% 0.63% 0.63% 0.58% 0.54% 0.50% 0.48%
50% 0.69% 0.70% 0.67% 0.62% 0.62% 0.63% 0.63% 0.64% 0.64% 0.64% 0.63% 0.62% 0.62% 0.57% 0.53% 0.50% 0.47%
60% 0.65% 0.65% 0.64% 0.60% 0.61% 0.61% 0.62% 0.62% 0.62% 0.62% 0.62% 0.61% 0.60% 0.56% 0.52% 0.49% 0.47%
70% 0.62% 0.61% 0.60% 0.58% 0.59% 0.60% 0.60% 0.61% 0.61% 0.61% 0.61% 0.60% 0.59% 0.54% 0.51% 0.48% 0.46%
75% 0.60% 0.59% 0.59% 0.58% 0.59% 0.59% 0.60% 0.61% 0.61% 0.61% 0.60% 0.59% 0.59% 0.54% 0.50% 0.48% 0.46%
90% 0.56% 0.55% 0.56% 0.57% 0.58% 0.59% 0.59% 0.60% 0.60% 0.60% 0.59% 0.59% 0.58% 0.53% 0.49% 0.47% 0.46%
100% 0.58% 0.56% 0.57% 0.58% 0.59% 0.59% 0.60% 0.60% 0.60% 0.60% 0.60% 0.59% 0.58% 0.53% 0.49% 0.47% 0.46%
120% 0.68% 0.67% 0.66% 0.63% 0.63% 0.63% 0.63% 0.63% 0.63% 0.63% 0.62% 0.61% 0.60% 0.55% 0.51% 0.49% 0.47%
140% 0.83% 0.81% 0.78% 0.70% 0.70% 0.69% 0.69% 0.68% 0.68% 0.68% 0.67% 0.66% 0.65% 0.59% 0.55% 0.52% 0.50%
160% 0.97% 0.96% 0.91% 0.79% 0.78% 0.77% 0.76% 0.74% 0.74% 0.74% 0.73% 0.72% 0.71% 0.65% 0.60% 0.57% 0.54%
Strikes in % of
forward swap

180% 1.12% 1.11% 1.04% 0.88% 0.86% 0.85% 0.83% 0.81% 0.81% 0.81% 0.80% 0.79% 0.78% 0.71% 0.66% 0.62% 0.58%
200% 1.26% 1.25% 1.16% 0.97% 0.95% 0.93% 0.91% 0.89% 0.89% 0.88% 0.87% 0.86% 0.84% 0.78% 0.72% 0.67% 0.63%
225% 1.44% 1.43% 1.32% 1.08% 1.06% 1.03% 1.01% 0.98% 0.98% 0.97% 0.96% 0.95% 0.93% 0.86% 0.79% 0.73% 0.69%
rates

250% 1.61% 1.60% 1.47% 1.18% 1.16% 1.13% 1.10% 1.07% 1.07% 1.06% 1.05% 1.03% 1.02% 0.94% 0.86% 0.80% 0.75%
275% 1.77% 1.76% 1.62% 1.29% 1.26% 1.23% 1.20% 1.16% 1.16% 1.15% 1.14% 1.12% 1.10% 1.02% 0.94% 0.87% 0.81%
300% 1.93% 1.92% 1.76% 1.39% 1.36% 1.33% 1.29% 1.25% 1.24% 1.24% 1.22% 1.21% 1.19% 1.10% 1.01% 0.93% 0.87%

0.1Y 0.3Y 0.5Y 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 15Y 20Y 25Y 30Y

10% 57.8% 57.1% 53.3% 44.6% 43.8% 43.0% 42.1% 41.2% 40.9% 40.5% 39.9% 39.2% 38.6% 36.4% 35.1% 34.3% 33.3%
20% 44.9% 44.3% 41.6% 35.2% 34.7% 34.0% 33.4% 32.8% 32.5% 32.2% 31.7% 31.2% 30.7% 28.8% 27.9% 27.3% 26.5%
25% 40.5% 40.0% 37.6% 32.1% 31.6% 31.1% 30.6% 30.0% 29.8% 29.5% 29.0% 28.6% 28.1% 26.4% 25.5% 25.0% 24.3%
30% 36.9% 36.4% 34.3% 29.6% 29.2% 28.7% 28.2% 27.8% 27.5% 27.3% 26.9% 26.4% 26.0% 24.4% 23.6% 23.1% 22.5%
40% 31.1% 30.7% 29.1% 25.6% 25.3% 24.9% 24.6% 24.2% 24.0% 23.8% 23.4% 23.0% 22.6% 21.2% 20.5% 20.2% 19.7%
50% 26.6% 26.2% 25.1% 22.6% 22.3% 22.0% 21.8% 21.5% 21.4% 21.2% 20.8% 20.5% 20.1% 18.8% 18.2% 18.0% 17.6%
60% 22.9% 22.5% 21.8% 20.2% 20.0% 19.8% 19.6% 19.4% 19.2% 19.1% 18.8% 18.4% 18.1% 16.9% 16.4% 16.2% 15.9%
70% 20.0% 19.5% 19.2% 18.3% 18.1% 18.0% 17.8% 17.7% 17.6% 17.4% 17.1% 16.8% 16.5% 15.3% 14.9% 14.8% 14.5%
75% 18.7% 18.2% 18.1% 17.5% 17.4% 17.2% 17.1% 17.0% 16.9% 16.7% 16.4% 16.1% 15.8% 14.7% 14.3% 14.2% 13.9%
90% 16.2% 15.7% 15.8% 15.9% 15.7% 15.6% 15.5% 15.4% 15.2% 15.1% 14.8% 14.5% 14.2% 13.2% 12.8% 12.7% 12.6%
100% 15.7% 15.2% 15.3% 15.3% 15.1% 14.9% 14.8% 14.7% 14.5% 14.4% 14.1% 13.8% 13.5% 12.5% 12.1% 12.1% 11.9%
120% 16.9% 16.5% 16.0% 15.1% 14.8% 14.5% 14.2% 14.0% 13.8% 13.7% 13.4% 13.1% 12.8% 11.8% 11.4% 11.3% 11.2%
140% 18.9% 18.5% 17.6% 15.6% 15.2% 14.7% 14.3% 13.9% 13.8% 13.6% 13.3% 13.0% 12.8% 11.8% 11.3% 11.2% 10.9%
160% 20.8% 20.4% 19.1% 16.3% 15.7% 15.2% 14.7% 14.2% 14.0% 13.8% 13.6% 13.3% 13.0% 12.0% 11.5% 11.3% 11.0%
180% 22.5% 22.1% 20.5% 17.0% 16.4% 15.7% 15.1% 14.6% 14.4% 14.2% 13.9% 13.6% 13.3% 12.3% 11.8% 11.5% 11.2%
200% 23.9% 23.5% 21.7% 17.7% 17.0% 16.3% 15.6% 15.0% 14.8% 14.6% 14.3% 14.0% 13.7% 12.7% 12.2% 11.8% 11.4%
225% 25.4% 25.1% 23.0% 18.4% 17.7% 16.9% 16.2% 15.4% 15.2% 15.0% 14.7% 14.4% 14.1% 13.1% 12.6% 12.2% 11.7%
250% 26.8% 26.4% 24.1% 19.1% 18.3% 17.5% 16.7% 15.9% 15.7% 15.5% 15.2% 14.8% 14.5% 13.6% 12.9% 12.5% 12.0%
275% 27.9% 27.6% 25.1% 19.7% 18.8% 18.0% 17.1% 16.3% 16.1% 15.9% 15.6% 15.2% 14.9% 13.9% 13.3% 12.8% 12.3%
300% 28.9% 28.6% 25.9% 20.2% 19.3% 18.4% 17.6% 16.7% 16.5% 16.2% 15.9% 15.6% 15.3% 14.3% 13.6% 13.1% 12.5%
forwards swaps 3.67% 3.70% 3.73% 3.79% 3.89% 3.98% 4.05% 4.13% 4.18% 4.22% 4.26% 4.29% 4.32% 4.28% 4.13% 3.98% 3.89%

