Didier Faivre Digital rate options and applications to exotic swaps Complex Products : digital, steepner, ratchet… Bermuda swaptions Callables Appendixes : Standard Deviations vs Volatilities SABR Correlation issues LGM Risk-Management of exotics
2 Digital rate options
Digital options: the payoff is binary, nil or fixed, in view of a barrier on an underlying security Example : digital on 3-month Euribor/ Strike (K) Mechanism At maturity of each 3 month period: If Euribor < K the buyer receives 0 If Euribor > K The buyer receives : Notional × « Pay out » × NJE/360 Profit/Loss chart:
Profit
Pay out K Premium Euribor
3 Digital rate options
Profit/Loss chart:
Profit
S Pay out
Premium Euribor
4 Digital rate options
example: sell a digital option with payoff equal to 2 % if the 3-month interest rate is higher than 4 % possible hedging modality: buy a call at 3 % and sell a call at 5 % with same initial nominal amount of the digital option (the approximation of a digital option by using a call spread).
5 Digital rate options
Intrinsic value
digital Call spread
3-month rate 0 3 % 4 % 5 % premium barrier
6 Digital rate options
The barrier is suitably replicated by reducing the spread of 200 to 20 bp (buying a call at 3.9% and selling a call at 4.10% for the bank selling the digital).
The equalisation of the payoff therefore implies multiplying the nominal amount by 10.
A digital option therefore corresponds to a leverage effect on an options spread.
7 Digital rate options
The pricing of the digital option is always consistent with this hedge method : Digital option strike K = (call (E3M, K-0.1%)-call (E3M, K+0.1%))/0.2% The spread between the two strikes is typically 2*10bp, 2*5bp or 2*20bp. We speak of static replication method, automatically consistent with the market smile without complex model.
8 Digital rate options
In practice : seller digital : (call (E3M, K-0.1%)-call (E3M, K))/0.1% Sur-replication buyer digital : (call (E3M, K)-call (E3M, K+0.1%))/0.1% Under-replication
Choice of half spread parameter is linked to gamma limits of trading desk.
9 Digital rate options
Intrinsic value 1+premium
E3M K 1 0.1%
K-0.1 % K 3-month rate 0 K+0.1 % E3M K 1 0.1% premium Sur-replication Under-replication 10 Digital :math justification for call spread technique
Let have and underlying with initial value S 0 and density S0 ,T, S
The price of a call (with premium paid at maturity T) is :
C S0 , K,T S K S0 ,T, S dS K Then
C S S0 ,T, S dS K S0 ,T, S dS K K K K
K S0 ,T, S K K S0 ,T, S K S0 ,T, S dS S0 ,T, S dS K K
11 Digital :math justification for call spread technique
If we derive a second time, we get :
2C S ,T, S K S ,T, K K 2 0 0
The equation
C S0 ,T, S dS P S K , K K
explains why we can always approximate a digital by a call spread divided by the spread of strikes, whatever the assumption on the underlying distribution density : S0 ,T, S
C C S , K ,T C S , K ,T P S K 0 0 K 2
12 Digital :math justification for call spread technique
Indeed, it’s important to see the previous demonstration is general and doesn’t rely on any assumption on .
If we now use the (Lognormal) B&S framework, we get :
C P S K N d K 2
In pratice, prices of digital using the call-spread method or the analytical B&S formula (lognormal or normal), leads to very different prices.
13 Digital rate options
Other pay-off : Let’s note :
Kdown Kup left Kdown Kdown 10bp right Kdown Kdown 10bp left Kup Kup 10bp right Kup Kup 10bp
14 Digital rate options
We have the following replication formulas :
left right Call E3M ,K down Call E3M ,K down 1) digital "c ap" digit E3 M K down right left K down K down left right Call E3M ,K up Call E3M ,K up 2)digital "f loor" digit E3M K up 1 right left K up K up
3)digital "in" digit E3 M Kdown , Kup
left right left right Call E3M , Kdown Call E3M , Kdown Call E3M , Kup Call E3M , Kup right left right left Kdown Kdown Kup Kup
4)digital"out" digit E3M Kdown , Kup
left right left right Call E3M , Kdown Call E3M , Kdown Call E3M , Kup Call E3M , Kup 1 right left right left Kdown Kdown Kup Kup
15 Use digital rate options for to exotic caps/floors
Cap Knock-in : Pay-off : E3M K 1 E3M B
Valuation : B K Buyer Cap with the strike set to B (the barrier) A buyer digital “Cap” with barrier B and Fixed rate = (B-K)
16 Use digital rate options for to exotic caps/floors
17 Use digital rate options for to exotic caps/floors
Floor Knock-in :
Pay-off : K E3M 1 E3M B B K Valuation : Buyer Floor with the strike set to B (the barrier) A buyer digital “floor” with barrier B and Fixed rate = (K-B)
18 Use digital rate options for to exotic caps/floors
19 Use digital rate options for to exotic caps/floors
Cap Knock-out : Pay-off : E3M K 1 E3M B Valuation : K B Buyer Cap with the strike set to K A seller Cap with the strike set to B (the barrier) A seller digital Cap with barrier B and Fixed rate = (B-K)
20 Use digital rate options for to exotic caps/floors
21 Use digital rate options for swap KO
Swap KO Bank pays r if index1 < B, index2 + m otherwise
Replication of receiver swap KO: Receiver fixleg rate r Payer floatleg on index2 Buyer cap Knock-In on index1 Buyer cap strike B on index1 Buyer digital Cap barrier B on index1 and Fixed rate (B+m)-r
22 Digital rate options : application to exotic swaps
23 Digital rate options : application to exotic swaps
Floor Knock-out : Pay-off : K E3M 1 E3M B Valuation : B K Buyer Floor with the strike set to K A seller Floor with the strike set to B (the barrier) A seller digital “Floor” with barrier B and Fixed rate = (K-B)
24 Digital rate options and applications to exotic swaps Complex Products : digital, steepner, ratchet… Bermuda swaptions Callables Appendixes : Standard Deviations vs Volatilities SABR Correlation issues LGM Risk-Management of exotics
25 Products based on digital on spread
Example 1 (4Y swap): Client receives 4.65% Pays Euribor3M +n/N*4.65% n : number of days on the interest period where spread CMS10Y-CMS2Y < B N : number of days on the interest period 4Y swap at pricing date (march 2005) : 3% B = 0.9% year 1, 0.85% year2, 0.8% year3, 0.75% year1 Example 2 (15Y amortizing swap) : Client receives Euribor3M Pays 1.9% If CMS10Y-CMS2Y > 0.70%, 4.95% otherwise
In both case : fixing of CMS is in arrears Client is long digitals on CMS10Y-CMS2Y spread that pays above strike CMS10Y-2Y spread around 120/110 bp at pricing date
26 Products based on digital on spread
Graph of forward spread CMS10-CMS2Y Euro for Example 1
5.00% 1.40%
4.50% 1.20%
4.00%
1.00% 3.50%
3.00% 0.80% CMS10Y 2.50% CMS2Y Series3 0.60% 2.00%
1.50% 0.40%
1.00%
0.20% 0.50%
0.00% 0.00% 0 0.5 1 1.5 2 2.5 3 3.5 4
27 Products based on digital on spread
Analysis : Both products were for debt liability management Structured Swap enables the client to switch to fixed/float with enhanced rate if CMS10Y-CMS2Y spread stays above some barriers : Client view on slope 10Y-2Y curve against the forward slope Rational : low spread = low growth + high inflation, high spread = higher growth + low inflation Issuance of debt + population ageing
Both product can be seen as vanilla swap + digital Analysis of the digitals Sensitivity to the curve slope Flattening of the curve implies higher probability of paying higher rates for the client, so client looses money on the swap To analyze sensitivity to volatilities and correlation, let’s move to a basic model
28 Products based on digital on spread
The idea is We know the basics of greeks on a single underlying digital, assuming this underlying normal or lognormal Whatever the model, it should have similar behaviour with respect to the parameters (volatilities or standard deviation + correlation)
Basic model to analyze a digital on CMS10Y-CMS2Y with maturity T : Step 1 : Calculate CMS10Y(0,T) and CMS2Y(0,T) at date T Step 2 : calculate implied volatilities or better, implied standard deviation for ATM swaptions on 10Y and 2Y of maturity T
29 Products based on digital on spread
Step 3 : let’s assume the spread is normal with :
mean m spread = CMS10Y(0,T)-CMS2Y(0,T)
2 2 Standard deviation : spread 1 2 2 1 2
1 , 2 Standard deviation of CMS10Y and CMS2Y Then the price of a digital with barrier B is given by :
m B B 0,T N spread spread T
N distribution function of standard gaussian variable mean 0, variance 1.
30 Products based on digital on spread
Basics behaviour of a digital with respect to the volatility (lognormal model) or the standard deviation (normal model) :
vega one year digital
3
2
1
0
a 60 70 80 90 100 110 120 130 140 g e v -1
-2
-3
-4 spot
31 Products based on digital on spread
To see the sensitivity to the standard deviation of each CMS, let’s calculate the derivative of the spread variance with respect to each standard deviation :
2 2 spread 2 spread 1 2 1 1 , 2 2 1 1 1 2 2
So even these derivatives are not obvious ; nevertheless, if the two standard deviation are close (usually it’s true) correlation not too high Then, the standard deviation of the spread is increasing function of each CMS standard deviation Anyway if both standard deviations increase, the standard deviation of the spread increase Standard deviation of the spread is decreasing function of the correlation
32 Products based on digital on spread
Digital vega of digital is positive out of the money, negative in the money So sensitivity of digital with respect to correlation and each standard deviation depends on moneyness If you are long a digital that pays above the barrier, you are short volatility if forward spread above the strike, so long correlation If you are long a digital that pays above the barrier, you are long volatility if forward spread below the strike, so short correlation
Conclusion : The global sensitivities will depend mainly on each digital moneyness and marginally on the discount factors
to understand the various sensitivities always try first to go back to basics, i.e black& Scholes type models…
33 Products based on digital on spread
2005 story : Both Investors and liability management clients long digital that pays above the strike For investors, strikes generally below initial forwards spreads (long correlation) For Liability management, strikes generally above initial forwards spreads (short correlation) More risky position for liability management than for investors LM clients buy digital out the money, Investors buy digital in the money More products sold to investors, so banks were initially short correlation on digitals
34 Steepener
Typical steepener : Client receives 4*Max(CMS10Y-CMS2Y;0) Client long spread option Client gains if slope increase Products stable with respect to a translation of the curve Client long the volatility of the spread, so short correlation
Banks can offset their short position on correlation due to digital by selling steepener
35 Other Complex products
Ratchets Definition The strike of the option or the coupon will depend on the preceding fixing. Common indexation is the following: Coupon (i) = Notional * max (or min) [E6M(i) + T1; Coupon(i-1) + T2] First coupon is E6M observed at the start date + T1
Example : Maturity 3Y; index E3M. Customer receives E3M Customer pays E3M - 25 bps on the first coupon and then Max(E3M-25; Coupon(i-1))
36 Other Complex products
Ratchets Pricing Very path-dependent products. Main issue is to address the exposure to volatility at a strike that is not yet known. Vega exposure is more on local volatilities than on straight B&S volatilities. Need stochastic volatility model for pricing (same for vol bonds)
Ratchets Pricing use Different types that can match various strategies. In general, they are used when customer has a view on market being either stable or oriented with a strong, long-term trend. In case of strong market moves, they can lead to heavy losses.
37 Other Complex products
Ratchets second type (current version) Also different types that can match various strategies. Example: Client pays E3M -0.30%+M, with margin M zero on the first year and afterwards previous margin + Max (0, 2%-E3M) It is a risky product. Similar to be short of floorlets strike 2% with cumulative effect in term of risk There is no need of stochastic vol model, because the product is a series of puts, but one needs proper modeling for long delay adjustment. Ratchet cap or floor are both level and slope curves products
38 Closed formulae : some limits
Many popular products with European features can be priced with closed formulae, in B&S framework All variant of Libor Legs, including caps & floors Standard Spread options
Nevertheless : Quants/Trading have to be careful with choice of parameters, including parameters you can observe on the markets as volatilities Examples : Non standard Libor Legs/CMS spread options
39 Closed formulae : some limits
Non Standard Libor Legs : You pay 6M Libor every 3M For pricing Cap/Floor on this Leg, needs implied vol For pricing the leg and so the underlying in the cap/floors (convexity adjustment for Non Standard Libor Leg) you need another vol How to deal : proposal Vol at strike for modeling Lognormal Behavior of Libor Vol ATM for modeling convexity adjustment in Leg/Cap/Floor In this way Call/Put Parity is automatically verified.
