Two Curves, One Price

TWO CURVES, ONE PRICE
Pricing & Hedging Interest Rate Derivatives
Using Different Yield Curves
For Discounting and Forwarding

Marco Bianchetti
Intesa San Paolo Bank, Risk Management – Market Risk – Pricing & Financial Modelling
marco.bianchetti intesasanpaolo.com

Quant Congress Europe 2009 – London,3-5 Nov. 2009

Summary

1. Context and Market Practices
Single-Curve Pricing & Hedging Interest-Rate Derivatives
From Single to Double-Curve Paradigm
2. Double-Curve Framework, No Arbitrage and Forward Basis
General Assumptions
Double-Curve Pricing Procedure
No Arbitrage and Forward Basis
3. Foreign-Currency Analogy, No Arbitrage and Quanto Adjustment
Spot and Forward Exchange rates
Quanto Adjustment
4. Double-Curve Pricing & Hedging Interest Rate Derivatives
Pricing> FRAs, Swaps, Caps/Floors/Swaptions
Hedging
5. No Arbitrage and Counterparty Risk
6. Other Approaches
7. Conclusions and References

“Two Curves, One Price” - Marco Bianchetti – Quant Congress Europe – London,3-5 Nov. 2009 p. 2

1: Context & Market Practices:
Single-Curve Pricing & Hedging IR Derivatives

Pre credit-crunch single curve market practice:

select a single set of the most convenient (e.g. liquid) vanilla interest rate
instruments traded on the market with increasing maturities; for instance, a
very common choice in the EUR market is a combination of short-term
EUR deposit, medium-term FRA/Futures on Euribor3M and medium-
long-term swaps on Euribor6M;
build a single yield curve C using the selected instruments plus a set of
bootstrapping rules (e.g. pillars, priorities, interpolation, etc.);
Compute, on the same curve C, forward rates, cashflows, discount
factors and work out the prices by summing up the discounted cashflows;
compute the delta sensitivity and hedge the resulting delta risk using the
suggested amounts (hedge ratios) of the same set of vanillas.


Market Evolution
80 EUR Basis swaps
1M
1M
vs
vs
3M
6M
Basis swaps (single
70
1M
3M
vs
vs
12M
6M
currency) as 2 swaps:
60 3M vs 12M
6M vs 12M
basis spread (bps)

50

40
1. Euribor 3MT vs RTM
3

30 2. Euribor 6MT vs RTM
6

3. BasisTM 6M = RTM − RTM
3 3 6
20

10

0
1Y

2Y

3Y

4Y

5Y

6Y

7Y

8Y

9Y

10Y

11Y

12Y

15Y

20Y

25Y

30Y
Quotations as of 16 Feb. 2009 (source: Reuters ICAPEUROBASIS)

Other market evidences:
the divergence between deposit (Euribor based) and OIS (Overnight
Indexed Swaps, Eonia based) rates;
The divergence between FRA rates and the corresponding forward rates
implied by consecutive deposits (see e.g. refs. [2], [6], [7].

Multiple-Curve Pricing & Hedging IR Derivatives

No market
Post credit-crunch multiple curve market practice: standard
…Eonia?

build a single discounting curve Cd using the preferred procedure;
build multiple distinct forwarding curves Cf1… Cfn using the preferred
distinct selections of vanilla interest rate instruments each homogeneous
in the underlying rate tenor (typically 1M, 3M, 6M, 12M);
compute the forward rates with tenor f on the corresponding forwarding
curve Cf and calculate the corresponding cashflows;
compute the corresponding discount factors using the discounting curve
Cd and work out prices by summing the discounted cashflows;
compute the delta sensitivity and hedge the resulting delta risk using the
suggested amounts (hedge ratios) of the corresponding set of vanillas.


Rationale
Apparently similar interest rate instruments with different underlying rate
tenors are characterised, in practice, by different liquidity and credit risk
premia, reflecting the different views and interests of the market
players.

Thinking in terms of more fundamental variables, e.g. a short rate, the
credit crunch has acted as a sort of symmetry breaking mechanism:
from a (unstable) situation in which an unique short rate process was able
to model and explain the whole term structure of interest rates of all tenors,
towards a sort of market segmentation into sub-areas corresponding to
instruments with different underlying rate tenors, characterised, in
principle, by distinct dynamics, e.g. different short rate processes.