96
 Tenor 10Y Swaptions standard deviation and volatility surfaces
15/10/08
0.1Y 0.3Y 0.5Y 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 15Y 20Y 25Y 30Y
10% 1.22% 1.39% 1.20% 1.04% 0.96% 0.92% 0.88% 0.86% 0.84% 0.82% 0.81% 0.79% 0.78% 0.63% 0.53% 0.48% 0.44%
20% 1.38% 1.40% 1.21% 1.06% 0.97% 0.93% 0.89% 0.87% 0.86% 0.84% 0.83% 0.81% 0.79% 0.65% 0.56% 0.51% 0.46%
25% 1.53% 1.40% 1.21% 1.05% 0.97% 0.92% 0.89% 0.87% 0.85% 0.84% 0.82% 0.81% 0.79% 0.66% 0.56% 0.52% 0.47%
30% 1.52% 1.39% 1.20% 1.05% 0.96% 0.92% 0.88% 0.86% 0.84% 0.83% 0.82% 0.80% 0.79% 0.65% 0.56% 0.52% 0.47%
40% 1.50% 1.36% 1.18% 1.03% 0.94% 0.89% 0.86% 0.83% 0.82% 0.80% 0.79% 0.78% 0.77% 0.64% 0.56% 0.51% 0.47%
50% 1.47% 1.32% 1.16% 1.01% 0.92% 0.87% 0.83% 0.80% 0.79% 0.78% 0.77% 0.75% 0.74% 0.62% 0.54% 0.50% 0.46%
60% 1.44% 1.29% 1.13% 0.99% 0.89% 0.84% 0.80% 0.77% 0.76% 0.74% 0.73% 0.72% 0.71% 0.60% 0.53% 0.49% 0.45%
70% 1.41% 1.25% 1.10% 0.96% 0.86% 0.81% 0.77% 0.74% 0.73% 0.71% 0.70% 0.69% 0.68% 0.58% 0.51% 0.47% 0.43%
75% 1.39% 1.23% 1.09% 0.95% 0.85% 0.79% 0.75% 0.73% 0.71% 0.70% 0.69% 0.67% 0.66% 0.57% 0.51% 0.47% 0.43%
90% 1.36% 1.19% 1.06% 0.93% 0.82% 0.76% 0.72% 0.69% 0.68% 0.66% 0.65% 0.64% 0.63% 0.54% 0.49% 0.45% 0.42%
100% 1.36% 1.18% 1.06% 0.93% 0.81% 0.75% 0.71% 0.68% 0.67% 0.65% 0.64% 0.63% 0.62% 0.54% 0.48% 0.45% 0.41%
120% 1.39% 1.21% 1.08% 0.95% 0.83% 0.78% 0.73% 0.71% 0.69% 0.68% 0.67% 0.66% 0.65% 0.56% 0.50% 0.46% 0.43%
140% 1.47% 1.30% 1.15% 1.01% 0.90% 0.85% 0.81% 0.78% 0.77% 0.76% 0.75% 0.74% 0.72% 0.62% 0.55% 0.51% 0.46%
Strikes in % of
forward swap

160% 1.59% 1.42% 1.25% 1.10% 1.00% 0.94% 0.91% 0.88% 0.87% 0.85% 0.84% 0.83% 0.82% 0.69% 0.61% 0.56% 0.51%
180% 1.73% 1.56% 1.36% 1.20% 1.10% 1.05% 1.01% 0.99% 0.97% 0.96% 0.95% 0.94% 0.92% 0.77% 0.68% 0.62% 0.56%
200% 1.87% 1.71% 1.48% 1.30% 1.21% 1.15% 1.12% 1.09% 1.08% 1.06% 1.05% 1.04% 1.02% 0.85% 0.74% 0.68% 0.62%
rates

225% 2.06% 1.89% 1.64% 1.43% 1.34% 1.29% 1.25% 1.23% 1.21% 1.20% 1.18% 1.17% 1.15% 0.96% 0.83% 0.76% 0.69%
250% 2.25% 2.08% 1.79% 1.57% 1.48% 1.42% 1.38% 1.36% 1.34% 1.33% 1.31% 1.30% 1.27% 1.06% 0.91% 0.84% 0.76%
275% 2.43% 2.26% 1.94% 1.70% 1.61% 1.55% 1.51% 1.49% 1.47% 1.45% 1.44% 1.42% 1.40% 1.16% 1.00% 0.91% 0.82%
300% 2.62% 2.44% 2.09% 1.83% 1.74% 1.68% 1.64% 1.61% 1.60% 1.58% 1.56% 1.54% 1.52% 1.25% 1.08% 0.99% 0.89%

0.1Y 0.3Y 0.5Y 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 15Y 20Y 25Y 30Y

10% 82.4% 75.9% 65.2% 55.8% 50.6% 47.9% 45.7% 44.0% 43.4% 42.9% 42.6% 42.4% 42.2% 42.4% 42.0% 39.7% 37.3%
20% 65.7% 60.2% 51.9% 44.4% 40.1% 37.8% 36.0% 34.6% 34.2% 33.7% 33.5% 33.3% 33.2% 33.4% 33.1% 31.3% 29.5%
25% 60.2% 55.1% 47.5% 40.7% 36.6% 34.5% 32.8% 31.5% 31.1% 30.7% 30.5% 30.3% 30.2% 30.4% 30.2% 28.6% 26.9%
30% 55.7% 50.9% 43.9% 37.6% 33.8% 31.8% 30.2% 29.0% 28.6% 28.2% 28.0% 27.8% 27.7% 28.0% 27.8% 26.4% 24.8%
40% 48.7% 44.2% 38.3% 32.8% 29.2% 27.4% 26.0% 24.9% 24.5% 24.2% 24.0% 23.9% 23.7% 24.1% 24.0% 22.8% 21.5%
50% 43.3% 39.0% 34.0% 29.2% 25.8% 24.1% 22.7% 21.7% 21.4% 21.1% 20.9% 20.8% 20.7% 21.1% 21.1% 20.0% 18.9%
60% 39.1% 34.9% 30.5% 26.2% 23.0% 21.4% 20.1% 19.2% 18.9% 18.6% 18.4% 18.3% 18.2% 18.6% 18.7% 17.8% 16.8%
70% 35.6% 31.6% 27.7% 23.8% 20.7% 19.1% 18.0% 17.1% 16.8% 16.5% 16.3% 16.2% 16.1% 16.6% 16.8% 16.0% 15.2%
75% 34.1% 30.1% 26.5% 22.8% 19.7% 18.2% 17.0% 16.2% 15.9% 15.6% 15.5% 15.3% 15.3% 15.8% 16.0% 15.2% 14.4%
90% 30.6% 26.7% 23.7% 20.4% 17.4% 16.0% 14.9% 14.1% 13.8% 13.5% 13.4% 13.3% 13.2% 13.8% 14.1% 13.4% 12.7%
100% 28.9% 25.1% 22.4% 19.3% 16.4% 15.0% 13.9% 13.2% 12.9% 12.6% 12.5% 12.4% 12.3% 12.9% 13.2% 12.6% 12.0%
120% 27.0% 23.5% 20.9% 18.0% 15.4% 14.1% 13.2% 12.5% 12.2% 12.0% 11.9% 11.8% 11.8% 12.2% 12.5% 11.9% 11.3%
140% 26.4% 23.3% 20.5% 17.7% 15.4% 14.2% 13.4% 12.8% 12.5% 12.3% 12.3% 12.2% 12.1% 12.5% 12.6% 11.9% 11.3%
160% 26.5% 23.7% 20.7% 17.9% 15.8% 14.7% 14.0% 13.4% 13.2% 13.0% 12.9% 12.9% 12.8% 13.0% 13.0% 12.3% 11.6%
180% 27.0% 24.4% 21.2% 18.3% 16.3% 15.4% 14.6% 14.1% 13.9% 13.7% 13.6% 13.6% 13.5% 13.7% 13.6% 12.8% 12.1%
200% 27.6% 25.2% 21.8% 18.7% 16.9% 16.0% 15.2% 14.7% 14.5% 14.4% 14.3% 14.2% 14.2% 14.3% 14.1% 13.3% 12.5%
225% 28.4% 26.1% 22.5% 19.3% 17.6% 16.7% 16.0% 15.4% 15.3% 15.1% 15.0% 15.0% 14.9% 15.0% 14.7% 13.9% 13.0%
250% 29.2% 27.0% 23.1% 19.9% 18.2% 17.3% 16.6% 16.1% 15.9% 15.8% 15.7% 15.6% 15.6% 15.6% 15.3% 14.4% 13.5%
275% 29.9% 27.8% 23.8% 20.4% 18.8% 17.9% 17.2% 16.7% 16.5% 16.4% 16.3% 16.2% 16.2% 16.1% 15.8% 14.9% 13.9%
300% 30.6% 28.5% 24.3% 20.9% 19.3% 18.5% 17.7% 17.2% 17.0% 16.9% 16.8% 16.8% 16.7% 16.6% 16.3% 15.3% 14.3%

forwards swaps 4.70% 4.71% 4.73% 4.82% 4.96% 5.03% 5.10% 5.18% 5.18% 5.17% 5.15% 5.10% 5.04% 4.21% 3.71% 3.62% 3.50%