40 Closed formulae : some limits
CMS spread option
Pay-off : m1CMS 1 (T ) m 2CMS 2 (T ) K , CMS i 1, 2 being of different maturities Easy to calculate and compute closed formulae if both CMS assumed lognormal or normal Which volatilities/standard deviations do take ? Main idea : Volatilities must depend upon so that specific degenerated m1,m2 , K cases are consistently treated. We know the price of the spread option if maturity, one of the volatilities or one of the weights is zero.
41 Closed formulae : some limits
Methodology : Step 1 : Choose K1 and K2 two strikes and evaluate the CMS call at K and the CMS put at 1 K2 Step 2 : Evaluate the volatilities 1 and 2 consistent with these prices. Step 3 : Compute the spread option price
How to choose K 1 a n d K 2 ? Proposal 1 :
m1CMS1( 0) K If m1 0 or 1 0 K2 m2
m2CMS2 ( 0) K If m2 0 or 2 0 K1 m1
42 Closed formulae : some limits
Proposal 2 :
m2CMS 2 (0) K K1 with s standard deviations now m1
m1CMS 1 ( 0) K K 2 m2
m1 T1 min(T1,T2 )m2 K1 CMS1 1 2 2 (m1CMS1 m2CMS2 K) Best choice( accordingm1 1) T1 to(m 22 dimensionnal2 ) T2 2 min(gaussianT1,T2 )lawm1 metric1m2
This way stable pricing mbetween2 2T2 normalmin( orT 1 lognormal,T2 )m2 choice K2 CMS2 2 2 2 (m1CMS1 m2CMS2 K) (m1 1) T1 (m2 2 ) T2 2 min(T1,T2 )m1 1m2
43 Complex products
Bermuda Swaptions
Market conditions
The more competition you have between the included swaptions, the more expensive the bermuda. - flat curve - in a steep curve: The value of the bermuda is close to the value of the first swaption. This value can be increased thanks to a step up rate.
44 Other Complex products
Flexicaps The buyer decides to be hedged on a limited number of fixings (1/2 usually). Either he decided himself when he wants the protection: liberty cap. Either it is automatic when the fixing is in the money: autocap.
45 Other Complex products
Flexicaps Example Maturity 10Y Index EURIBOR 3M Strike 6% Number of fixing 20 Automatic exercice IF E3M fixe above 6%, the buyer receives (E3M-6%) When number of time E3M above 6% reaches 20, cap disapears and buyer not protected Premium reduced by 30% (500 bps for vanilla, 350 for flexicap)
46 Other Complex products
Flexicaps Pricing The autocap is a good example of path-dependant product which can be priced with LGM model (for instance) and Monte-Carlo. The liberty cap is much more diffcult to price as it offers a choice to the buyer of the option. An autocap is cheaper than the corresponding liberty cap, itself cheaper than the vanilla cap. A liberty cap will be more expensive than the sum of the n most expensive caplets of the correponding vanilla cap.
47 Other Complex products
Flexicaps Use
Premium reduction A good product for a borrower who thinks a monetary crisis can occur but who believes these unlikely will not last for too long ( thanks to central banks).
48 Other Complex products
Flexicaps & Market conditions
Some conditions make them more attractive than vanilla caps: - a flat curve - high volatility - high correlation between forwards (if caps volatility are close to swaptions volatilities for example)
These parameters make them more likely to be in the money at the same time.
49 Other Complex products
Options with multiple underlyings
Definition Payoff depends on the observation of several different indexes (two in general).
50 Other Complex products
Options with multiple underlyings Examples They can appear as the conditions of a range : Range on a spread (customer bets on difference between Libor GBP and E3M being stable) Range with 2 conditions (bet on E3M staying within a range and 10Y swap, within another) Or as options themselves : Call/Put on the spread on two short-term rates in two currencies Call/Put on the spread between Long & Short Term rates in one currency (option on the slope of the curve) Or as European Digitals : A Cap 5% on E3M Knock-Out on CMS10 being above 7% (fixing by fixing)
51 Other Complex products
Options with multiple underlying Pricing These products are priced with closed-formed formulas derived from B&S framework. Though, their pricing involves an crucial additional parameter : the correlation between the two indices. This correlation parameter can be determined as an implicit output of LGM model (for 2 indices of a same currency), or from historical data (for 2 currencies). Very few market prices can sometimes provide the implicit correlation priced by the market.
52 Other Complex products
Options with multiple underlying Use
Options with 2 indices in different currencies are mostly speculation tools as they can leverage a convergence strategy (ex : Forward Range on Stibor - Euribor) Options with 2 indices in the same currency can be used as speculation products too, but also as a balance sheet hedge tool. They are often used in EMTN on spreads, where coupon needs to be floored.
53 Other Complex products
Global Cap Definition A Global Cap is a protection on the total amount of interest paid on a debt. This can be seen as an option on the average of E3M Example Maturity 5Y; index E3M; strike 5.50% Premium = 160 bp, compared to 215 for vanilla cap (25% premium reduction) In 5 years, if the average of E3M has been below 5.50%, owner of the cap doesn’t receive any payoff, even if some fixings have been above 5.50% If the average of the fixings has been 6%, owner of the option will receive at maturity 0.50% * 5 *365 /360.
54 Other Complex products
Global Cap Pricing The volatility of the average of the forwards is estimated with the correlation between the forwards (that can be inferred from the model used ), then option price is computed in the B&S framework. The main hypothesis is that local volatility is equal to B&S volatility.
55 Other Complex products
Global Cap Use Can yield to a significant premium reduction. Provides a hedge on the global hedge of a debt even if treasury flows are not perfectly matched (payoff happens at maturity) .
Market conditions : Compared to a vanilla cap, a global cap is cheap when : - Curve is steep - Volatility is high - Correlation between the forwards is small
56 Other Complex products
Maturity Cap Allows an financial institution to propose a floating rate loan while guaranteeing its customer that he won’t pay back his loan over more than a given period of time. In case capital has not been completely paid back after a given date, the seller of the option pays the remaining part of the debt.
57 Other Complex products
Example : Constant annuities are computed on a 15Y basis with a rate at 5.80%. If floating rates increase, each annuity will pay back less capital than forecasted (actually it can even not be enough to pay interest). The loan can then continue over an unlimited period of time. The ability to cap the maturity can be a marketing advantage.
58 Other Complex products
Maturity Cap Example: Loan of 1MEUR on a 15Y basis at 5.80% semi-annually. Constant 6-month annuity is 50 365 EUR. The guarantee loan won’t go over 18 years is worth 150 bps flat, on 20 years it is worth 110 bps.
59 Digital rate options and applications to exotic swaps Complex Products : digital, steepner, ratchet… Bermuda swaptions Callables Appendixes : Standard Deviations vs Volatilities SABR Correlation issues LGM Risk-Management of exotics
60 Bermudas under LGM
Bermudan swaption: option to enter in a swap at several exercise dates.
Bermuda Swaptions Utilization :
Bermuda swaptions are useful to hedge a callable paper or loan. They can also be packaged with a swap to get a callable swap. It gives a better rate than the boosted rate with a vanilla option.
61 Bermuda swaptions
Bermuda Swaptions
Pricing
PDE (Partial differential equation) or tree. At each nod of the tree, there will correspond an exercise date, the value of the option will be the max between the swap value and the value of the option itself. Minimum value: a bermuda swaption is more expensive than the most expensive swaption included. Max value: the corresponding cap/floor. In our example, the bermuda is cheaper than the 9Y floor into 1Y 4.80%.
62 Bermudas under LGM
Calibration: One wants to be consistent with the market price of all underlying swaptions, and especially with the most expensive ones.
Calibrate the diagonal of swaptions with strike the strike of the bermudan option.
63 Bermudas under LGM Example : Bermuda swaption 10Y in 5Y, strike 5% In 5Y pay-off of a 5Y, 5% strike, European swaption on 10Y swap In 6Y pay-off of a 6Y, 5% strike, European swaption on 9Y swap … In 14Y pay-off of 14Y, 5% strike, European swaption on 1Y swap
Calibration of LGM1F : for a given mean reversion, calibration of volatilities parameters to the previous european swaptions :
T 0 0 T1 5Y T2 6Y T15 14Y
64 Bermudas under LGM
constant between T and T 1 0 1 constant between T and 1 T2 2 And so on…
Calibration of LGM2F is similar as again all parameters except volatilities are choosen and not part of the calibration.
65 Bermudas under LGM
Exotic risk: To identify the part of « exotic » risk in the product (i.e. the risk orthogonal to the diagonal of swaptions), we use the following decomposition:
Bermuda = Most Expensive swaption + switch option
The most expensive swaption is, viewed from today, the one that is the most likely to be exercised. The switch option accounts for the future possibility to delay (or bring forward) exercise, if it turns to be more interesting to do so.
66 Bermudas under LGM
Effect of mean reversion on bermuda prices
The switch option value is determined by the way the remaining swaptions are evaluated within the model, from each exercise date.
In other words, the parameters of interest are the model forward volatilities of remaining swap rates, computed from each exercise date. The larger the mean reversion, the larger these forward volatilities and thus the larger the price of the switch option.
67 Bermudas under LGM
The larger the mean reversion, the larger the price of the bermudan swaption.
68 Bermudas under LGM
Example : CASA USD deal as of 25/03/2003 Deal characteristics : 30Y, no call 9Y. The bank pays fixed rate 6.65% quaterly 30/360 and receives LIBOR USD + 125 BP. Notional = 650 Mios USD. Bermudan option price (3% mean reversion) : 5.51 % / 35.8 Mios USD Callable Swap (3% mean reversion) : 3.16 % / 20.5 Mios USD
mean reversion option structure 0.01 5.17% 2.82% 0.03 5.51% 3.16% 0.05 5.85% 3.50% 69 Bermudas under LGM
Most expensive (expiry = 9Y / term 21Y) = 4.34 % / 28.2 Mios USD Switch option = bermudan option – most expensive = 1.17 % / 7.63 Mios USD Vega (1% parallel shift = 0.59 % / 3.87 Mios USD
70 Bermudas under LGM
Question: how to determine the mean reversion?
Market (e.g. on USD, for bermudas of short maturities). Statistical: use statistical methods to estimate the mean reversion from the short rate process itself (forget !) Historical vol of FRAs: compare historically the volatility of different FRAs (e.g. 1Y vs. 20Y).
71 Bermudas under LGM
Long swaption vs. short swaption: each day, calibration to a long swaption. Choose the mean reversion so that, in average, short swaptions are well priced by the model. LGM forward vol vs. market vol: each day, calibrate to a diagonal of swaptions and, from the expiry of the most expensive, compare LGM forward vols of remaining swaptions, to the market (normal) volatilities of swaptions with same underlying / time to maturity Back testing: consider a Bermuda. Each day, and for several values of mean reversion, vega hedge the bermuda, delta hedge the overall position and study the average / std deviation of daily P&L.
72 Bermudas under LGM
Historical study: long swaptions vs. short swaptions (or cap) Consider a set of historical datas (yield curves, vol curves) For each date, calibrate LGM to the diagonal of swaptions and determine the mean reversion such that the cap with same strike is well priced by the model
EUR : mean reversion breakeven to match 10Y 4.50% Cap USD : mean reversion breakeven to match 10Y 5.00% Cap
12.00% 7.00%
10.00% 6.00%
8.00% 5.00%
6.00% 4.00%
4.00% 3.00%
2.00% 2.00% Mar-03 May-03 Jun-03 Aug-03 Oct-03 Nov-03 May-03 Jul-03 Aug-03 Oct-03 Nov-03
73 Bermudas under LGM
Historical study: LGM forward vol vs. market vol Consider a set of historical datas (yield curves, vol curves). At each date: Calibrate the model to the diagonal of swaptions. From the expiry date of the most expensive swaption, compute the normal LGM forward vols of all remaining swaptions.
74 Bermudas under LGM
Compare them to (normal) market volatilities of swaptions with same underlying & time to maturity. Breakeven = mean reversion value such that the average of LGM forward vols matches the average of market normal vols.