Notice that market segmentation was already present (and well
understood) before the credit crunch (see e.g. ref. [3]), but not effective
due to negligible basis spreads.


2: Multiple-Curve Framework:
General Assumptions
1. There exist multiple different interest rate sub-markets
Mx, x = {d,f1 ,…,fn} characterized by the same currency and by distinct
bank accounts Bx and yield curves
C x := {T → Px ( t0 ,T ) ,T ≥ t0 } ,
2. The usual no arbitrage relation
Px ( t,T2 ) = Px ( t,T1 ) × Px ( t,T1,T2 )
holds in each interest rate market Mx.
3. Simple compounded forward rates are defined as usual for t≤T1< T2
Px ( t,T2 ) 1
Px ( t,T1,T2 ) = = ,
Px ( t,T1 ) 1 + Fx ( t;T1,T2 ) τx (T1,T2 )
T
4. FRA pricing under Qx 2 forward measure associated to numerairePx ( t,T2 )
FRAx ( t;T1,T2 , K ) = Px ( t,T2 ) τx (T1,T2 ) { EQx [ Lx (T1,T2 ) ] − K }
T2
t

= Px ( t,T2 ) τx (T1,T2 ) [ Fx ( t;T1,T2 ) − K ],


Pricing Procedure
1. assume Cd as the discounting curve and Cf as the forwarding curve;
2. calculate any relevant spot/forward rate on the forwarding curve Cf as
Pf ( t , Ti −1 ) − Pf ( t , Ti )
Ff ( t ;Ti −1 , Ti ) = , t ≤ Ti −1 < Ti ,
τ f (Ti −1 , Ti ) Pf ( t , Ti )
3. calculate cashflows ci, i = 1,...,n, as expectations of the i-th coupon payoff
Ti
πi with respect to the discounting Ti - forward measureQd
T
Qd i
ci := c ( t,Ti , πi ) = Et [ πi ];

4. calculate the price π at time t by discounting each cashflow ci using the
corresponding discount factor Pd ( t,Ti ) obtained from the discounting
curve Cd and summing,
n
T
π ( t,T ) = ∑ Qd i
P ( t,Ti ) Et [ πi ];
i =1
5. Price FRAs as
{ }
T
Qd 2
FRA ( t;T1,T2 , K ) = Pd ( t,T2 ) τ f (T1,T2 ) Et [ Ff (T1 ;T1,T2 ) ] − K


No Arbitrage and Forward Basis
Classic single-curve no arbitrage relations are broken: for instance, by
specifying the subscripts d and f as prescribed above we obtain the two eqs.
Pd ( t,T2 ) = Pd ( t,T1 ) Pf ( t,T1,T2 ) ,
1 Pf ( t,T2 )
Pf ( t,T1,T2 ) = = ,
1 + Ff ( t;T1,T2 ) τ f (T1,T2 ) Pf ( t,T1 )

that clearly cannot hold at the same time. No arbitrage is recovered by taking
into account the forward basis as follows
1 1
Pf ( t,T1,T2 ) = := ,
1 + Ff ( t;T1,T2 ) τf (T1,T2 ) 1 + [ Fd ( t;T1,T2 ) + BAfd ( t;T1,T2 ) ] τd (T1,T2 )

for which we obtain the following static expression in terms of discount factors

1 ⎡ Pf ( t,T1 ) Pd ( t,T1 ) ⎤
BAfd ( t;T1,T2 ) = ⎢ − ⎥.
τd (T1,T2 ) ⎢⎣ Pf ( t,T2 ) Pd ( t,T2 ) ⎥⎦


Forward Basis Curves
80 Forward Basis 0Y-3Y 10 Forward Basis 3Y-30Y
60 8
6
40
4
20
basis points

basis points
2
0 0

-20 -2
-4
-40 1M vs Disc 1M vs Disc
3M vs Disc -6 3M vs Disc
-60 6M vs Disc 6M vs Disc
12M vs Disc -8
12M vs Disc
-80 -10
May-09

May-10

May-11
Nov-09

Nov-10

Nov-11
Aug-09

Aug-10

Aug-11
Feb-09

Feb-10

Feb-11

Feb-12

Feb-12

Feb-15

Feb-18

Feb-21

Feb-24

Feb-27

Feb-30

Feb-33

Feb-36

Feb-39
Forward basis (bps) as of end of day 16 Feb. 2009, daily sampled 3M tenor forward rates calculated on
C1M, C3M , C6M , C12M curves against Cd taken as reference curve. Bootstrapping as described in ref. [2].
The richer term structure of the forward basis curves provides a sensitive
indicator of the tiny, but observable, statical differences between different
interest rate market sub-areas in the post credit crunch interest rate world, and
a tool to assess the degree of liquidity and credit issues in interest rate
derivatives' prices. Provided that…