97
 Tenor 10Y Swaptions standard deviation and volatility surfaces
22/10/09
0.1Y 0.3Y 0.5Y 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 15Y 20Y 25Y 30Y
10% 1.04% 1.24% 1.21% 1.17% 1.14% 1.09% 1.05% 1.01% 0.98% 0.95% 0.92% 0.90% 0.87% 0.73% 0.59% 0.49% 0.41%
20% 1.27% 1.24% 1.22% 1.18% 1.15% 1.11% 1.07% 1.04% 1.01% 0.97% 0.95% 0.92% 0.89% 0.77% 0.63% 0.53% 0.45%
25% 1.25% 1.23% 1.21% 1.18% 1.15% 1.11% 1.07% 1.03% 1.00% 0.97% 0.95% 0.92% 0.89% 0.77% 0.64% 0.54% 0.46%
30% 1.23% 1.21% 1.19% 1.16% 1.14% 1.10% 1.06% 1.02% 0.99% 0.96% 0.94% 0.91% 0.89% 0.77% 0.64% 0.55% 0.47%
40% 1.18% 1.17% 1.15% 1.13% 1.10% 1.06% 1.03% 0.99% 0.96% 0.93% 0.91% 0.89% 0.86% 0.75% 0.64% 0.55% 0.47%
50% 1.12% 1.12% 1.11% 1.09% 1.06% 1.02% 0.98% 0.95% 0.92% 0.89% 0.87% 0.85% 0.83% 0.73% 0.62% 0.55% 0.47%
60% 1.06% 1.06% 1.06% 1.04% 1.02% 0.97% 0.94% 0.90% 0.88% 0.85% 0.83% 0.81% 0.79% 0.70% 0.61% 0.54% 0.47%
70% 0.99% 1.00% 1.00% 0.99% 0.97% 0.93% 0.89% 0.85% 0.83% 0.80% 0.78% 0.77% 0.75% 0.67% 0.59% 0.53% 0.47%
75% 0.96% 0.97% 0.98% 0.97% 0.95% 0.90% 0.87% 0.83% 0.81% 0.78% 0.76% 0.75% 0.73% 0.65% 0.58% 0.53% 0.47%
90% 0.87% 0.89% 0.91% 0.92% 0.89% 0.84% 0.81% 0.77% 0.75% 0.72% 0.71% 0.69% 0.68% 0.61% 0.56% 0.52% 0.46%
100% 0.83% 0.86% 0.89% 0.90% 0.87% 0.82% 0.78% 0.75% 0.73% 0.70% 0.69% 0.67% 0.66% 0.60% 0.56% 0.52% 0.46%
120% 0.84% 0.88% 0.90% 0.92% 0.88% 0.84% 0.81% 0.79% 0.76% 0.73% 0.72% 0.70% 0.69% 0.62% 0.57% 0.53% 0.48%
140% 0.95% 0.98% 1.00% 1.01% 0.98% 0.94% 0.92% 0.89% 0.87% 0.83% 0.82% 0.80% 0.78% 0.70% 0.63% 0.57% 0.51%
Strikes in % of
forward swap

160% 1.10% 1.11% 1.13% 1.13% 1.10% 1.07% 1.05% 1.02% 1.00% 0.96% 0.94% 0.92% 0.89% 0.79% 0.69% 0.62% 0.54%
180% 1.25% 1.26% 1.27% 1.27% 1.23% 1.21% 1.19% 1.16% 1.13% 1.09% 1.07% 1.04% 1.01% 0.89% 0.77% 0.67% 0.58%
200% 1.41% 1.41% 1.41% 1.41% 1.37% 1.35% 1.33% 1.30% 1.27% 1.23% 1.20% 1.17% 1.13% 0.99% 0.84% 0.73% 0.63%
rates

225% 1.60% 1.60% 1.59% 1.58% 1.54% 1.52% 1.50% 1.47% 1.43% 1.39% 1.36% 1.32% 1.28% 1.12% 0.94% 0.81% 0.69%
250% 1.79% 1.78% 1.77% 1.75% 1.71% 1.69% 1.67% 1.64% 1.60% 1.55% 1.51% 1.47% 1.43% 1.24% 1.03% 0.88% 0.75%
275% 1.98% 1.96% 1.94% 1.92% 1.88% 1.86% 1.83% 1.80% 1.76% 1.70% 1.66% 1.62% 1.57% 1.36% 1.13% 0.95% 0.80%
300% 2.16% 2.13% 2.11% 2.09% 2.04% 2.02% 1.99% 1.96% 1.91% 1.85% 1.81% 1.76% 1.71% 1.48% 1.22% 1.03% 0.86%

0.1Y 0.3Y 0.5Y 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 15Y 20Y 25Y 30Y

10% 92.8% 89.0% 85.0% 78.9% 72.5% 67.5% 63.2% 59.3% 57.1% 54.8% 53.2% 51.6% 49.8% 48.4% 49.0% 48.8% 46.2%
20% 72.6% 69.8% 66.8% 62.1% 57.1% 53.0% 49.6% 46.5% 44.7% 42.9% 41.7% 40.4% 39.0% 38.0% 38.7% 38.7% 36.8%
25% 65.8% 63.4% 60.8% 56.6% 52.0% 48.2% 45.0% 42.2% 40.6% 39.0% 37.8% 36.7% 35.4% 34.5% 35.3% 35.4% 33.7%
30% 60.2% 58.1% 55.8% 52.0% 47.7% 44.2% 41.3% 38.6% 37.2% 35.7% 34.7% 33.6% 32.5% 31.7% 32.5% 32.8% 31.3%
40% 51.2% 49.7% 47.8% 44.7% 41.0% 37.9% 35.3% 33.0% 31.7% 30.4% 29.5% 28.6% 27.7% 27.1% 28.1% 28.6% 27.4%
50% 44.2% 43.0% 41.6% 39.0% 35.7% 32.9% 30.6% 28.5% 27.4% 26.3% 25.5% 24.8% 24.0% 23.6% 24.7% 25.3% 24.5%
60% 38.3% 37.6% 36.5% 34.3% 31.4% 28.8% 26.7% 24.9% 23.9% 22.9% 22.3% 21.6% 20.9% 20.7% 21.9% 22.8% 22.2%
70% 33.4% 33.1% 32.3% 30.5% 27.8% 25.4% 23.5% 21.9% 21.0% 20.0% 19.5% 19.0% 18.4% 18.3% 19.7% 20.7% 20.3%
75% 31.3% 31.1% 30.5% 28.8% 26.3% 24.0% 22.2% 20.6% 19.7% 18.8% 18.3% 17.8% 17.3% 17.3% 18.7% 19.8% 19.5%
90% 25.9% 26.2% 26.0% 24.9% 22.5% 20.4% 18.8% 17.5% 16.7% 15.9% 15.5% 15.1% 14.7% 14.8% 16.4% 17.7% 17.6%
100% 23.4% 24.0% 24.0% 23.1% 20.8% 18.9% 17.4% 16.2% 15.4% 14.6% 14.3% 13.9% 13.6% 13.8% 15.4% 16.7% 16.7%
120% 21.6% 22.2% 22.3% 21.5% 19.4% 17.7% 16.4% 15.4% 14.7% 14.0% 13.6% 13.3% 12.9% 13.0% 14.5% 15.6% 15.6%
140% 22.6% 22.9% 22.7% 21.9% 19.7% 18.2% 17.1% 16.1% 15.4% 14.7% 14.3% 13.9% 13.5% 13.5% 14.5% 15.4% 15.2%
160% 24.3% 24.3% 23.9% 22.9% 20.7% 19.3% 18.2% 17.2% 16.5% 15.8% 15.4% 14.9% 14.4% 14.2% 15.0% 15.6% 15.2%
180% 26.1% 25.8% 25.2% 24.0% 21.8% 20.5% 19.3% 18.3% 17.6% 16.9% 16.4% 15.9% 15.4% 15.1% 15.6% 15.9% 15.4%
200% 27.7% 27.2% 26.5% 25.2% 22.9% 21.5% 20.4% 19.4% 18.6% 17.9% 17.4% 16.8% 16.2% 15.8% 16.2% 16.4% 15.6%
225% 29.5% 28.8% 27.9% 26.5% 24.1% 22.7% 21.6% 20.5% 19.8% 19.0% 18.4% 17.8% 17.2% 16.7% 17.0% 16.9% 16.0%
250% 31.0% 30.2% 29.2% 27.6% 25.2% 23.8% 22.6% 21.5% 20.7% 19.9% 19.3% 18.7% 18.0% 17.5% 17.6% 17.4% 16.4%
275% 32.4% 31.5% 30.4% 28.7% 26.1% 24.8% 23.6% 22.4% 21.6% 20.8% 20.1% 19.5% 18.8% 18.2% 18.2% 17.9% 16.7%
300% 33.6% 32.6% 31.4% 29.6% 27.0% 25.6% 24.4% 23.2% 22.4% 21.5% 20.9% 20.2% 19.5% 18.8% 18.7% 18.3% 17.1%

forwards swaps 3.53% 3.60% 3.70% 3.89% 4.17% 4.36% 4.53% 4.68% 4.75% 4.80% 4.83% 4.87% 4.90% 4.42% 3.68% 3.19% 2.87%