75 Bermudas under LGM
Expiry of most expensive swaption LGM forward vol
LGM forward vol : Today T T2 Te 1 nd
Market vol :
T2 – T1 Tend – T2
76 Bermudas under LGM
LGM forward vol vs. market vol: empirical results on EUR
EUR, 10Y and 20Y non call 1Y receiver bermudas Mean reversion breakevens for several strikes
EUR 10 Y : mean reversion breakevens for several strikes EUR 20 Y : mean reversion breakevens for several strikes 25.00% 14.00%
12.00% 20.00% 10.00%
15.00% 8.00%
6.00% 10.00% 4.00%
5.00% 2.00%
0.00% 0.00% Mar-03 May-03 Jun-03 Aug-03 Oct-03 Nov-03 Mar-03 May-03 Jun-03 Aug-03 Oct-03 Nov-03 strike = 4.50 strike = 5.00 strike = 5.50 strike = 3.50 strike = 4.00 strike = 4.50 strike = 5.00 strike = 5.50
77 Bermudas under LGM
LGM forward vol vs. market vol: empirical results on USD
USD, 10Y and 20Y non call 1Y receiver bermudas Mean reversion breakevens for several strikes
USD 10 Y : mean reversion breakevens for several strikes USD 20 Y : mean reversion breakevens for several strikes 25.00% 25.00%
20.00% 20.00%
15.00% 15.00%
10.00% 10.00%
5.00% 5.00% May-03 Jul-03 Aug-03 Oct-03 Nov-03 May-03 Jul-03 Aug-03 Oct-03 Nov-03 strike = 4.50 strike = 5.00 strike = 5.50 strike = 6.00 strike = 4.00 strike = 4.50 strike = 5.00 strike = 5.50 strike = 6.00
78 Bermudas under LGM
LGM forward vol vs. market vol: some conclusions Results: Slightly high mean reversion breakevens (up to 15 – 20%), especially on USD Strike dependent (especially for 10Y bermudas) Depends on maturity
79 Bermudas under LGM
What to conclude? Avoid being massively short of forward vol. If it is the case, be conservative (i.e. use a mean reversion greater than 15%), or better, use a stochastic volatility model.
What if doing the same study with LGM 2 Factors?
80 Bermudas under LGM
Risk management back testing Consider a set of historical datas (yield curves, vol curves) Consider a bermuda. For each date of the data set, compute the vega hedge, delta hedge the overall position and compute the P&L at the end of the day. Repeat the operation for several values of mean reversion and compare the results it terms of average and standard deviation of the daily P&L. Try to identify robustness properties on the mean reversion.
81 Bermudas under LGM
25Y non call 5Y 15Y non call 3Y (2 call dates) Average of daily P&L Average of daily P&L EUR historical datas Simulated datas 30 1 L 20 L 0.5 & & P P y y l l i 10 i 0 a a d d 0% 5% 10% 15% 20% 25% f f o o e 0 e -0.5 g g a -5% 0% 5% 10% 15% 20% 25% a r r e e v -10 v -1 A A
-20 -1.5 mean reversion mean reversion
82 Bermudas under LGM
LGM 1 Factor vs. LGM 2 Factors Question: is a one factor model appropriate to price and risk manage bermudas? In other words: to what extent decorrelation between rates affects the price of a bermuda ? To understand the effects involved, let us consider a receiver bermuda with only 2 exercise dates:
Today T Te 2 nd
Exercise in T1 Delay exercise to T2
83 Bermudas under LGM
The payoff of the bermuda at date T1 looks like:
84 Bermudas under LGM
LGM 1 Factor vs. LGM 2 Factors (continued)
Given that the vols of the swap rates S1(T1) an S2(T2) are fixed by the calibration, the price of our 2-exercise-dates bermuda depends mainly on 2 things:
The forward vol of S2(T2) between dates T1 and T2 (the larger this forward volatility, the larger the price of the bermuda).
The correlation between swap rate S1(T1) and forward swap rate S2(T1) (the larger this correlation, the smaller the price of the bermuda).
85 Bermudas under LGM
Suppose that we choose model parameters such that the 1 factor and 2 factors model provide the same forward volatilities. Under LGM 1 Factor, the correlation is equal to 1 whereas it is stricly smaller than 1 for LGM 2 Factors. Therefore, the LGM 1 Factor price of the bermuda will be smaller than the 2 Factors price. In other words, to match LGM-1F & LGM-2F bermuda prices, one has to compensate by increasing the one factor forward vol (i.e. by increasing the 1F mean reversion).
86 Bermudas under LGM
LGM 1 Factor vs. LGM 2 Factors (Example) USD, as of 23/012004
T1 = 2Y / T2 = 4Y / Tend = 15Y Strike = 5.00%
swaption T1 = 4.24% / swaption T2 = 4.00% LGM-2F parameters: = 0.02 / = 0.34 / = 2.28 / = -0.26 The correlation effect is marginal (it accounts for only 7 bp).
87 Bermudas under LGM
LGM-2F bermuda price = 5.17% , therefore switch option = 0.93%
LGM-2F correlation S1(T1) / S2(T1) = 0.977 LGM-1F mean reversion that matches 2F forward vol = 4.00% LGM-1 bermuda price for this mean reversion = 5.10% The conclusion holds for more than 2 exercise dates. On all examples we considered, the discrepancies remain below 10 bp. Basically, what matters is what happens around the most expensive swaptions.
88 Conclusion / Summary
Bermuda = swaption + exotic risk (switch option) Exotic risk depends on forward volatility Under LGM, forward volatility depends on mean reversion Mean reversion is chosen historically, using several approaches LGM one factor is appropriate for bermudas Typical mean reversion 1.5% to 5%, nearly market implied parameter
89 Amortizing bermudan swaptions
Standard diagonal calibration on vanilla swaptions is not satisfactory. Idea: Swap with variable notional = basket of vanilla swaps with various end dates.
90 Amortizing bermudan swaptions
Example : a 3Y annual amortizing swap with notionals 2 mios EUR on first year, 1.5 mios EUR on second year, and 1 mios EUR on third year, is the sum of three vanilla swaps (3Y with notional 1 + 2Y with notional 0.5 + 1Y with notional 0.5) .
The idea is the following: For each exercise date, compute the basket of (forward) vanilla swaps equivalent to the (forward) amortizing swap. For each exercise date, consider the corresponding basket of swaptions (with same coefficients as in the basket of swaps), with strike the strike of the bermudan, and calibrate the model to the market price of this basket.
91 Amortizing bermudan swaptions
Pros: OK for pricing. European amortizing swaptions model price is independant of the choice of mean reversion.
92 Amortizing bermudan swaptions
For hedging, the gamma replication may be inacurrate : as the swaptions of the basket have fixed strikes, there might be a mismatch between the gamma of the basket of swaptions and the gamma of the amortizing swaption, especially when the amortizing swaption turns to be ATM.
93 Bermuda swaptions aproximation
Let us consider a N-years Bermuda swaption with Ti, 1 The idea is to add to the most expensive a correction based on the other swaptions prices and which is proportional to the probability of the most expensive to be out of the money 94 Bermuda swaption The probability for the most expensive swaption to be in the money is N(d1) With the previous notations, the approximation for a Bermuda Swaption price is: V˜ Vj* + (1- Pj*) S i<>j* Vi Pi S i<>j* Pi 95 Digital rate options and applications to exotic swaps Complex Products : digital, steepner, ratchet… Bermuda swaptions Callables Appendixes : Standard Deviations vs Volatilities SABR Correlation issues LGM Risk-Management of exotics 96 Callables products Bank pays 0.2% + (CMS20Y-CMS2Y)YEN Receives Libor YEN6M Callable every 6 months after one year At every call date, after exchange of current cash flow Bank PV = PV future cash flow + call option>=0 Cancel option = Max(cancel at the current date, all future call) This is the decision rule between immediate cancel and keep the product Callable structures enable to create the possibility of pick-up for the client whicj is short the call option 97 Exotic products: definition The main exotic feature of IR exotic products is their illiquidity: a lack of inter-bank market for the most exotic. Most of them use non quoted parameters such as correlation. These risks need a specific approach (“worst case”). 98 Exotic products: strategy Importance of marketing: you need to identify a risk or an opportunity for a customer. Being able to handle large volumes on vanilla products. Strong interest of using historical data (for marketing and risk purposes) and a strong analysis of illiquid Greeks on illiquid risks. Simulation of portfolio on different scenarios (VaR). 