Bad Curves
6% 3M curves 0Y-30Y 4 Forward Basis 3Y-30Y

2
5%
0

basis points
4% -2

3% -4

-6
1M vs Disc
2% 3M vs Disc
zero rates -8 6M vs Disc
forward rates 12M vs Disc
1% -10
Feb-09
Feb-11
Feb-13
Feb-15
Feb-17
Feb-19
Feb-21
Feb-23
Feb-25
Feb-27
Feb-29
Feb-31
Feb-33
Feb-35
Feb-37
Feb-39

Feb-12

Feb-15

Feb-18

Feb-21

Feb-24

Feb-27

Feb-30

Feb-33

Feb-36

Feb-39
Left: 3M zero rates (red dashed line) and forward rates (blue continuous line). Right: forward basis. Linear
interpolation on zero rates has been used. Numerical results from QuantLib (www.quantlib.org).

…smooth yield curves are used…Non-smooth bootstrapping techniques, e.g.
linear interpolation on zero rates (still a diffused market practice), produce zero
curves with no apparent problems, but ugly forward curves with a sagsaw
shape inducing, in turn, strong and unnatural oscillations in the forward basis
(see [2]).

3: Foreign Currency Analogy:
Spot and Forward Exchange Rates
A second issue regarding no arbitrage arises in the double-curve framework:
{ [ Ff (T1 ;T1,T2 ) ] − K }
T
Qd 2
FRA ( t;T1,T2 , K ) = Pd ( t,T2 ) τ f (T1,T2 ) Et
≠ Pd ( t,T2 ) τ f (T1,T2 ) [ Ff (T1 ;T1,T2 ) − K ]
1. Double-curve-double-currency:
⇒ d = domestic, f = foreign
c d ( t ) = x fd ( t ) c f ( t ) ,
X fd ( t , T ) Pd ( t , T ) = x fd ( t ) P f ( t , T ),
x fd ( t 0 ) = x fd , 0.

2. Double-curve-single-currency:
⇒ d = discounting, f=forwarding
cd ( t ) = x fd ( t ) c f ( t ) ,
X fd ( t , T ) Pd ( t , T ) = x fd ( t ) Pf ( t , T ) ,
x fd ( t 0 ) = 1.
Picture of no arbitrage definition of the forward exchange rate.
Circuitation (round trip) ⇒ no money is created or destructed.

3: Foreign Currency Analogy:
Quanto Adjustment
1. Assume a lognormal martingale dynamic for the Cf (foreign) forward rate
dFf ( t ;T1,T2 )
= σ f ( t ) dW fT2 ( t ) , Q T2 ↔ Pf ( t ,T2 ) ↔ C f ;
f
Ff ( t ;T1,T2 )
2. since x fd ( t ) Pf ( t ,T ) is the price at time t of a Cd (domestic) tradable
asset, the forward exchange rate must be a martingale process
dX fd ( t ,T2 ) T T
= σ X ( t ) dWX 2 ( t ) , Qd 2 ↔ Pd ( t ,T2 ) ↔ C d ,
X fd ( t ,T2 )
with dW fT2 ( t ) dWX 2 ( t ) = ρ fX ( t ) dt ;
T

3. by changing numeraire from Cf to Cd we obtain the modified dynamic
dFf ( t ;T1,T2 )
= µ f ( t ) dt + σ f ( t ) dW fT2 ( t ) , Qd 2 ↔ Pd ( t ,T2 ) ↔ C d ,
T
Ff ( t ;T1,T2 )
µ f ( t ) = −σ f ( t ) σ X ( t ) ρ fX ( t ) ;
4. and the modified expectation including the (additive) quanto-adjustment
T
Qd 2
Et [ L f (T1,T2 ) ] = Ff ( t ;T1,T2 ) + QAfd ( t ;T1, σ f , σ X , ρ fX ),
⎡ T1 ⎤
QAfd ( t ;T1, σ f , σ X , ρ fX ) = Ff ( t ;T1,T2 ) ⎢ exp
⎣ ∫t µ f ( s ) ds − 1 ⎥ .
⎦