98
 Volatility cubes

➢ Volatility surfaces practice:

✓ These surfaces being defined for all reference rates (Euribor1M, 3M,
…12M) and all swap (1Y,…10Y, 30Y), any vanilla cap or Swaptions can be
priced
✓ The set of volatility surfaces is called a volatility cube
✓ In practice, more complex models (SABR) are used to get on previous
surfaces and used to get the volatilities for any Caplet/floorlet/swaption
✓ The purposes of these models are:
o Greeks calculations
o To avoid numerical problem due to non proper interpolation methods

99
 Volatility cubes

➢ Volatility surface for a given reference rate, the Euribor3M:

✓ To be able to price a Caplet on Euribor3M for any strike and any
maturity
✓ A volatility surface description for EURIBOR3M:

100
 Volatility surface for Euribor3M, as of 31/1/06

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 of
Caplets/Floorlets 101
 Volatility cubes: Tenor definition

➢ Tenor definition:
✓ 3M in 6M and 3M in 5Y FRA are two different underlying
✓ 10Y/10Y forward swap rate and 10Y/1Y forward swap rate are two
different underlying
✓ 1M, 2M, 3M …12M deposit rates and 1Y, 2Y, …, 50Y swap rates called
tenor to avoid misunderstandings

➢ Underlying = a combination of tenor and maturity:

✓ 10Y/5Y forward swap rate is the forward swap rate for a 10Y starting in
5Y

102
 Volatility surface and standard deviation surface

➢ Relationship between volatilities and standard deviation for vanilla

options:
✓ Same price for vanilla options (caps or Swaptions, depending on the
tenor), for:
o lognormal model (volatilities) and
o Gaussian model (standard deviations)

➢ Volatility surfaces examples:

✓ Volatilities, the standard deviations as of 1/12/06, EUR for 3 tenors:
o Deposit 3M
o Swap 5Y
o Swap 10Y
✓ forward values for the 3 tenors at each option maturity (vertical axis of
volatility surface)

103
 3M Volatility surface, standard deviation surface
3M 31 90 182 365 731 1096 1461 1826 2192 2557 2922 3287 3653 5479 7305 9131 10958
0.50% 12.83% 16.22% 20.02% 32.67% 35.20% 36.28% 35.99% 35.67% 35.58% 35.44% 35.26% 35.03% 34.77% 32.73% 31.03% 31.07% 31.64%
1.00% 9.55% 12.28% 15.25% 24.93% 26.95% 27.99% 27.88% 27.69% 27.71% 27.68% 27.58% 27.45% 27.29% 25.76% 24.31% 24.26% 24.62%
2.00% 6.23% 8.31% 10.56% 17.28% 18.76% 19.87% 19.95% 19.89% 19.99% 20.05% 20.03% 19.99% 19.94% 18.85% 17.62% 17.52% 17.68%
2.50% 5.30% 7.15% 9.27% 15.13% 16.42% 17.57% 17.69% 17.65% 17.73% 17.80% 17.79% 17.77% 17.74% 16.74% 15.57% 15.46% 15.58%
3.00% 4.80% 6.41% 8.49% 13.79% 14.89% 16.05% 16.16% 16.10% 16.15% 16.18% 16.17% 16.15% 16.11% 15.15% 14.01% 13.91% 14.00%
3.50% 4.74% 6.09% 8.18% 13.20% 14.13% 15.22% 15.28% 15.17% 15.14% 15.10% 15.07% 15.02% 14.95% 13.96% 12.85% 12.77% 12.86%
4.00% 4.94% 6.12% 8.21% 13.18% 14.01% 14.97% 14.94% 14.75% 14.62% 14.48% 14.42% 14.32% 14.21% 13.14% 12.05% 12.00% 12.11%
4.50% 5.23% 6.34% 8.43% 13.50% 14.28% 15.10% 14.97% 14.71% 14.48% 14.24% 14.14% 13.98% 13.81% 12.64% 11.57% 11.57% 11.71%
5.00% 5.54% 6.63% 8.72% 13.96% 14.74% 15.44% 15.22% 14.91% 14.59% 14.26% 14.12% 13.90% 13.68% 12.39% 11.36% 11.39% 11.57%
6.00% 6.09% 7.23% 9.33% 14.97% 15.80% 16.31% 15.97% 15.58% 15.16% 14.72% 14.51% 14.20% 13.90% 12.42% 11.44% 11.52% 11.75%
7.00% 6.55% 7.76% 9.89% 15.89% 16.79% 17.19% 16.77% 16.33% 15.86% 15.35% 15.09% 14.74% 14.37% 12.77% 11.83% 11.93% 12.18%
8.00% 6.94% 8.21% 10.37% 16.69% 17.66% 17.98% 17.51% 17.03% 16.53% 15.98% 15.69% 15.30% 14.90% 13.22% 12.30% 12.39% 12.66%
9.00% 7.27% 8.59% 10.79% 17.38% 18.41% 18.67% 18.16% 17.66% 17.13% 16.56% 16.25% 15.83% 15.41% 13.66% 12.75% 12.84% 13.12%
10.00% 7.55% 8.93% 11.15% 17.98% 19.07% 19.28% 18.74% 18.22% 17.68% 17.09% 16.75% 16.32% 15.88% 14.08% 13.18% 13.25% 13.54%
11.00% 7.79% 9.22% 11.47% 18.51% 19.65% 19.82% 19.25% 18.72% 18.16% 17.56% 17.21% 16.76% 16.31% 14.47% 13.56% 13.63% 13.92%
12.00% 8.01% 9.47% 11.75% 18.98% 20.15% 20.30% 19.71% 19.17% 18.60% 17.98% 17.62% 17.16% 16.69% 14.82% 13.92% 13.97% 14.27%
13.00% 8.20% 9.70% 12.00% 19.39% 20.61% 20.73% 20.12% 19.56% 18.99% 18.36% 17.99% 17.52% 17.04% 15.15% 14.23% 14.28% 14.58%
14.00% 8.37% 9.90% 12.22% 19.77% 21.02% 21.12% 20.48% 19.92% 19.34% 18.71% 18.33% 17.85% 17.36% 15.44% 14.52% 14.57% 14.87%