99 Exotic products : interest To find the product which match the exact need and expectations of the customer. In order to decrease the variance of a portfolio by accepting a lack in the expected return (more important bid/offer than the vanilla products). And to minimize the future hedging cost. 100 Exotic products : non-hedgeable risk Three kind of risk can be hedge with vanilla products : Parameters such as correlation between CMS, quanto correlation Mean reversion if models use like for Bermuda Correlation between forward rates, model depending No obvious solution Measure your risk wih good mapping Limit control Buy and sell risk to stay within limits 101 Digital rate options and applications to exotic swaps Complex Products : digital, steepner, ratchet… Bermuda swaptions Callables Appendixes : Standard Deviations vs Volatilities SABR Correlation issues LGM Risk-Management of exotics 102 Standard deviation versus volatility The diffusion process of a rate (short rate, zero-coupon, swap, whatever..) is typically : dr = (…)dt + s dz (« normal models »), or dr/r = (…)dt + s ’dz (« lognormal models ») It’s important to see the difference between s and s ’ : s is a standard deviation (often called volatility in working papers!) s ’ is a volatilty ! 103 Standard deviation versus volatility Central banks of developed countries tend to move short rates by 25 or 50 bp a few times in a year, whatever the level of short/long term rates So standard deviation is typically 0.50% to 1.30% Volatility is very well approximated (ATM) by : s ’ = s /r(0) r(0) initial value of r at date 0 Standard deviation is more stable across time for one currency So volatility tends to increase when rates go down, tends to decrease when rates go-up 104 Standard deviation versus volatility It’s important to keep in mind that what makes the price of interest rates vanilla options (caps/floors/swaptions) is not volatility, it’s standard deviation Of course same thing for exotics! So when comparing two currencies, in terms of cheapness of interest derivatives, look at standard deviations, not volatilities Same thing for two type of rates (deposits vs swaps, short maturity swaps vs long term swaps, short maturity options vs long maturity options…) 105 Standard deviation versus volatility Example as of 7/6/2005 (implied ATM volatilities) : maturity EURIBOR12M CMS10Y CMS2Y standard deviation Volatility standard deviation Volatility standard deviation Volatility EURIBOR 12M Euribor 12M CMS10Y CMS10Y CMS2Y CMS2Y 0.53 2.16% 3.41% 2.35% 0.46% 21.22% 0.61% 17.82% 0.55% 23.46% 1.53 2.54% 3.66% 2.74% 0.59% 23.13% 0.62% 17.13% 0.62% 22.58% 2.53 2.92% 3.87% 3.09% 0.63% 21.52% 0.62% 16.36% 0.64% 21.00% 3.53 3.23% 4.05% 3.39% 0.64% 19.87% 0.63% 15.81% 0.65% 19.37% 4.53 3.51% 4.20% 3.65% 0.65% 18.68% 0.63% 15.45% 0.65% 18.14% 5.53 3.74% 4.33% 3.88% 0.66% 17.74% 0.63% 15.08% 0.65% 17.08% 6.53 3.95% 4.43% 4.08% 0.66% 16.89% 0.63% 14.68% 0.64% 16.11% 7.54 4.11% 4.50% 4.22% 0.66% 16.11% 0.62% 14.30% 0.64% 15.35% 8.54 4.23% 4.55% 4.31% 0.65% 15.43% 0.61% 13.93% 0.62% 14.75% 9.54 4.29% 4.58% 4.37% 0.63% 14.86% 0.59% 13.57% 0.61% 14.21% 10.53 4.33% 4.61% 4.42% 0.62% 14.43% 0.58% 13.28% 0.60% 13.81% 11.53 4.38% 4.62% 4.46% 0.61% 14.04% 0.57% 13.06% 0.59% 13.53% 12.54 4.41% 4.63% 4.49% 0.60% 13.69% 0.56% 12.84% 0.58% 13.26% 13.54 4.43% 4.63% 4.52% 0.59% 13.35% 0.55% 12.62% 0.57% 12.98% 14.54 4.45% 4.62% 4.54% 0.57% 13.00% 0.54% 12.41% 0.56% 12.73% 15.54 4.45% 4.61% 4.53% 0.56% 12.81% 0.53% 12.29% 0.55% 12.59% 16.54 4.44% 4.59% 4.52% 0.56% 12.75% 0.52% 12.25% 0.55% 12.54% 17.54 4.43% 4.57% 4.51% 0.56% 12.70% 0.52% 12.21% 0.55% 12.47% 18.54 4.41% 4.55% 4.50% 0.55% 12.65% 0.51% 12.16% 0.54% 12.42% 19.55 4.38% 4.53% 4.46% 0.55% 12.61% 0.51% 12.12% 0.53% 12.41% 106 Standard deviation versus volatility 5.00% 25.00% 4.50% 4.00% 20.00% 3.50% EURIBOR12M CMS10Y 3.00% 15.00% CMS2Y standard deviation EURIBOR 12M standard deviation CMS10Y 2.50% standard deviation CMS2Y Volatility Euribor 12M 2.00% 10.00% Volatility CMS10Y Volatility CMS2Y 1.50% 1.00% 5.00% 0.50% 0.00% 0.00% 0 5 10 15 20 25 107 Standard deviation versus volatility maturity stdev/rates stdev/rates stdev/rates V olatility V olatility V olatility EURIBOR12M CMS10Y CMS2Y Euribor 12M CMS10Y CMS2Y 0.53 21.20% 17.75% 23.41% 21.22% 17.82% 23.46% 1.53 23.05% 16.96% 22.43% 23.13% 17.13% 22.58% 2.53 21.42% 16.10% 20.80% 21.52% 16.36% 21.00% 3.53 19.75% 15.47% 19.13% 19.87% 15.81% 19.37% 4.53 18.56% 15.03% 17.88% 18.68% 15.45% 18.14% 5.53 17.61% 14.59% 16.80% 17.74% 15.08% 17.08% 6.53 16.76% 14.14% 15.83% 16.89% 14.68% 16.11% 7.54 15.98% 13.72% 15.06% 16.11% 14.30% 15.35% 8.54 15.30% 13.32% 14.45% 15.43% 13.93% 14.75% 9.54 14.73% 12.93% 13.92% 14.86% 13.57% 14.21% 10.53 14.30% 12.62% 13.51% 14.43% 13.28% 13.81% 11.53 13.91% 12.37% 13.22% 14.04% 13.06% 13.53% 12.54 13.55% 12.12% 12.94% 13.69% 12.84% 13.26% 13.54 13.22% 11.89% 12.66% 13.35% 12.62% 12.98% 14.54 12.87% 11.66% 12.39% 13.00% 12.41% 12.73% 15.54 12.67% 11.52% 12.25% 12.81% 12.29% 12.59% 16.54 12.61% 11.44% 12.18% 12.75% 12.25% 12.54% 17.54 12.55% 11.35% 12.09% 12.70% 12.21% 12.47% 18.54 12.49% 11.27% 12.03% 12.65% 12.16% 12.42% 19.55 12.45% 11.19% 12.00% 12.61% 12.12% 12.41% 108 Standard deviation versus volatility 26.00% 24.00% 22.00% 20.00% stdev/rates EURIBOR12M 18.00% stdev/rates CMS10Y stdev/rates CMS2Y Volatility Euribor 12M 16.00% Volatility CMS10Y Volatility CMS2Y 14.00% 12.00% 10.00% 8.00% 0 5 10 15 20 25 109 Standard deviation versus volatility Remark : the approximation s ’ = s /r(0) is less good for CMS than for EURIBOR12M because implied volatilities were calculated before convexity adjustment. 110 Standard deviation versus volatility For risk management purpose, it’s important to have the possibility to adjust your smile modelling between : Full normal models (standard deviation constant when rates move) Full lognormal models (volatilities constant when rates move) Any intermediate version between the two above SABR model is a good choice 111 Digital rate options and applications to exotic swaps Complex Products : digital, steepner, ratchet… Bermuda swaptions Callables Appendixes : Standard Deviations vs Volatilities SABR Correlation issues LGM Risk-Management of exotics 112 SABR SABR: Stochastic, Alpha, Beta, Rho Dynamic of underlying : dF F dW d dZ dW , dZ dt with F forward rate or swap rate 113 SABR Call on forward rate, under proper measure, priced by Black- Scholes F 1 2 Ln BS T t C B t,T K F 2N d K N d d 1,2 1 2 BS T t With 4 parameters, we get the closed form formula for the implied Black & Scholes volatility : BS K,F 114 SABR x BS K,F 1 2 4 1 2 F 1 4 F y x FK 2 1 log log 24 K 1920 K 1 2 2 1 2 3 2 1 2 T 24 FK 1 4 1 24 and FK 2 1 F x FK 2 log K 1 2 x x 2 x y x log 1 115 SABR At the money formula: 1 2 2 1 2 3 2 F ,F 1 2 T BS F 1 24 F 2 2 4 F 1 24 One can choose between , , , parametrisation or replace by the ATM –normal or lognormal volatility. 116 SABR: the parameters : ATM volatility sigmaBeta 1% 2% 2,50% 3% 4% 45,00 40,00 35,00 30,00 25,00 20,00 15,00 10,00 5,00 - 0,0% 2,0% 4,0% 6,0% 8,0% 10,0% 12,0% 117 SABR: the parameters Beta For Beta=1, lognormal model. For beta = 0, normal model Enables to know how ATM vol moves when forward moves. d dF ATM 1 ATM F 118 SABR: the parameters For beta = 1, ATM vol doesn’t move when forward moves Beta = 1 25,00 24,50 24,00 23,50 Fwd 3% 23,00 Fwd 4% 22,50 Fwd 5% 22,00 21,50 21,00 20,50 20,00 0,0% 2,0% 4,0% 6,0% 8,0% 10,0% 12,0% 119 SABR: the parameters For beta =0 , ATM vol moves with F (constant standard deviation) Be ta = 0 25,00 23,00 21,00 19,00 Fw d 3% 17,00 Fw d 4% 15,00 Fw d 5% 13,00 11,00 9,00 7,00 5,00 0,0% 2,0% 4,0% 6,0% 8,0% 10,0% 12,0% 120 SABR: the parameters • Alpha or « vovol »: volatility convexity The bigger is Alpha the more pronouced is the convexity. Alp h a 1 0 % 2 0 % 3 0 % 3 0 ,0 0 2 5 ,0 0 2 0 ,0 0 1 5 ,0 0 1 0 ,0 0 5 ,0 0 1 ,0 % 2 ,0 % 3 ,0 % 4 ,0 % 5 ,0 % 6 ,0 % 7 ,0 % 8 ,0 % 9 ,0 % 1 0 ,0 % 1 1 ,0 % 121 SABR: the parameters • Rho : correlation between the volatility and the underlying causes what we call a Vanna skew : it’s the slope of the tangent line At the Money. Rho 0,1 0 -0,2 30,00 28,00 26,00 24,00 22,00 20,00 18,00 16,00 14,00 12,00 10,00 1,0% 2,0% 3,0% 4,0% 5,0% 6,0% 7,0% 8,0% 9,0% 10,0% 11,0% 122 SABR: risk management Delta: C BS BS BS F F BS F Different from classic delta ! Choose Beta to have stable hedge, so predict smile dynamic (trader work and skill !). Vega: it’s the sensivity of the price to change in the volatility. 123 SABR: risk management We can consider an ATM-LogNormal-Vega, an ATM-Normal-Vega or a Sigmabeta-Vega, depending on the shifted volatility type. Drawback of SABR : gives sensitivities with respect to the model parameters. Sometimes traders need their exposure by strike, need add B&S risk to SABR risk. 124 SABR: risk management Volga: C Hedged with strangles. If long vovol, buy straddle, sell strangle. 2 Vanna: C qui ressemble à C F hedged with collars. 125 SABR : calibration SABR model should provide a good fit to the observed implied volatility curves. Since SABR parameters have different and complementary effects on the smile, the calibration become easier and intuitive. Nevertheless, an exception has to be made for Beta and Rho. Indeed, those parameters have the same impact on the smile, and more precisely they both impact its skew. d ln ATM 1 d ln F 126 SABR : calibration To avoid over-parameterisation, the Beta is not calibrated but fixed from the smile roll Numerical calibration for other parameters 127 SABR : example : EUR 28/10/05 S I G M A 1 M 3 M 6 M 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 Y 8 Y 9 Y 1 0 Y 1 5 Y 2 0 Y 2 5 Y 3 0 Y 1 M 1 . 