4: Pricing & Hedging IR Derivatives:
Pricing Plain Vanillas [1]
1. FRA: FRA ( t;T1,T2 , K ) = Pd ( t,T2 ) τ f (T1,T2 )
× [ Ff ( t;T1,T2 ) + QAfd ( t,T1, σ f , σX , ρfX ) − K ]
m
2. Swaps:
Swap ( t;T , S, K ) = −∑ Pd ( t, S j ) τd ( S j −1, S j )K j
n j =1
+∑ Pd ( t, ST ) τ f (Tj −1,Tj )[ Ff ( t;Ti −1,Ti ) + QAfd ( t,Ti −1, σ f ,i , σX ,i , ρfX ,i ) ].
i =1
n
3. Caps/Floors:
CF ( t;T , K , ω ) = ∑ Pd ( t,Ti ) τd (Ti −1,Ti )
i =1
×Black [ Ff ( t;Ti −1,Ti ) + QAfd ( t,Ti −1, σ f ,i , σX ,i , ρfX ,i ), Ki , µf ,i , v f ,i , ωi ],

4. Swaptions: Swaption ( t;T , S, K , ω ) = Ad ( t, S )
×Black [ S f ( t;T , S ) + QAfd ( t,T , S, ν f , νY , ρfY ), K , λf , v f , ω ].


Pricing Plain Vanillas [2]
100 Quanto Adjustment (additive)
80
60
Quanto adj. (bps)

40
Numerical scenarios for the (additive)
20
quanto adjustment, corresponding to
0 three different combinations of (flat)
-20 volatility values as a function of the
-40 correlation. The time interval is fixed
-60
Sigma_f = 10%, Sigma_X = 2.5% to T1-t=10 years and the forward rate
Sigma_f = 20%, Sigma_X = 5% to 3%.
-80 Sigma_f = 30%, Sigma_X = 10%
-100
-1.0 -0.8 -0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6 0.8 1.0
Correlation

We notice that the adjustment may be not negligible. Positive correlation
implies negative adjustment, thus lowering the forward rates. The
standard market practice, with no quanto adjustment, is thus not arbitrage
free. In practice the adjustment depends on market variables not directly
quoted on the market, making virtually impossible to set up arbitrage positions
and locking today positive gains in the future.

Hedging
1. Given any portfolio of interest rate derivatives with price Π ( t,T , Rmkt ),
compute delta risk with respect to both curves Cd and Cf :
∆π ( t,T , Rmkt ) = ∆d ( t,T , Rd ) + ∆π ( t,T , Rf )
π mkt
f
mkt

Nd
∂Π ( t,T , Rmkt ) ∂Π ( t,T , Rmkt )
Nf
= ∑ ∂Rd (Tj )
mkt
+∑ mkt
(Tj )
,
j =1 j =1 ∂Rf

2. eventually aggregate it on the subset of most liquid market instruments
(hedging instruments);

3. calculate hedge ratios:
∂Π ( t,T , Rmkt ) mkt
hx , j = δx , j ,
∂Rx (Tj )
mkt

mkt
mkt ∂πx , j ( t )
δx , j = , x = f , d.
∂Rx (Tj )
mkt


5: No Arbitrage and Counterparty Risk:
A Simple Credit Model (adapted from ref. [6])
Both the forward basis and the quanto adjustment discussed above find a
simple financial explanation in terms of counterparty risk.
If we identify:
Pd(t,T) = default free zero coupon bond,
Pf(t,T) = risky zero coupon bond emitted by a risky counterparty for
maturity T and with recovery rate Rf ,
τ(t)>t = (random) counterparty default time observed at time t,
qd ( t,T ) = EtQd {1[ τ ( t )>T ] } = default probability after time T expected at time t,

we obtain the following expressions
Pf ( t,T ) = Pd ( t,T ) R ( t; t,T, Rf ),
1 ⎡ Pd ( t,T1 ) R ( t; t,T1, Rf ) ⎤
Ff ( t;T1,T2 ) = ⎢ − 1⎥ ,
τf (T1,T2 ) ⎢⎣ Pd ( t,T2 ) R ( t; t,T2, Rf ) ⎥⎦

where: R ( t;T1,T2, Rf ) = Rf + ( 1 − Rf ) EtQd [ qd (T1,T2 ) ].