3M 31 90 182 365 731 1096 1461 1826 2192 2557 2922 3287 3653 5479 7305 9131 10958
0.50% 0.17% 0.22% 0.27% 0.52% 0.57% 0.59% 0.59% 0.60% 0.60% 0.61% 0.61% 0.61% 0.61% 0.58% 0.53% 0.51% 0.50%
1.00% 0.17% 0.24% 0.29% 0.51% 0.56% 0.59% 0.60% 0.60% 0.61% 0.62% 0.62% 0.62% 0.62% 0.59% 0.55% 0.53% 0.52%
2.00% 0.16% 0.23% 0.28% 0.48% 0.53% 0.56% 0.57% 0.58% 0.59% 0.60% 0.60% 0.61% 0.61% 0.58% 0.54% 0.52% 0.51%
2.50% 0.15% 0.22% 0.28% 0.46% 0.51% 0.55% 0.56% 0.57% 0.58% 0.59% 0.59% 0.60% 0.60% 0.57% 0.53% 0.51% 0.50%
3.00% 0.15% 0.21% 0.28% 0.46% 0.50% 0.55% 0.56% 0.56% 0.57% 0.58% 0.59% 0.59% 0.59% 0.57% 0.52% 0.50% 0.49%
3.50% 0.16% 0.22% 0.29% 0.47% 0.52% 0.56% 0.57% 0.57% 0.58% 0.59% 0.59% 0.59% 0.59% 0.56% 0.51% 0.50% 0.49%
4.00% 0.18% 0.23% 0.31% 0.51% 0.55% 0.59% 0.60% 0.60% 0.60% 0.60% 0.60% 0.60% 0.60% 0.57% 0.51% 0.50% 0.49%
4.50% 0.20% 0.26% 0.34% 0.55% 0.59% 0.63% 0.63% 0.63% 0.63% 0.63% 0.62% 0.62% 0.62% 0.58% 0.52% 0.51% 0.51%
5.00% 0.23% 0.29% 0.37% 0.60% 0.65% 0.68% 0.68% 0.67% 0.67% 0.66% 0.66% 0.65% 0.65% 0.60% 0.54% 0.53% 0.53%
6.00% 0.28% 0.34% 0.44% 0.71% 0.76% 0.79% 0.79% 0.77% 0.76% 0.75% 0.74% 0.73% 0.72% 0.66% 0.60% 0.59% 0.59%
7.00% 0.34% 0.40% 0.50% 0.82% 0.88% 0.91% 0.90% 0.88% 0.87% 0.85% 0.84% 0.82% 0.81% 0.73% 0.67% 0.66% 0.66%
8.00% 0.43% 0.46% 0.57% 0.93% 1.00% 1.02% 1.01% 0.99% 0.97% 0.95% 0.94% 0.92% 0.90% 0.81% 0.75% 0.74% 0.74%
9.00% 0.53% 0.51% 0.63% 1.03% 1.11% 1.14% 1.12% 1.09% 1.07% 1.05% 1.04% 1.02% 1.00% 0.90% 0.83% 0.81% 0.82%
10.00% 0.62% 0.56% 0.70% 1.14% 1.22% 1.25% 1.22% 1.20% 1.18% 1.15% 1.13% 1.11% 1.09% 0.98% 0.90% 0.89% 0.89%
11.00% 0.71% 0.62% 0.76% 1.24% 1.33% 1.35% 1.33% 1.30% 1.28% 1.25% 1.23% 1.20% 1.18% 1.06% 0.98% 0.97% 0.97%
12.00% 0.81% 0.67% 0.82% 1.33% 1.44% 1.46% 1.43% 1.40% 1.37% 1.34% 1.32% 1.30% 1.27% 1.14% 1.06% 1.04% 1.04%
13.00% 0.90% 0.72% 0.88% 1.43% 1.54% 1.56% 1.53% 1.50% 1.47% 1.44% 1.41% 1.38% 1.36% 1.22% 1.13% 1.11% 1.12%
14.00% 0.99% 0.76% 0.93% 1.52% 1.64% 1.66% 1.63% 1.59% 1.56% 1.53% 1.50% 1.47% 1.44% 1.30% 1.21% 1.19% 1.19%

104
 5Y Volatility surface, standard deviation surface
5Y 31 90 182 365 731 1096 1461 1826 2192 2557 2922 3287 3653 5479 7305 9131 10958
0.50% 45.98% 48.06% 45.86% 40.03% 39.81% 39.12% 37.56% 37.05% 36.99% 36.91% 36.70% 36.46% 36.20% 34.95% 33.75% 32.59% 31.74%
1.00% 34.26% 35.97% 34.54% 30.64% 30.66% 30.30% 29.29% 29.01% 29.00% 28.97% 28.82% 28.66% 28.46% 27.48% 26.48% 25.50% 24.82%
2.00% 21.75% 23.11% 22.62% 20.95% 21.27% 21.28% 20.90% 20.87% 20.89% 20.90% 20.80% 20.69% 20.55% 19.84% 19.06% 18.31% 17.85%
2.50% 17.62% 18.90% 18.76% 17.91% 18.32% 18.46% 18.28% 18.33% 18.35% 18.35% 18.26% 18.15% 18.03% 17.40% 16.70% 16.04% 15.67%
3.00% 14.36% 15.62% 15.78% 15.61% 16.08% 16.29% 16.27% 16.36% 16.36% 16.36% 16.26% 16.15% 16.03% 15.46% 14.83% 14.27% 13.98%
3.50% 12.17% 13.42% 13.79% 14.03% 14.48% 14.70% 14.76% 14.87% 14.84% 14.81% 14.69% 14.57% 14.44% 13.91% 13.36% 12.88% 12.68%
4.00% 11.69% 12.85% 13.10% 13.21% 13.54% 13.68% 13.74% 13.80% 13.72% 13.65% 13.51% 13.37% 13.22% 12.70% 12.23% 11.85% 11.73%
4.50% 12.70% 13.71% 13.57% 13.07% 13.21% 13.19% 13.15% 13.13% 12.99% 12.88% 12.70% 12.52% 12.35% 11.83% 11.44% 11.15% 11.10%
5.00% 14.12% 15.05% 14.56% 13.39% 13.33% 13.14% 12.93% 12.81% 12.61% 12.44% 12.22% 12.01% 11.81% 11.29% 10.97% 10.77% 10.76%
6.00% 16.80% 17.68% 16.71% 14.49% 14.15% 13.69% 13.17% 12.84% 12.57% 12.32% 12.04% 11.77% 11.52% 10.98% 10.76% 10.65% 10.66%
7.00% 19.00% 19.87% 18.58% 15.64% 15.14% 14.51% 13.75% 13.29% 12.97% 12.69% 12.39% 12.09% 11.82% 11.26% 11.07% 10.97% 10.95%
8.00% 20.81% 21.69% 20.16% 16.67% 16.07% 15.32% 14.39% 13.83% 13.51% 13.21% 12.90% 12.59% 12.31% 11.73% 11.53% 11.42% 11.35%
9.00% 22.32% 23.22% 21.49% 17.57% 16.89% 16.06% 15.00% 14.38% 14.04% 13.74% 13.42% 13.11% 12.83% 12.23% 12.01% 11.87% 11.76%
10.00% 23.61% 24.52% 22.63% 18.35% 17.62% 16.71% 15.55% 14.88% 14.55% 14.24% 13.92% 13.60% 13.32% 12.71% 12.47% 12.30% 12.15%
11.00% 24.73% 25.65% 23.63% 19.05% 18.26% 17.30% 16.05% 15.34% 15.01% 14.70% 14.38% 14.06% 13.77% 13.16% 12.90% 12.70% 12.51%
12.00% 25.72% 26.65% 24.50% 19.66% 18.83% 17.83% 16.51% 15.76% 15.43% 15.12% 14.80% 14.48% 14.19% 13.56% 13.28% 13.06% 12.84%
13.00% 26.59% 27.53% 25.28% 20.20% 19.34% 18.30% 16.92% 16.14% 15.81% 15.50% 15.18% 14.86% 14.57% 13.93% 13.63% 13.39% 13.14%
14.00% 27.37% 28.32% 25.97% 20.70% 19.80% 18.72% 17.29% 16.49% 16.15% 15.85% 15.53% 15.21% 14.92% 14.27% 13.95% 13.69% 13.42%

5Y 31 90 182 365 731 1096 1461 1826 2192 2557 2922 3287 3653 5479 7305 9131 10958
0.50% 0.65% 0.68% 0.77% 0.68% 0.68% 0.68% 0.66% 0.65% 0.66% 0.66% 0.66% 0.66% 0.65% 0.62% 0.58% 0.54% 0.51%
1.00% 0.68% 0.72% 0.75% 0.67% 0.68% 0.68% 0.66% 0.66% 0.67% 0.67% 0.67% 0.67% 0.67% 0.64% 0.60% 0.56% 0.53%
2.00% 0.61% 0.66% 0.65% 0.61% 0.63% 0.63% 0.63% 0.64% 0.64% 0.65% 0.65% 0.65% 0.65% 0.62% 0.58% 0.55% 0.52%
2.50% 0.55% 0.60% 0.60% 0.58% 0.60% 0.61% 0.61% 0.62% 0.62% 0.63% 0.63% 0.63% 0.63% 0.60% 0.57% 0.53% 0.51%
3.00% 0.50% 0.54% 0.55% 0.55% 0.57% 0.59% 0.59% 0.60% 0.61% 0.61% 0.61% 0.61% 0.61% 0.59% 0.55% 0.52% 0.50%
3.50% 0.45% 0.50% 0.52% 0.53% 0.55% 0.57% 0.58% 0.59% 0.59% 0.59% 0.59% 0.59% 0.59% 0.57% 0.54% 0.51% 0.49%
4.00% 0.46% 0.51% 0.52% 0.53% 0.55% 0.56% 0.57% 0.58% 0.58% 0.59% 0.58% 0.58% 0.58% 0.55% 0.52% 0.50% 0.48%
4.50% 0.54% 0.58% 0.58% 0.56% 0.57% 0.58% 0.58% 0.59% 0.59% 0.59% 0.58% 0.58% 0.57% 0.55% 0.52% 0.50% 0.49%
5.00% 0.63% 0.67% 0.65% 0.60% 0.61% 0.61% 0.60% 0.60% 0.60% 0.60% 0.59% 0.58% 0.57% 0.55% 0.53% 0.51% 0.50%
6.00% 0.82% 0.87% 0.82% 0.72% 0.71% 0.70% 0.68% 0.67% 0.66% 0.65% 0.64% 0.63% 0.62% 0.59% 0.57% 0.55% 0.54%
7.00% 1.01% 1.06% 1.00% 0.84% 0.83% 0.80% 0.77% 0.75% 0.74% 0.72% 0.71% 0.70% 0.68% 0.65% 0.63% 0.61% 0.60%
8.00% 1.19% 1.25% 1.16% 0.97% 0.94% 0.91% 0.86% 0.84% 0.82% 0.81% 0.80% 0.78% 0.77% 0.73% 0.71% 0.69% 0.67%
9.00% 1.37% 1.43% 1.32% 1.09% 1.06% 1.02% 0.96% 0.93% 0.91% 0.90% 0.88% 0.87% 0.85% 0.81% 0.79% 0.76% 0.74%
10.00% 1.54% 1.60% 1.48% 1.21% 1.17% 1.12% 1.05% 1.02% 1.00% 0.99% 0.97% 0.95% 0.94% 0.89% 0.86% 0.84% 0.81%
11.00% 1.70% 1.77% 1.63% 1.33% 1.28% 1.23% 1.15% 1.11% 1.09% 1.08% 1.06% 1.04% 1.02% 0.98% 0.94% 0.91% 0.88%
12.00% 1.86% 1.93% 1.78% 1.44% 1.39% 1.33% 1.24% 1.20% 1.18% 1.16% 1.14% 1.12% 1.10% 1.06% 1.02% 0.98% 0.95%
13.00% 2.02% 2.10% 1.93% 1.55% 1.50% 1.43% 1.33% 1.28% 1.27% 1.25% 1.23% 1.21% 1.19% 1.14% 1.09% 1.06% 1.02%
14.00% 2.17% 2.25% 2.07% 1.66% 1.60% 1.53% 1.42% 1.37% 1.35% 1.33% 1.31% 1.29% 1.27% 1.21% 1.17% 1.13% 1.09%