2 3 % 1 . 1 8 % 1 . 3 0 % 2 . 0 3 % 2 . 5 5 % 2 . 5 4 % 2 . 5 3 % 2 . 5 2 % 2 . 4 4 % 2 . 3 7 % 2 . 3 3 % 2 . 2 9 % 2 . 2 6 % 2 . 1 0 % 2 . 0 5 % 2 . 0 0 % 1 . 9 7 % 3 M 1 . 4 8 % 1 . 4 3 % 1 . 6 5 % 2 . 3 0 % 2 . 5 7 % 2 . 6 1 % 2 . 5 8 % 2 . 4 7 % 2 . 3 9 % 2 . 3 4 % 2 . 3 0 % 2 . 2 7 % 2 . 2 4 % 2 . 1 0 % 2 . 0 5 % 2 . 0 1 % 1 . 9 8 % 6 M 1 . 9 3 % 1 . 8 8 % 2 . 1 2 % 2 . 3 4 % 2 . 5 8 % 2 . 5 4 % 2 . 5 0 % 2 . 4 6 % 2 . 3 9 % 2 . 3 3 % 2 . 2 9 % 2 . 2 7 % 2 . 2 5 % 2 . 0 9 % 2 . 0 4 % 1 . 9 9 % 1 . 9 7 % 9 M 2 . 4 1 % 2 . 3 6 % 2 . 4 0 % 2 . 4 8 % 2 . 5 7 % 2 . 5 6 % 2 . 5 2 % 2 . 4 6 % 2 . 3 8 % 2 . 3 1 % 2 . 2 9 % 2 . 2 7 % 2 . 2 6 % 2 . 1 4 % 2 . 0 6 % 2 . 0 2 % 1 . 9 9 % 1 Y 2 . 4 9 % 2 . 4 4 % 2 . 4 4 % 2 . 6 6 % 2 . 6 0 % 2 . 5 8 % 2 . 5 3 % 2 . 4 6 % 2 . 4 2 % 2 . 3 9 % 2 . 3 5 % 2 . 3 2 % 2 . 2 8 % 2 . 1 5 % 2 . 0 8 % 2 . 0 4 % 2 . 0 1 % 2 Y 2 . 6 9 % 2 . 6 4 % 2 . 6 4 % 2 . 6 3 % 2 . 5 6 % 2 . 5 3 % 2 . 4 6 % 2 . 4 0 % 2 . 3 8 % 2 . 3 5 % 2 . 3 2 % 2 . 3 0 % 2 . 2 7 % 2 . 1 8 % 2 . 1 1 % 2 . 0 6 % 2 . 0 3 % 3 Y 2 . 6 2 % 2 . 6 0 % 2 . 6 0 % 2 . 5 9 % 2 . 5 3 % 2 . 4 7 % 2 . 4 2 % 2 . 3 7 % 2 . 3 5 % 2 . 3 1 % 2 . 3 0 % 2 . 2 8 % 2 . 2 6 % 2 . 1 9 % 2 . 1 2 % 2 . 0 8 % 2 . 0 4 % 4 Y 2 . 5 9 % 2 . 5 7 % 2 . 5 7 % 2 . 5 5 % 2 . 4 9 % 2 . 4 4 % 2 . 3 9 % 2 . 3 3 % 2 . 3 2 % 2 . 3 0 % 2 . 2 8 % 2 . 2 6 % 2 . 2 4 % 2 . 1 8 % 2 . 1 3 % 2 . 0 8 % 2 . 0 4 % 5 Y 2 . 5 6 % 2 . 5 4 % 2 . 5 4 % 2 . 5 1 % 2 . 4 3 % 2 . 3 8 % 2 . 3 2 % 2 . 3 0 % 2 . 2 8 % 2 . 2 4 % 2 . 2 5 % 2 . 2 3 % 2 . 2 2 % 2 . 1 7 % 2 . 1 2 % 2 . 0 7 % 2 . 0 3 % 7 Y 2 . 4 5 % 2 . 4 4 % 2 . 4 4 % 2 . 3 9 % 2 . 2 9 % 2 . 2 6 % 2 . 2 1 % 2 . 2 0 % 2 . 1 9 % 2 . 1 6 % 2 . 1 7 % 2 . 1 6 % 2 . 1 5 % 2 . 1 1 % 2 . 0 6 % 2 . 0 2 % 1 . 9 8 % 1 0 Y 2 . 3 0 % 2 . 2 9 % 2 . 2 9 % 2 . 2 7 % 2 . 1 5 % 2 . 1 3 % 2 . 1 0 % 2 . 0 7 % 2 . 0 7 % 2 . 0 7 % 2 . 0 7 % 2 . 0 7 % 2 . 0 7 % 2 . 0 1 % 1 . 9 5 % 1 . 9 1 % 1 . 8 8 % 1 5 Y 2 . 0 6 % 2 . 0 4 % 2 . 0 4 % 2 . 0 2 % 1 . 9 4 % 1 . 9 2 % 1 . 9 1 % 1 . 8 9 % 1 . 8 9 % 1 . 9 0 % 1 . 9 0 % 1 . 9 0 % 1 . 9 1 % 1 . 8 5 % 1 . 7 9 % 1 . 7 4 % 1 . 7 2 % 2 0 Y 1 . 8 4 % 1 . 8 3 % 1 . 8 3 % 1 . 8 1 % 1 . 7 6 % 1 . 7 7 % 1 . 7 7 % 1 . 7 7 % 1 . 7 8 % 1 . 7 9 % 1 . 8 0 % 1 . 8 1 % 1 . 8 2 % 1 . 7 6 % 1 . 6 8 % 1 . 6 5 % 1 . 6 4 % 2 5 Y 1 . 7 6 % 1 . 7 4 % 1 . 7 4 % 1 . 7 2 % 1 . 6 9 % 1 . 7 0 % 1 . 7 1 % 1 . 7 2 % 1 . 7 3 % 1 . 7 4 % 1 . 7 5 % 1 . 7 6 % 1 . 7 7 % 1 . 7 1 % 1 . 6 6 % 1 . 6 4 % 1 . 6 3 % 3 0 Y 1 . 6 8 % 1 . 6 7 % 1 . 6 7 % 1 . 6 4 % 1 . 6 2 % 1 . 6 4 % 1 . 6 6 % 1 . 6 8 % 1 . 6 9 % 1 . 7 0 % 1 . 7 1 % 1 . 7 2 % 1 . 7 3 % 1 . 7 0 % 1 . 6 6 % 1 . 6 5 % 1 . 6 4 % A L P H A 1 M 3 M 6 M 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 Y 8 Y 9 Y 1 0 Y 1 5 Y 2 0 Y 2 5 Y 3 0 Y 1 M 1 5 . 0 0 % 1 5 . 0 0 % 1 5 . 0 0 % 2 6 . 3 3 % 4 9 . 0 0 % 5 7 . 6 7 % 6 6 . 6 7 % 7 5 . 0 0 % 7 5 . 0 0 % 7 5 . 0 0 % 7 5 . 0 0 % 7 5 . 0 0 % 7 5 . 0 0 % 7 0 . 0 0 % 6 5 . 0 0 % 6 5 . 0 0 % 6 5 . 0 0 % 3 M 1 7 . 5 0 % 1 7 . 5 0 % 1 7 . 5 0 % 2 8 . 3 3 % 5 0 . 0 0 % 5 8 . 3 3 % 6 6 . 6 7 % 7 5 . 0 0 % 7 5 . 0 0 % 7 5 . 0 0 % 7 5 . 0 0 % 7 5 . 0 0 % 7 5 . 0 0 % 7 0 . 0 0 % 6 5 . 0 0 % 6 5 . 0 0 % 6 5 . 0 0 % 6 M 2 0 . 0 0 % 2 0 . 0 0 % 2 0 . 0 0 % 2 9 . 2 2 % 4 7 . 6 7 % 5 4 . 2 2 % 5 9 . 7 8 % 6 5 . 3 3 % 6 4 . 8 7 % 6 4 . 4 0 % 6 3 . 9 3 % 6 3 . 4 7 % 6 3 . 0 0 % 5 9 . 8 3 % 5 6 . 6 7 % 5 6 . 6 7 % 5 6 . 6 7 % 9 M 2 3 . 0 0 % 2 3 . 0 0 % 2 3 . 0 0 % 3 0 . 7 8 % 4 6 . 3 3 % 5 0 . 1 1 % 5 2 . 8 9 % 5 5 . 6 7 % 5 4 . 7 3 % 5 3 . 8 0 % 5 2 . 8 7 % 5 1 . 9 3 % 5 1 . 0 0 % 4 9 . 6 7 % 4 8 . 3 3 % 4 8 . 3 3 % 4 8 . 3 3 % 1 Y 3 4 . 0 0 % 3 4 . 0 0 % 3 4 . 0 0 % 3 8 . 0 0 % 4 6 . 0 0 % 4 6 . 0 0 % 4 6 . 0 0 % 4 6 . 0 0 % 4 4 . 6 0 % 4 3 . 2 0 % 4 1 . 8 0 % 4 0 . 4 0 % 3 9 . 0 0 % 3 9 . 5 0 % 4 0 . 0 0 % 4 0 . 0 0 % 4 0 . 0 0 % 2 Y 3 9 . 5 0 % 3 9 . 5 0 % 3 9 . 5 0 % 4 0 . 2 9 % 4 1 . 8 8 % 4 2 . 7 1 % 4 2 . 5 4 % 4 2 . 3 8 % 4 1 . 3 8 % 4 0 . 3 8 % 3 9 . 3 8 % 3 8 . 3 8 % 3 7 . 3 8 % 3 7 . 6 3 % 3 7 . 8 8 % 3 7 . 8 8 % 3 7 . 8 8 % 3 Y 4 1 . 5 0 % 4 1 . 5 0 % 4 1 . 5 0 % 4 0 . 5 8 % 3 8 . 7 5 % 3 9 . 4 2 % 3 9 . 0 8 % 3 8 . 7 5 % 3 8 . 1 5 % 3 7 . 5 5 % 3 6 . 9 5 % 3 6 . 3 5 % 3 5 . 7 5 % 3 5 . 7 5 % 3 5 . 7 5 % 3 5 . 7 5 % 3 5 . 7 5 % 4 Y 4 0 . 5 0 % 4 0 . 5 0 % 4 0 . 5 0 % 3 8 . 8 8 % 3 5 . 6 3 % 3 6 . 1 3 % 3 5 . 6 3 % 3 4 . 1 3 % 3 4 . 9 3 % 3 4 . 7 3 % 3 4 . 5 3 % 3 4 . 3 3 % 3 4 . 1 3 % 3 3 . 8 8 % 3 3 . 6 3 % 3 3 . 6 3 % 3 3 . 6 3 % 5 Y 3 7 . 0 0 % 3 7 . 0 0 % 3 7 . 0 0 % 3 5 . 8 3 % 3 3 . 5 0 % 3 2 . 8 3 % 3 2 . 1 7 % 3 1 . 5 0 % 3 1 . 7 0 % 3 1 . 9 0 % 3 2 . 1 0 % 3 2 . 3 0 % 3 2 . 5 0 % 3 2 . 0 0 % 3 1 . 5 0 % 3 1 . 5 0 % 3 1 . 5 0 % 7 Y 3 2 . 0 0 % 3 2 . 0 0 % 3 2 . 0 0 % 3 1 . 9 3 % 3 1 . 8 0 % 3 1 . 4 0 % 3 1 . 0 0 % 3 0 . 6 0 % 3 0 . 5 4 % 3 0 . 4 8 % 3 0 . 4 2 % 3 0 . 3 6 % 3 0 . 3 0 % 2 9 . 9 0 % 2 9 . 5 0 % 2 9 . 5 0 % 2 9 . 5 0 % 1 0 Y 2 8 . 5 0 % 2 8 . 5 0 % 2 8 . 5 0 % 2 8 . 7 5 % 2 9 . 2 5 % 2 9 . 2 5 % 2 9 . 2 5 % 2 9 . 2 5 % 2 8 . 8 0 % 2 8 . 3 5 % 2 7 . 9 0 % 2 7 . 4 5 % 2 7 . 0 0 % 2 6 . 7 5 % 2 6 . 5 0 % 2 6 . 5 0 % 2 6 . 5 0 % 1 5 Y 2 5 . 5 0 % 2 5 . 5 0 % 2 5 . 5 0 % 2 6 . 3 5 % 2 8 . 0 6 % 2 7 . 8 1 % 2 7 . 5 6 % 2 7 . 3 1 % 2 6 . 8 0 % 2 6 . 2 9 % 2 5 . 7 8 % 2 5 . 2 6 % 2 4 . 7 5 % 2 4 . 5 6 % 2 4 . 3 8 % 2 4 . 3 8 % 2 4 . 3 8 % 2 0 Y 2 5 . 0 0 % 2 5 . 0 0 % 2 5 . 0 0 % 2 5 . 6 3 % 2 6 . 8 8 % 2 6 . 3 8 % 2 5 . 8 8 % 2 5 . 3 8 % 2 4 . 8 0 % 2 4 . 2 3 % 2 3 . 6 5 % 2 3 . 0 8 % 2 2 . 5 0 % 2 2 . 3 8 % 2 2 . 2 5 % 2 2 . 2 5 % 2 2 . 2 5 % 2 5 Y 2 4 . 5 0 % 2 4 . 5 0 % 2 4 . 5 0 % 2 4 . 9 0 % 2 5 . 6 9 % 2 4 . 9 4 % 2 4 . 1 9 % 2 3 . 4 4 % 2 2 . 8 0 % 2 2 . 1 6 % 2 1 . 5 3 % 2 0 . 8 9 % 2 0 . 2 5 % 2 0 . 1 9 % 2 0 . 1 3 % 2 0 . 1 3 % 2 0 . 1 3 % 3 0 Y 2 4 . 0 0 % 2 4 . 0 0 % 2 4 . 0 0 % 2 4 . 1 7 % 2 4 . 5 0 % 2 3 . 5 0 % 2 2 . 5 0 % 2 1 . 5 0 % 2 0 . 8 0 % 2 0 . 1 0 % 1 9 . 4 0 % 1 8 . 7 0 % 1 8 . 0 0 % 1 8 . 0 0 % 1 8 . 0 0 % 1 8 . 0 0 % 1 8 . 0 0 % R H O 1 M 3 M 6 M 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 Y 8 Y 9 Y 1 0 Y 1 5 Y 2 0 Y 2 5 Y 3 0 Y 1 M 3 3 . 0 0 % 3 3 . 0 0 % 3 3 . 0 0 % 3 3 . 6 7 % 3 5 . 0 0 % 3 0 . 0 0 % 2 5 . 0 0 % 2 0 . 0 0 % 1 8 . 0 0 % 1 6 . 0 0 % 1 4 . 0 0 % 1 2 . 0 0 % 1 0 . 0 0 % 5 . 2 5 % 0 . 5 0 % 0 . 5 0 % 0 . 5 0 % 3 M 3 3 . 0 0 % 3 3 . 0 0 % 3 3 . 0 0 % 3 3 . 6 7 % 3 5 . 0 0 % 3 0 . 0 0 % 2 5 . 0 0 % 2 0 . 0 0 % 1 8 . 0 0 % 1 6 . 0 0 % 1 4 . 0 0 % 1 2 . 0 0 % 1 0 . 0 0 % 5 . 2 5 % 0 . 5 0 % 0 . 5 0 % 0 . 5 0 % 6 M 3 3 . 0 0 % 3 3 . 0 0 % 3 3 . 0 0 % 3 2 . 5 6 % 3 1 . 6 7 % 2 7 . 0 0 % 2 2 . 3 3 % 1 7 . 6 7 % 1 1 . 3 0 % 9 . 6 0 % 7 . 9 0 % 6 . 2 0 % 8 . 1 7 % 1 . 2 5 % - 0 . 3 0 % - 3 . 5 0 % - 1 . 3 0 % 9 M 3 3 . 0 0 % 3 3 . 0 0 % 3 3 . 0 0 % 3 1 . 4 4 % 2 8 . 3 3 % 2 4 . 0 0 % 1 9 . 6 7 % 1 5 . 3 3 % 1 1 . 3 0 % 9 . 6 0 % 7 . 9 0 % 6 . 2 0 % 6 . 3 3 % 1 . 2 5 % - 1 . 2 0 % - 3 . 5 0 % - 3 . 2 0 % 1 Y 3 1 . 0 0 % 3 1 . 0 0 % 3 1 . 0 0 % 2 9 . 0 0 % 2 5 . 0 0 % 2 1 . 0 0 % 1 7 . 0 0 % 1 3 . 0 0 % 1 1 . 3 0 % 9 . 6 0 % 7 . 9 0 % 6 . 2 0 % 4 . 5 0 % 1 . 2 5 % - 2 . 0 0 % - 3 . 5 0 % - 5 . 0 0 % 2 Y 2 3 . 0 0 % 2 3 . 0 0 % 2 3 . 0 0 % 2 2 . 5 8 % 2 1 . 7 5 % 1 7 . 4 2 % 1 3 . 0 8 % 8 . 7 5 % 7 . 3 8 % 6 . 0 0 % 4 . 6 3 % 3 . 2 5 % 1 . 8 8 % - 0 . 9 0 % - 3 . 8 0 % - 5 . 0 0 % - 6 . 3 0 % 3 Y 2 0 . 5 0 % 2 0 . 5 0 % 2 0 . 5 0 % 1 9 . 8 3 % 1 8 . 5 0 % 1 3 . 8 3 % 9 . 1 7 % 4 . 5 0 % 3 . 4 5 % 2 . 4 0 % 1 . 3 5 % 0 . 3 0 % - 0 . 8 0 % - 3 . 1 0 % - 5 . 5 0 % - 6 . 5 0 % - 7 . 5 0 % 4 Y 2 0 . 0 0 % 2 0 . 0 0 % 2 0 . 0 0 % 1 8 . 4 2 % 1 5 . 2 5 % 1 0 . 2 5 % 5 . 2 5 % 0 . 2 5 % - 0 . 5 0 % - 1 . 2 0 % - 1 . 9 0 % - 2 . 7 0 % - 3 . 4 0 % - 5 . 3 0 % - 7 . 3 0 % - 8 . 0 0 % - 8 . 8 0 % 5 Y 2 0 . 0 0 % 2 0 . 0 0 % 2 0 . 0 0 % 1 7 . 3 3 % 1 2 . 0 0 % 6 . 6 7 % 1 . 3 3 % - 4 . 0 0 % - 4 . 4 0 % - 4 . 8 0 % - 5 . 2 0 % - 5 . 6 0 % - 6 . 0 0 % - 7 . 5 0 % - 9 . 0 0 % - 9 . 5 0 % - 1 0 . 0 0 % 7 Y 2 0 . 0 0 % 2 0 . 0 0 % 2 0 . 0 0 % 1 6 . 1 5 % 8 . 4 4 % 3 . 8 9 % - 0 . 7 0 % - 5 . 2 0 % - 5 . 5 0 % - 5 . 8 0 % - 6 . 2 0 % - 6 . 5 0 % - 6 . 8 0 % - 8 . 3 0 % - 9 . 8 0 % - 1 0 . 3 0 % - 1 0 . 8 0 % 1 0 Y 1 8 . 5 0 % 1 8 . 5 0 % 1 8 . 5 0 % 1 3 . 3 7 % 3 . 1 0 % - 0 . 3 0 % - 3 . 6 0 % - 7 . 0 0 % - 7 . 2 0 % - 7 . 4 0 % - 7 . 6 0 % - 7 . 8 0 % - 8 . 0 0 % - 9 . 5 0 % - 1 1 . 0 0 % - 1 1 . 5 0 % - 1 2 . 0 0 % 1 5 Y 1 3 . 0 0 % 1 3 . 0 0 % 1 3 . 0 0 % 9 . 4 4 % 2 . 3 3 % - 0 . 3 0 % - 3 . 6 0 % - 8 . 0 0 % - 7 . 2 0 % - 7 . 4 0 % - 7 . 6 0 % - 7 . 8 0 % - 9 . 9 0 % - 1 1 . 3 0 % - 1 2 . 6 0 % - 1 3 . 1 0 % - 1 3 . 