5: No Arbitrage and Counterparty Risk:
A Simple Credit Model [2]

If Ld(T1,T2), Lf(T1,T2) are the risk free and the risky Xibor rates underlying the
corresponding derivatives, respectively, we obtain:

Pd ( t,T1 )
FRAf ( t;T1,T2, K ) = − [ 1 + K τ f (T1,T2 ) ] Pd ( t,T2 ) ,
R ( t;T1,T2, Rf )
1 Pd ( t,T1 ) ⎡ R ( t; t,T1, Rf ) ⎤
BAfd ( t;T1,T2 ) = ⎢ − 1⎥ ,
τd (T1,T2 ) Pd ( t,T2 ) ⎢⎣ R ( t; t,T2, Rf ) ⎥⎦
1 Pd ( t,T1 ) ⎡ 1 R ( t; t,T1, Rf ) ⎤
QAfd ( t;T1,T2 ) = ⎢ − ⎥,
τ f (T1,T2 ) Pd ( t,T2 ) ⎢⎣ R ( t;T1,T2, Rf ) R ( t; t,T2, Rf ) ⎥⎦

That is, the forward basis and the quanto adjustment expressed in terms of risk
free zero coupon bonds Pd(t,T) and of the expected recovery rate.


6: Pros & Cons, Other Approaches:

PROs CONs
Simple and familiar framework, no Unobservable exchange rate and
additional effort, just analogy. parameters.
Straightforward interpretation in Plain vanilla prices acquire volatility
terms of counterparty risk. and correlation dependence.

M. Henrard: “ab-initio parsimonious” model [5]
F. Mercurio: generalised Libor Market Model [6]
M. Morini: full credit model [7]
F. Kijima et al: DLG model [8]


7: Conclusions
1. We have reviewed the pre and post credit crunch market practices for
pricing & hedging interest rate derivatives.
2. We have shown that in the present double-curve framework standard
single-curve no arbitrage conditions are broken and can be recovered
taking into account the forward basis; once a smooth bootstrapping
technique is used, the richer term structure of the calculated forward basis
curves provides a sensitive indicator of the tiny, but observable, statical
differences between different interest rate market sub-areas.
3. Using the foreign-currency analogy we have computed the no arbitrage
generalised double-curve-single-currency market-like pricing
expressions for basic interest rate derivatives, including a quanto
adjustment arising from the change of numeraires naturally associated to
the two yield curves. Numerical scenarios show that the quanto
adjustment can be non negligible.
4. Both the forward basis and the quanto adjustment have a simple
interpretation in terms of counterparty risk, using a simple credit model
with a risk-free and a risky zero coupond bonds.

7: Main references

[1] M. Bianchetti, “Two Curves, One Price: Pricing & Hedging Interest Rate Derivatives Using
Different Yield Curves for Discounting and Forwarding”, (2009), available at SSRN:
http://ssrn.com/abstract=1334356.
[2] F. Ametrano, M. Bianchetti, “Bootstrapping the Illiquidity: Multiple Yield Curves Construction
For Market Coherent Forward Rates Estimation”, in “Modeling Interest Rates: Latest
Advances for Derivatives Pricing”, edited by F. Mercurio, Risk Books, 2009.
[3] B. Tuckman, P. Porfirio, “Interest Rate Parity, Money Market Basis Swaps, and Cross-
Currency Basis Swaps”, Lehman Brothers, Jun. 2003.
[4] W. Boenkost, W. Schmidt, “Cross currency swap valuation”, working Paper, HfB--Business
School of Finance & Management, May 2005.
[5] M. Henrard, ”The Irony in the Derivatives Discounting - Part II: The Crisis”, working paper, Jul.
2009, available at SSRN: http://ssrn.com/abstract=1433022.
[6] F. Mercurio, "Post Credit Crunch Interest Rates: Formulas and Market Models", working
paper, Bloomberg, 2009, available at SSRN: http://ssrn.com/abstract=1332205.
[7] M. Morini, "Credit Modelling After the Subprime Crisis", Marcus Evans course, 2008.
[8] M. Kijima, K. Tanaka, T. Wong, “A Multi-Quality Model of Interest Rates”, Quantitative
Finance, 2008.
[9] D. Brigo, F. Mercurio, "Interest Rate Models - Theory and Practice", 2nd ed., Springer, 2006.


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