105
 10Y Volatility surface, standard deviation surface
10Y 31 90 182 365 731 1096 1461 1826 2192 2557 2922 3287 3653 5479 7305 9131 10958
0.50% 48.19% 50.74% 47.54% 39.63% 39.13% 38.55% 37.79% 37.04% 36.95% 36.85% 36.72% 36.58% 36.44% 34.91% 33.15% 31.43% 29.72%
1.00% 36.21% 38.28% 36.11% 30.66% 30.39% 30.05% 29.56% 29.08% 29.03% 28.97% 28.88% 28.78% 28.67% 27.47% 26.03% 24.67% 23.34%
2.00% 23.37% 24.99% 24.02% 21.42% 21.39% 21.32% 21.13% 20.94% 20.93% 20.90% 20.85% 20.78% 20.70% 19.85% 18.79% 17.81% 16.91%
2.50% 19.09% 20.60% 20.06% 18.49% 18.53% 18.54% 18.44% 18.35% 18.35% 18.33% 18.28% 18.23% 18.16% 17.41% 16.48% 15.65% 14.90%
3.00% 15.61% 17.07% 16.91% 16.21% 16.30% 16.36% 16.33% 16.32% 16.31% 16.29% 16.24% 16.19% 16.13% 15.48% 14.66% 13.95% 13.32%
3.50% 12.93% 14.40% 14.55% 14.49% 14.59% 14.67% 14.68% 14.71% 14.69% 14.67% 14.62% 14.57% 14.51% 13.93% 13.22% 12.62% 12.10%
4.00% 11.51% 12.99% 13.19% 13.34% 13.39% 13.44% 13.44% 13.48% 13.45% 13.41% 13.36% 13.30% 13.24% 12.72% 12.11% 11.60% 11.17%
4.50% 11.83% 13.17% 13.02% 12.76% 12.70% 12.68% 12.62% 12.63% 12.57% 12.51% 12.45% 12.38% 12.32% 11.84% 11.31% 10.89% 10.52%
5.00% 13.07% 14.28% 13.67% 12.64% 12.46% 12.33% 12.18% 12.12% 12.03% 11.95% 11.87% 11.80% 11.73% 11.27% 10.81% 10.45% 10.11%
6.00% 15.78% 16.90% 15.62% 13.18% 12.80% 12.46% 12.13% 11.89% 11.75% 11.63% 11.53% 11.44% 11.36% 10.89% 10.51% 10.19% 9.83%
7.00% 18.08% 19.18% 17.47% 14.03% 13.52% 13.06% 12.59% 12.20% 12.04% 11.91% 11.80% 11.70% 11.61% 11.11% 10.72% 10.37% 9.96%
8.00% 19.98% 21.08% 19.05% 14.86% 14.28% 13.73% 13.17% 12.69% 12.52% 12.38% 12.27% 12.16% 12.08% 11.53% 11.11% 10.71% 10.23%
9.00% 21.57% 22.69% 20.39% 15.63% 14.99% 14.38% 13.76% 13.20% 13.03% 12.89% 12.77% 12.67% 12.59% 11.99% 11.53% 11.08% 10.54%
10.00% 22.93% 24.06% 21.55% 16.31% 15.63% 14.97% 14.30% 13.68% 13.52% 13.37% 13.26% 13.16% 13.07% 12.44% 11.94% 11.44% 10.84%
11.00% 24.10% 25.25% 22.56% 16.91% 16.20% 15.51% 14.79% 14.13% 13.96% 13.82% 13.71% 13.61% 13.53% 12.86% 12.32% 11.77% 11.13%
12.00% 25.14% 26.29% 23.45% 17.45% 16.72% 15.99% 15.24% 14.54% 14.38% 14.24% 14.13% 14.02% 13.94% 13.25% 12.67% 12.08% 11.40%
13.00% 26.06% 27.22% 24.24% 17.94% 17.18% 16.42% 15.65% 14.91% 14.75% 14.61% 14.50% 14.40% 14.32% 13.60% 12.99% 12.37% 11.65%
14.00% 26.88% 28.05% 24.94% 18.38% 17.60% 16.82% 16.01% 15.25% 15.09% 14.96% 14.85% 14.75% 14.66% 13.92% 13.29% 12.63% 11.88%

10Y 31 90 182 365 731 1096 1461 1826 2192 2557 2922 3287 3653 5479 7305 9131 10958
0.50% 0.70% 0.74% 0.83% 0.69% 0.69% 0.69% 0.68% 0.67% 0.67% 0.67% 0.67% 0.66% 0.66% 0.62% 0.56% 0.52% 0.48%
1.00% 0.74% 0.86% 0.81% 0.69% 0.69% 0.69% 0.68% 0.68% 0.68% 0.68% 0.68% 0.68% 0.67% 0.63% 0.58% 0.54% 0.50%
2.00% 0.68% 0.74% 0.71% 0.64% 0.65% 0.65% 0.65% 0.65% 0.65% 0.65% 0.65% 0.65% 0.65% 0.62% 0.57% 0.53% 0.49%
2.50% 0.62% 0.68% 0.66% 0.61% 0.62% 0.63% 0.63% 0.63% 0.63% 0.63% 0.63% 0.63% 0.63% 0.60% 0.56% 0.52% 0.48%
3.00% 0.56% 0.61% 0.61% 0.58% 0.59% 0.60% 0.60% 0.61% 0.61% 0.61% 0.61% 0.61% 0.61% 0.58% 0.54% 0.50% 0.47%
3.50% 0.50% 0.55% 0.56% 0.56% 0.57% 0.58% 0.58% 0.59% 0.59% 0.59% 0.59% 0.59% 0.59% 0.56% 0.53% 0.49% 0.47%
4.00% 0.47% 0.53% 0.54% 0.55% 0.56% 0.57% 0.57% 0.58% 0.58% 0.58% 0.58% 0.58% 0.58% 0.55% 0.51% 0.48% 0.46%
4.50% 0.51% 0.57% 0.57% 0.56% 0.56% 0.57% 0.57% 0.57% 0.57% 0.57% 0.57% 0.57% 0.57% 0.54% 0.51% 0.48% 0.46%
5.00% 0.60% 0.66% 0.63% 0.59% 0.58% 0.58% 0.58% 0.58% 0.58% 0.58% 0.58% 0.57% 0.57% 0.55% 0.51% 0.49% 0.47%
6.00% 0.79% 0.85% 0.79% 0.67% 0.66% 0.65% 0.63% 0.63% 0.62% 0.62% 0.61% 0.61% 0.61% 0.58% 0.55% 0.52% 0.50%
7.00% 0.99% 1.05% 0.96% 0.78% 0.75% 0.73% 0.71% 0.70% 0.69% 0.69% 0.68% 0.68% 0.67% 0.64% 0.61% 0.58% 0.55%
8.00% 1.18% 1.24% 1.13% 0.88% 0.86% 0.83% 0.80% 0.78% 0.77% 0.77% 0.76% 0.76% 0.75% 0.71% 0.68% 0.64% 0.61%
9.00% 1.36% 1.43% 1.29% 0.99% 0.96% 0.93% 0.89% 0.86% 0.86% 0.85% 0.85% 0.84% 0.84% 0.79% 0.75% 0.71% 0.67%
10.00% 1.53% 1.61% 1.44% 1.10% 1.06% 1.02% 0.99% 0.95% 0.94% 0.94% 0.93% 0.93% 0.92% 0.87% 0.82% 0.77% 0.73%
11.00% 1.70% 1.78% 1.60% 1.20% 1.16% 1.12% 1.08% 1.03% 1.03% 1.02% 1.02% 1.01% 1.00% 0.95% 0.89% 0.84% 0.79%
12.00% 1.86% 1.95% 1.75% 1.31% 1.26% 1.21% 1.17% 1.12% 1.11% 1.10% 1.10% 1.09% 1.09% 1.03% 0.97% 0.91% 0.85%
13.00% 2.03% 2.12% 1.89% 1.41% 1.36% 1.31% 1.25% 1.20% 1.19% 1.19% 1.18% 1.18% 1.17% 1.10% 1.04% 0.97% 0.91%
14.00% 2.18% 2.28% 2.03% 1.51% 1.45% 1.40% 1.34% 1.28% 1.28% 1.27% 1.26% 1.26% 1.25% 1.18% 1.11% 1.04% 0.96%