5 0 % 2 0 Y 7 . 0 0 % 7 . 0 0 % 7 . 0 0 % 5 . 1 8 % 1 . 5 5 % - 0 . 3 0 % - 3 . 6 0 % - 9 . 0 0 % - 7 . 2 0 % - 7 . 4 0 % - 7 . 6 0 % - 7 . 8 0 % - 1 1 . 8 0 % - 1 3 . 0 0 % - 1 4 . 3 0 % - 1 4 . 6 0 % - 1 5 . 0 0 % 2 5 Y 3 . 0 0 % 3 . 0 0 % 3 . 0 0 % 2 . 2 6 % 0 . 7 8 % - 0 . 3 0 % - 3 . 6 0 % - 1 0 . 0 0 % - 7 . 2 0 % - 7 . 4 0 % - 7 . 6 0 % - 7 . 8 0 % - 1 3 . 6 0 % - 1 4 . 8 0 % - 1 5 . 9 0 % - 1 6 . 2 0 % - 1 6 . 5 0 % 3 0 Y 1 . 0 0 % 1 . 0 0 % 1 . 0 0 % 0 . 6 7 % 0 . 0 0 % - 0 . 3 0 % - 3 . 6 0 % - 1 1 . 0 0 % - 7 . 2 0 % - 7 . 4 0 % - 7 . 6 0 % - 7 . 8 0 % - 1 5 . 5 0 % - 1 6 . 5 0 % - 1 7 . 5 0 % - 1 7 . 8 0 % - 1 8 . 0 0 % B E T A 1 M 3 M 6 M 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 Y 8 Y 9 Y 1 0 Y 1 5 Y 2 0 Y 2 5 Y 3 0 Y 1 M 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 3 M 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 6 M 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 9 M 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 1 Y 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 2 Y 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 3 Y 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 4 Y 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 5 Y 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 7 Y 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 1 0 Y 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 1 5 Y 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 2 0 Y 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 2 5 Y 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 3 0 Y 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 0 . 4 Vertical axis : maturity of option, Horizontal axis: underlying 128 SABR: conclusion and drawback Good parametrisation of market smile Easy numerical calibration Parameters interpretation in term of risk-management Main drawback : each underlying modeled separately : so can be inconsistent for some exotic products main application of risk-management is for vanilla products. 129 Digital rate options and applications to exotic swaps Complex Products : digital, steepner, ratchet… Bermuda swaptions Callables Appendixes : Standard Deviations vs Volatilities SABR Correlation issues LGM Risk-Management of exotics 130 Correlations issues Example of correlation matrix : in red tenor, expiry = maturity of the options expiry 1y 3m 6m 1y 2y 5y 10y 15y 20y 30y 3m 1.00 0.90 0.74 0.67 0.46 0.43 0.41 0.38 0.37 6m 0.90 1.00 0.79 0.71 0.49 0.44 0.42 0.38 0.38 1y 0.74 0.79 1.00 0.87 0.83 0.76 0.73 0.68 0.62 2y 0.67 0.71 0.87 1.00 0.89 0.86 0.79 0.72 0.69 5y 0.46 0.49 0.83 0.89 1.00 0.92 0.91 0.85 0.84 10y 0.43 0.44 0.76 0.86 0.92 1.00 0.93 0.90 0.88 15y 0.41 0.42 0.73 0.79 0.91 0.93 1.00 0.92 0.92 20y 0.38 0.38 0.68 0.72 0.85 0.90 0.92 1.00 0.95 30y 0.37 0.38 0.62 0.69 0.84 0.88 0.92 0.95 1.00 expiry 2y 3m 6m 1y 2y 5y 10y 15y 20y 30y 3m 1.00 0.92 0.79 0.73 0.57 0.54 0.53 0.50 0.50 6m 0.92 1.00 0.83 0.76 0.60 0.56 0.54 0.51 0.51 1y 0.79 0.83 1.00 0.89 0.85 0.79 0.77 0.73 0.66 2y 0.73 0.76 0.89 1.00 0.89 0.86 0.81 0.76 0.72 5y 0.57 0.60 0.85 0.89 1.00 0.92 0.91 0.88 0.87 10y 0.54 0.56 0.79 0.86 0.92 1.00 0.93 0.91 0.89 15y 0.53 0.54 0.77 0.81 0.91 0.93 1.00 0.93 0.93 20y 0.50 0.51 0.73 0.76 0.88 0.91 0.93 1.00 0.95 30y 0.50 0.51 0.66 0.72 0.87 0.89 0.93 0.95 1.00 expiry 5y 3m 6m 1y 2y 5y 10y 15y 20y 30y 3m 1.00 0.96 0.88 0.85 0.76 0.75 0.74 0.72 0.72 6m 0.96 1.00 0.91 0.87 0.79 0.77 0.76 0.74 0.74 1y 0.88 0.91 1.00 0.93 0.87 0.85 0.83 0.82 0.75 2y 0.85 0.87 0.93 1.00 0.89 0.87 0.85 0.82 0.78 5y 0.76 0.79 0.87 0.89 1.00 0.94 0.93 0.91 0.91 10y 0.75 0.77 0.85 0.87 0.94 1.00 0.95 0.93 0.92 15y 0.74 0.76 0.83 0.85 0.93 0.95 1.00 0.95 0.94 20y 0.72 0.74 0.82 0.82 0.91 0.93 0.95 1.00 0.96 30y 0.72 0.74 0.75 0.78 0.91 0.92 0.94 0.96 1.00 131 Correlations issues expiry 7y 3m 6m 1y 2y 5y 10y 15y 20y 30y 3m 1.00 0.97 0.90 0.88 0.81 0.80 0.79 0.78 0.78 6m 0.97 1.00 0.93 0.89 0.83 0.82 0.81 0.80 0.79 1y 0.90 0.93 1.00 0.93 0.88 0.86 0.85 0.84 0.79 2y 0.88 0.89 0.93 1.00 0.89 0.87 0.85 0.84 0.81 5y 0.81 0.83 0.88 0.89 1.00 0.94 0.93 0.92 0.92 10y 0.80 0.82 0.86 0.87 0.94 1.00 0.95 0.94 0.93 15y 0.79 0.81 0.85 0.85 0.93 0.95 1.00 0.96 0.95 20y 0.78 0.80 0.84 0.84 0.92 0.94 0.96 1.00 0.96 30y 0.78 0.79 0.79 0.81 0.92 0.93 0.95 0.96 1.00 expiry 10y 3m 6m 1y 2y 5y 10y 15y 20y 30y 3m 1.00 0.98 0.91 0.90 0.83 0.82 0.81 0.80 0.80 6m 0.98 1.00 0.94 0.91 0.85 0.84 0.83 0.82 0.82 1y 0.91 0.94 1.00 0.94 0.88 0.87 0.86 0.85 0.82 2y 0.90 0.91 0.94 1.00 0.89 0.87 0.86 0.85 0.83 5y 0.83 0.85 0.88 0.89 1.00 0.94 0.93 0.93 0.93 10y 0.82 0.84 0.87 0.87 0.94 1.00 0.95 0.94 0.93 15y 0.81 0.83 0.86 0.86 0.93 0.95 1.00 0.96 0.95 20y 0.80 0.82 0.85 0.85 0.93 0.94 0.96 1.00 0.96 30y 0.80 0.82 0.82 0.83 0.93 0.93 0.95 0.96 1.00 132 Correlations issues Let’s note T ,m the value at date t of CMS m years CMS t fixed at T Two type of correlations between rates, let’s say CMS for example : Instantaneous correlations : T ,l T ,m dCMS t , dCMS t T ,l T ,l T ,m T ,m dCMS t , dCMS t dCMS t , dCMS t Term correlations : T ,l T ,m Cov CMST ,CMST T ,l T ,m Var CMST Var CMST 133 Correlations issues Remarks Term correlations can be seen as a kind of average of instantaneous correlations (not true as volatilities are not constant) : the instantaneous historical correlation of forwards swaps at horizon ? of maturities ? + t and ? + t’ giving and estimate of the correlation for maturity T-?. Think to the risk managament of a 5Y CMS 10Y/2Y spread option, first you will be exposed to the correlation between 10Y/2Y CMS in 5Years, in 4 years, …in 1years and finally to the correlation between CMS10Y and 2Y. 134 Correlations issues For derivatives pricing/risk management (example : spread options pricing), of course only term correlations are relevant. 135 Correlations issues Calculations : Method 1: using the fact that correlations between zero-coupon rates can be calculated exactly by analytical formula, express the swap as a function of zero-coupon rates and set the sensitivities of a swap as deterministic numbers Method 2 : B t,T B t,Tm swap t,T ,m S t,T ,m m B t,Ti i 1 136 Correlations issues in LGM2F A sketch of calculations : 1 2 dB t,T 1 t,T B t,T dW t 2 t,T B t,T dW t 1 2 dB t,Tm 1 t,Tm B t,Tm dWt 2 t,Tm B t,Tm dWt 1 d B t ,T B t ,T m 1 t ,T B t ,T 1 t ,T m B t ,T m dW t 2 2 t ,T B t ,T 2 t ,T m B t ,T m dW t 137 Correlations issues in LGM2F m 1 t,Ti B t,Ti dS t,T ,m 1 t,T B t,T 1 t,Tm B t,Tm i 1 1 m dW t S t,T ,m B t,T B t,Tm B t,Ti i 1 m 2 t,Ti B t,Ti 2 t,T B t,T 2 t,Tm B t,Tm i 1 2 m dW t B t,T B t,Tm B t,Ti i 1 138 Correlations issues in LGM2F Formula : T 2 T 2 2 1 t 2 t 2 2 t 2 t Var Ln S T ,m f m 2 e dt f m 2 e dt 0 0 T 1 t 2 t t 2 f m f m e dt 0 T 2 T 2 1 t 2 t 2 t 2 t Cov Ln S T ,m ,Ln S T ,n f m f n 2 e dt f m f n 2 e dt 0 0 T 1 t 2 t t f m f n f m f n e dt 0 139 Correlations issues in LGM2F By setting again some stochastic numbers as constant, we get again the correlations The two methods give exactly the same results ! 140 Correlations issues in LGM2F If you take two independants brownians motions, and look at term correlations between two CMS (for instance CMS2Y and CMS10Y, a canonical example for spreads products) these correlations tend to degenerate very quick to a value very close to 1 By choosing a correlation between the two brownians very negative (typically -0.8), you can address this problem 141 Correlations issues in LGM2F Example with constant volatilities, just for illustration purpose : T1 2 0.15 0.34 T2 10 correl -0.8 sigma1 0.65% sigma2 1.17% Correl swaps 2Y/swap10Y 1.0000 0.9800 0.9600 LGMwithout correl 0.9400 LGMwith correl 0.9200 0.9000 0.8800 0.8600 0.8400 0.8200 0.25 2.25 4.25 6.25 8.25 10.25 12.25 14.25 16.25 18.25 142 Digital rate options and applications to exotic swaps Complex Products : digital, steepner, ratchet… Bermuda swaptions Callables Appendixes : Standard Deviations vs Volatilities SABR Correlation issues LGM Risk-Management of exotics 143 IR models : reminder on Models of the short rate Vasicek model (1977) dr a(b r)dt , dzt z standard brownian motion s standard deviation of the short rate b long term level of r, a mean reversion Analytical formulas for today and any future date yield curve Analytical formulas for european options on coupon bearing bonds Possibility of negative rates (normal model) 144 IR models : reminder on Models of the short rate Cox Ingersoll (1985) dr a(b r)dt rdz , t Rates are always nonnegative As the short rate increase, its standard déviation increases 145 IR models : reminder on Models of the short rate Main drawback of these models : modeling only the dynamic of the short rate : Yield curve today and in the future depends only on the short rate today Models parameters Impossibility to fit the yield curve today ! Risk of mispricing ! 146 How to solve this problem ? Hull & White extended version of Vasicek model (1990) : dr=(?(t)-ar)dt+sdz Closed formulas for all vanilla derivatives (caps, swaptions, european bond options) ?(t) enables to fit exactly the today yield curve Easy Tree implementation for american, some callable products 147 How to solve this problem ? Ho & Lee (1985 ) and Heath-Jarrow Morton (1987-1992 ) Ho & Lee is essentially a discrete version of HJM See Jamshidian (1991) for discrete version Focus on HJM, and in fact LGM 148 HJM framework : a few equations… Main idea : Use the market yield curve as a starting point and model its dynamic over time, under the constraint that no arbitrage is possible The general equation of the model + the absence of arbitrage opportunities leads to the existence of a risk- neutral probability Q under which the dynamics of zero- coupon prices is : dB(t,T ) r dt (t,T ), dW B(t,T ) t t t,T vector of local volatilities Wt multidimensional brownian motion, components have 0 average, independan ts or not 149 HJM framework : a few equations… The general framework also gives : If we note, L n B thte, Tforward f t,T spot rate we get : T t t f t,T f 0,T s,T , dWs s,T , s,T 0 0 s,T T s,T Especially, for the short rate , we have : rt f t,t t t rt f 0,t s,t dWs s,t s,t ds 0 0 150 HJM framework : a few equations… By eliminating the short rate in the starting equation, one also gets another very important equation : t t B 0,T 1 2 2 B t,T exp s,T s,t dWs s,T s,t ds B 0,t 0 2 0 151 LGM framework HJM is a very general framework : for practical implementation and use, need more specifications LGM (Linear Gaussian Markov model) 152 LGM framework A gaussian HJM is a model on zero-coupon bonds : the zero- coupon follow lognormal laws under Q. In other words, the volatility of the zero-coupon bonds is deterministic under Q, and thus under all probabilities. Need for markovian models for getting simpler numerical procedures (trees, PDE or Monte-Carlo) It can be shown that to have gaussian and Markovian feature, volatility must be restricted to exponentialy decaying-functions : i t,T fi (t)exp gi T t 153 LGM framework In practice, a good choice for volatilities is : t,T 1 t,T , , n t,T with k t k 1 exp k T t k called mean-reversion parameters, are positive constant k as the instantaneous volatilities, piecewise constant k 154 LGM framework The control the amortizing of the volatility : the larger , the k k smaller the volatility induced by factor k. 155 LGM-1F model features LGM-1F properties Gaussian instantaneous forward rates The model is fully determined by the mean reversion and the deterministic volatility (t) (supposed piecewise constant) If the mean reversion is positive, forwards of long maturity will be less volatilile than forwards of short maturity. Lognormal discount factors: dB(t,T ) r dt t,T dW B(t,T ) t t t t,T 1 e T t 156 LGM-1F model features LGM-1F properties Gaussian short rate, mean reverting Forward Libor are shifted lognormal (constant = 1/coverage shift) 157 LGM-1F model features Reconstruction formula for zero-coupon bonds : the whole dynamics of the curve can be summarized by a single gaussian state variable : X t rt f 0,t See previous slide to see that is a gaussian variable. X t Remark that : 158 LGM-1F model features Then : t 1 e T t T t 2 (s)e 2 t s ds, t, T e u t du 0 t Fundamental equation for all numerical methods. This is why we speak of Linear models : zero-coupon bonds can always be seen as exponential of Linear sum of Gaussian state variables (whatever the number of factors) So zero-coupon rates are Linear sum of these state variables 159 LGM-1F model : vanilla pricing Analytical formula for vanilla products Straitghtforward B&S formula for caps & floors as Libor are shiftted Lognormal 160 LGM-1F model : vanilla pricing For swaptions, let’s define call bond option as a derivative of pay-off : n ci B T,Ti KB T,T0 i 1 where K 0, T T0 T1 Tn i 1, ,n ,ci 0 T is the maturity of the option, T 0 is the start date of calibration product 161 LGM-1F model : vanilla pricing The pricing of call and put Bond option is analytical in LGM1F : n bs CallBO T ,T0 , K, Ti 1 n , ci 1 n B 0,T0 ci BScall Fi ,0, i , Ki ,T i 1 n bs PutBO T ,T0 , K, Ti 1 n , ci 1 n B 0,T0 ci BSput Fi ,0, i , Ki ,T i 1 T0 T Ti T B 0,Ti bs e e Fi , i T B 0,T0 T x0 and Ki defined by : n B x B ( x ) c T ,Ti 0 K, and K T ,Ti 0 i B (x ) i B (x ) i 1 T ,T0 0 T ,T0 0 162 LGM-1F model : vanilla pricing We use the reconstruction formula for these calculations, especially for the last equations. 163 LGM-1F model : vanilla pricing Then a payer swaption can be seen as a modified put option : Payer swaption ~ payer swaption Te , T0 ,Tn , K PutBO Te ,T0 , K,(Ti )1 n, ci 1 n ci i K for i 1, ,n -1 cn 1 n K ~ K 1 Receiver swaption is obtained as a call bond option with same parameters. 164 LGM-1F model : calibration Bootstrap calibration procedure Bootstrap calibration, fast & exact Procedure = calibration on a set of caplets/swaptions (possibly mixed) with strictly increasing expiries (i.e. one instrument by expiry date) Choose a mean reversion , for example to match market bermuda price as bermuda are liquid 165 LGM-1F model : calibration i = so called instantaneous volatility between Ti 1 and Ti , used to match the price of option ' of maturity Ti on underlying of maturity Ti instrument (libor for caplet or swap for a swaption ) for option of maturity Ti , starts at Ti , ' ends at Ti 1 2 i 3 ….. ' T1 ' T2 T1 T0 0 T2 T3 …… 166 LGM-1F model : calibration Step 1 : calibration 1 of in order to match market price 1 of the option (swaption or caplet) of maturity T1 , on instrument (ex : 3M T ' 1 or 10Y) such that : T start date 1 T1 T1 ' T0 0 End date T1 167 LGM-1F model : calibration 2 Step 2 : calibration of in order to match 2 market price of the option (swaption or caplet) of maturity T 2 T ' 2 ,on instrument such that : T2 start date ' T0 0 T1 T2 End date T2 168 LGM-1F model : calibration Each instrument provides the variance of the state variable up to its expiry date The short rate volatility (t), supposed piecewise constant, is then deduced iteratively 169 LGM-1F model calibration One can iterate the above process to calibrate the mean reversion to a basket of instruments (e.g. to a cap) Depending on the product, we can use a diagonal of swaption (9Y in 1Y, 8Y in 2Y, …1Y in 9Y), for instance for bermudean swaptions ; the strike being the strike of bermudean swaptions The set of the calibration is choosen for each exotic product 170 LGM-1F model : calibration example Calibration of LGM1F on ATM caplets 23/05/05 EURO 0.15 0 . 25 0 . 35 Expiry EURIBOR 3MBS_Vol sigma_i sigma_i 0.252 2.15% 6.92% 0.16% 0.16% 0.16% 0.504 2.20% 11.52% 0.34% 0.35% 0.36% 1.000 2.43% 20.25% 0.70% 0.73% 0.76% 2.000 2.83% 20.73% 0.78% 0.86% 0.93% 3.002 3.11% 20.44% 0.92% 1.04% 1.15% 4.000 3.37% 19.25% 0.97% 1.13% 1.28% 5.000 3.61% 18.14% 1.04% 1.23% 1.41% 6.000 3.83% 17.17% 1.09% 1.32% 1.52% 7.002 4.00% 16.25% 1.12% 1.37% 1.59% 8.000 4.15% 15.61% 1.17% 1.44% 1.68% 9.000 4.25% 15.02% 1.18% 1.48% 1.73% 10.000 4.31% 14.50% 1.19% 1.50% 1.77% 15.002 4.44% 12.85% 1.26% 1.62% 1.93% 20.000 4.35% 12.18% 1.33% 1.72% 2.06% 25.000 4.14% 11.85% 1.37% 1.78% 2.12% 30.000 4.05% 11.75% 1.46% 1.88% 2.25% 171 LGM-1F model : calibration example LGM1F and market smiles as of 23/5/2005 calibration on ATM 3M caplets 48.00% 43.00% 38.00% 33.00% LGM 1Y LGM 3Y LGM 5Y 28.00% market 1Y market 3Y market 5Y 23.00% 18.00% 13.00% 8.00% 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 172 LGM-1F model features Effect of mean reversion on non calibrated instruments Result: If calibration to a short underlying (e.g. a caplet), the price of a long underlying instrument (e.g. a 15Y-term swaption) of same expiry will decrease as the mean reversion increases Analogously, if calibration to a long swaption, the price of a caplet of same expiry will increase as lambda increases. 173 LGM-1F model features Effect of mean reversion on non calibrated instruments (continued) Example : calibrate to a caplet (expiry = 2Y, term = 1Y), and graph the cumulative vol of B(t,2Y+x) / B(t, 2Y) on [0, 2Y] for various values of mean reversion . calibrate to a long swaption (expiry = 2Y, term = 20Y), and graph the cumulative vol of B(t,2Y+x) / B(t, 2Y) on [0, 2Y] for various values of mean reversion . 174 LGM-1F model features Calibration on caplet Calibration on a 20Y-term swaption Cumulative vol of B(t, T+x) / B(t,T) Cumulative vol of B(t, T+x) / B(t,T) 25.00% 20.00% lambda = 0.01 20.00% lambda = 0.05 lambda = 0.10 15.00% lambda = 0.10 15.00% 10.00% 10.00% 5.00% 5.00% 0.00% 0.00% 0 5 10 15 20 0 5 10 15 20 x (years) x (years) 175 LGM-1F model features Effect of mean reversion on forward volatilities Assume that we have calibrated to the swaption (Tstart, Tend) (see below). Then the cumulative volatility of the model remains the same for any value of the mean reversion. 