106
 Volatility cubes

➢ Volatility surfaces practice:

✓ If these surfaces defined for all reference rates (Euribor1M, 3M, …12M)
and all swap (1Y,…10Y, 30Y), any vanilla cap or Swaptions can be
priced
✓ set of volatility surfaces called a volatility cube
✓ In practice, more complex models (SABR) are used to get on previous
surfaces and used to get the volatilities for any Caplet/floorlet/swaption
o Greeks calculations
o to avoid numerical problem due to non proper interpolation methods

107
 CMS

➢ CMS definition
✓ CMS Rate n years = rate at which one would exchange at time T’ (T '  T ) the
Swap Rate of maturity n years, this Swap Rate being fixed at time T:

t T T’
S(T, Tn) =
CMSnY(T,T,T n)

CMSnY(t,T,Tn)

Fixing of the swap rate of Cash Flows

Pricing at date t of the CMS maturity nY: SnY(T,Tn) exchange
nY which will be fixed at T
and paid at date T’ :
Ex : A client wants to receive in 5Y 100M€ * fixing (in 5Y) of the swap rate 10Y
CMSnY(t,T,Tn)
Fixing of Swap rate 10Y in 5Y = Fixing of CMS10Y in 5Y by definition

By default T’ = T+ 2 business days (fixing in arrears), then

dependence of CMS with respect to T’ neglected

108
 CMS

➢ Everyday, fixing of Fixed Rate Swap for every maturities (1Y,

2Y, …10Y, …30Y)
✓ To buy at date 0 a CMS n years at date T to buy the right to get at T the
fixing rate at date T of a n years swap ending at Tn
✓ Quotation at date 0 of this operation called: CMS (0 ,T ,Tn )

➢ Settlement at T of the CMS:

✓ Client pays = CMS (0 ,T ,Tn )
✓ Client receives the fixing of the swap rate n years = CMS (T , T , Tn ) = S (T , Tn )

109
 CMS, CMS options

CMS fixing as of
30/06/2009
(USD, JPY, EUR)
Daily fixing of CMS as for
Libor, Euribor

110
 CMS, CMS options

CMS conventions for main markets:

111
 CMS, CMS options

Fixing CMS EURO 2Y and 10Y

6.5% 2.5%

6.0%
2.0%
5.5%

5.0% 1.5%

4.5%

1.0%
4.0%

3.5%
0.5%

3.0%

2.5% 0.0%

2.0%
-0.5%
1.5%

1.0% -1.0%
01/01/99 01/01/00 31/12/00 31/12/01 01/01/03 01/01/04 31/12/04 31/12/05 01/01/07 01/01/08 31/12/08

CMS 2Y CMS 10Y spread CMS10Y-CMS2Y

112
 CMS, CMS options

➢ Pay-off at date T of a call (Caplet CMS) of strike K on CMS:

max(CMS (T , T , Tn ) − K ,0) = max(S (T , Tn ) − K ,0)

➢ Pay-off at date T of a put (floorlet CMS) of strike K on CMS:

max(K − CMS (T , T , Tn ),0) = max(K − S (T , Tn ),0)

➢ S(T,Tn) = fixing of n years Swap Rate at date T = CMS(T,T,Tn)

113
 CMS, CMS options

➢ Pricing of CMS and CMS options not straightforward derivation

of swap forward and Swaptions…
✓ call/put parity formula:

Price at date 0 of (call-put) on CMS n years fixed at T

= price at 0 of CMS n years that will be fixed at T minus strike

Call and put premium paid at T

Call(0, CMS (T , T , Tn ), K , T ) − Put(0, CMS (T , T , Tn ), K , T ) = CMS (0, T , Tn ) − K

114
 CMS, CMS options

➢ Analytical formula for CMS (approximation):

CMS (0, T,Tn ) = SwapFwd (0, T , Tn ) (1 +  (0, T , n,  ))  SwapFWd
 
( ( ) )
1 +  (0, T , n,  ) = 1 + exp  2T − 1 1 −
nSwapFwd

 (1 + SwapFwd / f() (1 + SwapFwd / f )n f
− )
1 

f = 1/frequency of the swap on the fixed leg
n  f = total number of coupons
exemple : CMS10Y, semi annuel, n = 10 f = 2, n  2 = 20
T = horizon of the CMS
 = volatility (ATM) of the underlying swap to the CMS

1 +  is called convexity adjustment

➢ Convexity adjustment can be justified by the fact that one who has
to pay a CMS rate is gamma negative (seller of a an option)

115
 CMS leg

➢ Price of a leg with schedule (Ti , d i )1im indexed on CMS (fixing of

CMS (in arrears) at date Ti ):

m
 CMS (0, Ti , Ti + n)d i B(0, Ti ) =
i =1
m
 SwapFwd (0, Ti , Ti + n) (1 +  (0, Ti , n,  i )d i B(0, Ti )
i =1

with :
  nSwapFwd (0, Ti , Ti + n ) 
 ( ( ) )
(1 +  (0, Ti , n,  i ) = 1 + exp  i T − 1 1 −
 
( )
2
  (1 + SwapFwd (0, T , T + n ) / f ) (1 + SwapFwd (0, T , T + n ) / f )n f − 1  
  i i i i 

116
 CMS Swap

➢ CMS Swap:
✓ Clients exchange a floating leg indexed on CMS –
margin against a floating leg indexed on Euribor:
o Usually, client wants to receive CMS and pay Euribor
(especially Life insurance companies)
o Ex.: CMS10YEUR-100bps against Euribor3M during 5Y,
payment BasisACT360, payment frequency 3M
▪ Every quarter, client pays Euribor3M and receives CMS10Y-100bps
▪ Euribor3M is fixed at the beginning of the quarter and CMS10Y is
fixed at the end (in arrears)

✓ CMS Swap pricing: the margin is calculated when the

CMS swap is settled.
o present value of leg indexed on CMS10Y-margin = present
value of leg indexed on Euribor
✓ Present value of the CMS Leg: discounted cash flows
of “anticipated value” of each fixing
117
 Margin of a CMS Swap

➢ CMS Swap = Swap where CMS + margin is paid against

EURIBOR
➢ Example:
✓ EURIBOR6M Leg payment schedule is (T j , d j )1im
✓ CMS Leg payment schedule is (T j , d j )1 jm '
~ ~

✓ Margin defined by:

 m
( ) ( )
m'
  B(0, Ti )d i CMS (0, Ti , Ti + n ) − B 0, T j d j FRA 0, T j −1 , T j −1 + 6M
~ ~ ~ ~
 i =1 j =1  PV CMSLeg − PV EURIBOR6MLeg
margin =  m =
 B(0, Ti )d i
  Physical LEVEL of CMSLeg
 
 i =1 

118
 CMS Swap : Examples as of 30/12/2009

10Y Swap : CMS10Y-132 against EURIBOR3M

10Y Swap : CMS10Y-107 against EURIBOR3M Without convexity adjustment (Vol=0)

10Y Swap : CMS5Y-101 against EURIBOR3M

10Y Swap : CMS5Y-88 against EURIBOR3M Without convexity adjustment (Vol=0)