176 LGM-1F model features But because of the exponential shape of the volatility, the forward volatility increases as the mean reversion increases. In the LGM-1F framework, the mean reversion controls the “repartition” of the volatility through time. mid-curve vol Forward vol Today = T 0 Tf Tst Ten art d 177 LGM-1F model features Effect of mean reversion on forward volatilities (continued) Calibration on a 20Y caplet. The graph below represents the instantaneous vol of B(t, 20Y) / B(t, 21Y) for several values of mean reversion. 0.030 lambda = 0.05 0.025 lambda = 0.1 lambda = 0.2 ) 1 0.020 lambda = 0.5 + i > - - - 0.015 i ( l o 0.010 v 0.005 0.000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 years 178 LGM1F and negative rates Negatives rates ? We give the probabilities of 2Y, 5Y, 10Y zero-coupon rates being negative, on the Euro, US, Japan swap market, in march 2002. 179 LGM1F and negative rates Euro 2-year rate 0.8% 7.0% 0.7% 6.5% 0.6% 6.0% 0.5% 5.5% 0.4% 5.0% 0.3% 4.5% 0.2% 4.0% 0.1% 3.5% 0.0% 3.0% 0 4 8 12 16 20 Maturity P(R < 0) under risk-neutral measure Forward ZC Rate (right scale) 180 LGM1F and negative rates US 2-year rate 1.0% 8.5% 0.9% 8.0% 0.8% 7.5% 0.7% 7.0% 0.6% 6.5% 0.5% 6.0% 0.4% 5.5% 0.3% 5.0% 0.2% 4.5% 0.1% 4.0% 0.0% 3.5% 0 3 6 9 12 15 Maturity P(R < 0) under risk-neutral measure Forward ZC Rate (right scale) 181 LGM1F and negative rates Japanese 2-year rate 35.0% 3.5% 30.0% 3.0% 25.0% 2.5% 20.0% 2.0% 15.0% 1.5% 10.0% 1.0% 5.0% 0.5% 0.0% 0.0% 0 2 4 6 8 10 Maturity P(R < 0) under risk-neutral measure Forward ZC Rate (right scale) 182 LGM1F and negative rates Euro 5-year rate 0.8% 7.0% 0.7% 6.5% 0.6% 6.0% 0.5% 5.5% 0.4% 5.0% 0.3% 4.5% 0.2% 4.0% 0.1% 3.5% 0.0% 3.0% 0 4 8 12 16 20 Maturity P(R < 0) under risk-neutral measure Forward ZC Rate (right scale) 183 LGM1F and negative rates US 5-year rate 1.0% 8.5% 0.9% 8.0% 0.8% 7.5% 0.7% 7.0% 0.6% 6.5% 0.5% 6.0% 0.4% 5.5% 0.3% 5.0% 0.2% 4.5% 0.1% 4.0% 0.0% 3.5% 0 3 6 9 12 15 Maturity P(R < 0) under risk-neutral measure Forward ZC Rate (right scale) 184 LGM1F and negative rates Japanese 5-year rate 5.0% 5.0% 4.5% 4.5% 4.0% 4.0% 3.5% 3.5% 3.0% 3.0% 2.5% 2.5% 2.0% 2.0% 1.5% 1.5% 1.0% 1.0% 0.5% 0.5% 0.0% 0.0% 0 2 4 6 8 10 Maturity P(R < 0) under risk-neutral measure Forward ZC Rate (right scale) 185 LGM1F and negative rates Euro 10-year rate 0.8% 7.0% 0.7% 6.5% 0.6% 6.0% 0.5% 5.5% 0.4% 5.0% 0.3% 4.5% 0.2% 4.0% 0.1% 3.5% 0.0% 3.0% 0 4 8 12 16 20 Maturity P(R < 0) under risk-neutral measure Forward ZC Rate (right scale) 186 LGM1F and negative rates US 10-year rate 1.0% 8.5% 0.9% 8.0% 0.8% 7.5% 0.7% 7.0% 0.6% 6.5% 0.5% 6.0% 0.4% 5.5% 0.3% 5.0% 0.2% 4.5% 0.1% 4.0% 0.0% 3.5% 0 3 6 9 12 15 Maturity P(R < 0) under risk-neutral measure Forward ZC Rate (right scale) 187 LGM1F and negative rates Japanese 10-year rate 5.0% 5.0% 4.5% 4.5% 4.0% 4.0% 3.5% 3.5% 3.0% 3.0% 2.5% 2.5% 2.0% 2.0% 1.5% 1.5% 1.0% 1.0% 0.5% 0.5% 0.0% 0.0% 0 2 4 6 8 10 Maturity P(R < 0) under risk-neutral measure Forward ZC Rate (right scale) 188 LGM1F and negative rates Negative probabilities not very important, except 2Y Yen on short term Be careful nevertheless for very long products Standard deviation in 2005 similar than in 2002 Long term swaps lower in USD, even more in EUR Short term swaps lower in EUR, higher in USD Yen curve mostly unchanged 189 Digital rate options and applications to exotic swaps Complex Products : digital, steepner, ratchet… Bermuda swaptions Callables Appendixes : Standard Deviations vs Volatilities SABR Correlation issues LGM Risk-Management of exotics 190 Risk-Management for exotics : principles Vanilla risk Management Each underlying (tenor + option maturity : 10Y in 2Y, 10Y in 6Y, 3m in 5Y…) has specific SABR parameters So some limits for exotics Nevertheless, SABR beta has generally the same value for all underlying Exotics Risk-management Calibration of a model for each exotic Choice of proper vanilla products for calibration The vanilla products are for example limits of the exotics in some cases (typical example : bermuda swaptions calibrated on diagonal) Adequation choice of the model for each exotics (number of factors) Some rules (conservative risk-management/pricing) for exotics parameters like mean reversion 191 Risk-Management for exotics : principles To calculate the greeks (vs interest rates buckets or volatilities) For each exotic,the specific model for is recalibrated on the same set of vanilla products when a market parameters (buckets or volatilities) is bumped Before recalibration, new prices of vanilla products are calculated, using constant SABR parameters That’s why so important to have speed and exact calibration method (to avoid excess time calculation) and numerical noise Parameters that are not part of calibration, like mean reversions in a LGM model are kept constant in all the process SABR beta : The SABR Beta is especially important as it sets the intermediate rule between full normal (beta = 0) and full lognormal model (beta = 1) SABR Beta, generally common value for all underlying, says how ATM volatilities moves when rates moves This creates very important consistency feature between Vanilla risk- management and Exotics risk-management 192 Inflation : a few definitions ■ Difference between CPI and DRI Index and DRI ■ CPI: consumer price index ■ DRI: daily reference index. j 1 DRJ CPI x CPI CPI j m 3 m 2 m 3 NBDm July 117.20 01/06/05 01/09/05 ■Various indexes traded: INDEX FRENCH XTOB EUR XTOB EUR ITXTOB SP US BLOOMBERG FRCPXTOB CPTFEMU CPTFIEU ITCPI SPIPC CPURNSA REUTERS OATINFLATION01 OATEI01 HICPFIX PUBLISHER INSEE EUROSTAT EUROSTAT ISTAT INDE BLS 193 INFLATION SWAP CUZeroRVcouponES curves 3.5 3.3 3.1 2.9 2.7 2.5 2.3 2.1 1.9 Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 1 1 1 1 2 2 2 2 2 FRF EUR EUR ITL ESP •The bid/offer spread is between 1 and 8 bp according to the maturity and the market configuration. •The market size is about 50M € •European inflation incl. tobacco is less liquid and can not be dealt between two French counterparts ( Evin Law). 194 INFLATION SWAP CUForwardRVEScurves 3.6 3.4 3.2 3.0 2.8 2.6 2.4 2.2 2.0 EUR EUR XTOB France Spain Italy 1.8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 EUR 2.290 2.321 2.270 2.238 2.204 2.218 2.264 2.232 2.248 2.272 2.290 2.312 2.338 2.368 2.393 2.416 2.441 2.469 2.498 2.528 2.553 2.573 2.589 2.626 2.656 2.657 2.680 2.671 2.594 EUR XTOB 2.130 2.159 2.111 2.099 2.107 2.135 2.161 2.122 2.141 2.168 2.186 2.210 2.240 2.275 2.309 2.339 2.366 2.390 2.412 2.429 2.444 2.458 2.473 2.514 2.546 2.547 2.570 2.561 2.484 France 2.036 2.128 2.176 2.152 2.153 2.179 2.206 2.159 2.191 2.256 2.295 2.323 2.342 2.353 2.359 2.371 2.384 2.399 2.415 2.429 2.451 2.471 2.494 2.544 2.575 2.562 2.534 2.526 2.535 Spain 3.330 3.194 3.029 2.947 2.921 2.886 2.812 2.704 2.695 2.709 2.704 2.709 2.720 2.738 2.761 2.782 2.798 2.809 2.815 2.816 2.817 2.823 2.835 2.877 2.903 2.898 2.928 2.937 2.890 Italy 2.410 2.459 2.421 2.389 2.349 2.353 2.387 2.342 2.336 2.330 2.323 2.329 2.347 2.375 2.400 2.420 2.439 2.457 2.475 2.493 2.508 2.517 2.523 2.551 2.557 2.535 2.555 2.563 2.521 195 INFLATION SWAP CUEURR VCPIE andS expected forwards 240 3.2 220 3.0 200 2.8 180 2.6 160 2.4 140 2.2 120 June CPI June ZC Forwards 2.0 100 1.8 0Y 1Y 2Y 3Y 4Y 5Y 6Y 7Y 9Y 10Y 11Y 12Y 13Y 14Y 15Y 16Y 17Y 18Y 19Y 20Y 21Y 22Y 24Y 25Y 26Y 27Y 28Y 29Y 30Y June CPI 115.10117.50120.00 122.59125.18127.81130.50133.29 139.06142.04145.12148.29151.57 154.96158.49162.15165.94 169.87173.93178.12182.45186.91 196.24201.17206.29211.55216.99 222.54228.07 June ZC 2.08992.1097 2.12622.12242.11782.11602.1186 2.12382.12552.12942.13412.1399 2.14712.15562.16522.1754 2.18602.19672.20752.21802.2283 2.24802.25862.26972.27992.2903 2.29962.3057 Forw ards 2.130 2.159 2.111 2.099 2.107 2.135 2.122 2.141 2.168 2.186 2.210 2.240 2.275 2.309 2.339 2.366 2.390 2.412 2.429 2.444 2.473 2.514 2.546 2.547 2.570 2.561 2.484 196 Inflation swaps ■ NB: ■ There is no calculation basis on the inflation leg. It is an index performance between two unadjusted dates. ■ The calculation basis on the fixed leg is 30/360 on unadjusted dates. ■ To make it simple, on the interbank market of European inflation swaps, the DIR is not interpolated but dealt on m-2 or m-3 197 Inflation swaps ■ The flows are the following: DIR i DIR T * Fixed Rate at maturity Max 1; 0 DIR Initial DIR Initial against euribor 3M ■For example the fixed rate (real rate) is eThequ a«lOATito 1. 0format1% on» Fswaprench inflation ex-tobacco at 10 years. 198