5Y Swap : CMS10Y-180 against EURIBOR3M

5Y Swap : CMS10Y-163 against EURIBOR3M Without convexity adjustment (Vol=0)

5Y Swap : CMS5Y-136 against EURIBOR3M

5Y Swap : CMS5Y-128 against EURIBOR3M Without convexity adjustment (Vol=0)

Margin is both an increasing function of implied volatilities and the curve’s slope

119
 Caplet/Floorlet CMS

➢ Pay-off at date T of a call (Caplet CMS) of strike K on CMS:

max(CMS (T , T , Tn ) − K ,0) = max(S (T , Tn ) − K ,0)

➢ Pay-off at date T of a put (floorlet CMS) of strike K on CMS:

max(K − CMS (T , T , Tn ),0) = max(K − S (T , Tn ),0)

➢ S(T,Tn) = fixing of n years Swap Rate at date T = CMS(T,T,Tn))

120
 Caplet/Floorlet CMS

➢ Pricing of CMS and CMS Options not straightforward

derivation of Forward Swap and Swaptions…
✓ Call/Put Parity formula:
Price at date 0 of (Caplet-Floorlet) on CMS n years fixed at T
=
Price at 0 of CMS n years that will be fixed at T minus strike

Caplet(0, CMS (T , T , Tn ), K , T ) − Floorlet(0, CMS (T , T , Tn ), K , T ) = CMS (0, T , Tn ) − K

Caplet and Floorlet premium assumed to be paid at T in this

formula

121
Pay-off Caplets CMS vs. Cash Settlement Payer Swaption

➢ Caplet CMS and Cash settlement Payer Swaption:

✓ Pay-off Caplet on CMS n years, at T = Max(S (T , Tn ) − K ,0)
n
 Max(S (T , Tn ) − K ,0)
1
✓ Pay-off Swaption Cash Settlement = 
i =1 (1 + S (T , Tn ))
i

Cash Level

✓ Non linear dependence of level cash with respect to CMS

✓ No perfect matching between two types of pay-off

✓ Cash Level= 7.72 if 10Y swap rate = 5.00%

122
 Replication for Caplets, Floorlets CMS : key ideas
➢ Pay-off Caplets CMS pay-off vs. payer swaption pay-off for two hedge ratios:
5% cash settlement payer swaption on 10Y vs. caplets CMS10Y strike 5%
7.72 caplets for 1 swaption
50%
pay-off as % of notional

45%
40%
35%
30%
25%
20%
15%
10%
5%
0%
4.5% 5.0% 5.5% 6.0% 6.5% 7.0% 7.5% 8.0% 8.5% 9.0% 9.5% 10.0%
10Y swap rate at maturity
pay-off swaption pay-off caplets cms

5% cash settlement payer swaption on 10Y vs. caplets CMS10Y strike 5%

6 caplets for 1 swaption
50%
pay-off as % of notional

45%
40%
35%
30%
25%
20%
15%
10%
5%
0%
4.5% 5.0% 5.5% 6.0% 6.5% 7.0% 7.5% 8.0% 8.5% 9.0% 9.5% 10.0%
10Y swap rate at maturity
pay-off swaption pay-off caplets cms

123
 Replication for Caplets, Floorlets CMS : key ideas
➢
5% cash settlement payer swaption on 10Y vs. caplets CMS10Y strike 5%
7.72 caplets for 1 swaption
50%
pay-off as % of notional

45%
40%
35%
30%
25%
20%
15%
10%
5%
0%
4.5% 5.0% 5.5% 6.0% 6.5% 7.0% 7.5% 8.0% 8.5% 9.0% 9.5% 10.0%
10Y swap rate at maturity
pay-off swaption pay-off caplets cms

➢ If number of Caplets for 1 payer swaption = 7.72 = Cash Level(5%, 10Y)

Pay-off 7.72 Caplets > Pay-off 1 Payer Swaption

124
 Replication for Caplets, Floorlets CMS : key ideas
➢ More generally, one Caplet = static basket of cash settlement Swaptions
with increasing strike starting at 5%

Caplet (10Y ,5% ) =  wic  payer swaption (10Y , K i )

➢ Properties:
(1) w1c  1 / 7.72  13%  w2c
(2) 0  w2c  ...  wnc (increasing positive weights from i = 2)
(3) K1 = 5%  K 2  ...K n
✓ Static Weights independent of market conditions
(Yield Curve, Volatilities)
✓ All Swaptions with same maturity: maturity of the
Caplet

125
 Replication for Caplets, Floorlets CMS : key ideas
➢ Pay-off Floorlets CMS pay-off vs. receiver Swaptions pay-off for two hedge
ratios:
5% cash settlement receiver swaption on 10Y vs. floorlets CMS10Y strike 5%
7.72 floorlets for 1 swaption
35%
pay-off as % of notional

30%
25%
20%
15%
10%
5%
0%
1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% 5.5% 6.0% 6.5% 7.0% 7.5% 8.0%
10Y swap rate at maturity
pay-off swaption pay-off floorlets cms

5% cash settlement receiver swaption on 10Y vs. floorlets CMS10Y strike 5%

9 floorlets for 1 swaption
35%
pay-off as % of notional

30%

25%

20%

15%

10%

5%

0%
1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% 5.5% 6.0% 6.5% 7.0% 7.5% 8.0%
10Y swap rate at maturity
pay-off swaption pay-off floorlets cms

126
 Replication for Caplets, Floorlets CMS : key ideas

➢ 5% cash settlement receiver swaption on 10Y vs. floorlets CMS10Y strike 5%

9 floorlets for 1 swaption
35%
pay-off as % of notional

30%

25%

20%

15%

10%

5%

0%
1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% 5.5% 6.0% 6.5% 7.0% 7.5% 8.0%
10Y swap rate at maturity
pay-off swaption pay-off floorlets cms

➢ If number of Floorlets for 1 payer swaption = 7.72 = Cash Level(5%, 10Y)

Pay-off 7.72 Floorlets < Pay-off 1 Receiver Swaption

127
 Replication for Caplets, Floorlets CMS : key ideas

➢ More generally, one Floorlet = static basket of Cash Settlement Swaptions

with increasing strike starting at 5%

Floorlet (10Y ,5% ) =  wif  receiver s waption (10Y , K i )

✓ Properties:
(1) w1f = w1c  1 / 7.72  13%
(2) w2f  w3f    wnf  0(increasing negative weights from i = 2)
(3) K1' = 5%  K 2'  ...K n'
✓ Static Weights independent of market conditions (yield Curve,
Volatilities)
✓ All Swaptions with same maturity: maturity of the Floorlet

128
 Caplet CMS pricing: replication

CMS-K
CMS-K

w1SP1+w2SP2
w1SP1 + w2 SP2 K1< K2<K3…

i, wi 0
i, wi  0
w1SP1

w1SP1
Swap Rate at
Maturity = CMS
SP = payer swaption Same maturity and tenor for Caplet CMS and (Cash Settlement) Payer Swaptions
Ex.: Caplet on CMS10Y in 5Y strike 5%, Payer Swaptions on 10Y in 5Y, strikes 5% < 5.25% < 5.50% <…

129
 FLOORLET CMS pricing: replication

w1R1+w2SR2

w1SR1
K1> K2>K3…

CMS-K

K3 K2 K=K1 Swap rate at

Maturity = CMS
SR = receiver swaption
Same maturity and tenor for Floorlet CMS and (Cash Settlement) Receiver Swaptions
Ex.: Floorlet on CMS10Y in 5Y strike 5%, Receiver Swaptions on 10Y in 5Y, strikes 5% > 4.75% > 4.50% >…

130
Replication for Caplets, Floorlets CMS : key ideas

131
 Replication for Caplets, Floorlets CMS : key ideas
➢ Key ideas:
✓ Caplet (floorlet) CMS priced as a static basket of cash settlement
payer (receiver) Swaptions
✓ CMS priced by call/put parity
✓ Replication basket + call/put parity on swaptions/options on CMS

CMS = forward swap rate

+ static basket of out the money payer Swaptions
- static basket of out the money receiver Swaptions

✓ Pricing of CMS, CMS option: Replication method better than

Analytical convexity Adjustment (takes into account the full
smile)

132
 CMS: Examples as of 30/12/2009

Forwards 10Y Swap Rates and CMS 10Y, EUR

5.60%
0.46%

5.40%
0.41%

5.20% 0.36%

5.00% 0.31%

0.26%
4.80%
0.21%
4.60%
0.16%
4.40%
0.11%

4.20% 0.06%

4.00% 0.01%
1 2 3 4 5 6 7 8 9 10
Forward Swap Forward with ATM Convexity adjustment CMS 10Y by replication ATM Convexity Adjustment Adjustment by replication

133