Bootstrapping the interest-rate term structure - Marco Marchioro

Bootstrapping the interest-rate term structure 

Marco Marchioro 

www.marchioro.org 

October 20 th , 2012 

Advanced Derivatives, Interest Rate Models 2010-2012 c○ Marco Marchioro

Bootstrapping the interest-rate term structure 1 

Summary (1/2) 

• Market quotes of deposit rates, IR futures, and swaps 

• Need for a consistent interest-rate curve 

• Instantaneous forward rate 

• Parametric form of discount curves 

• Choice of curve nodes 



Summary (2/2) 

• Bootstrapping quoted deposit rates 

• Bootstrapping using quoted interest-rate futures 

• Bootstrapping using quoted swap rates 

• QuantLib, bootstrapping, and rate helpers 

• Derivatives on foreign-exchange rates 

• Sensitivities of interest-rate portfolios (DV01) 

• Hedging portfolio with interest-rate risk 



Major liquid quoted interest-rate derivatives 

For any given major currency (EUR, USD, GBP, JPY, ...) 

• Deposit rates 

• Interest-rate futures (FRA not reliable!) 

• Interest-rate swaps 



Quotes from Financial Times 



Consistent interest-rate curve 

We need a consistent interest-rate curve in order to 

• Understand the current market conditions (e.g. forward rates) 

• Compute the at-the-money strikes for Caps, Floor, and Swaptions 

• Compute the NPV of exotic derivatives 

• Determine the “fair” forward currency-exchange rate 

• Hedge portfolio exposure to interest rates 

• ... (many more reasons) ... 



One forward rate does not fit all (1/2) 

Assume a continuously compounded discount rate from a flat rate r 

D(t) = e 

−r t 

Matching exactly the implied discount for the first deposit rate 

1 

1 + T1 r fix(1) 

and for the second deposit rate 

1 

1 + T1 r fix(2) 

(1) 

= D(T1) = e −r T1 (2) 

= D(T2) = e −r T2 (3) 


Yielding 

and 


One forward rate does not fit all (2/2) 

r = 1 

T1 

r = 1 

T2 

log � 

� 

1 + T1 rfix(1) log � 

� 

1 + T2 rfix(2) which would imply two values for the same r. Hence, 

a single constant rate is not consistent with all market quotes! 

Advanced Derivatives, Interest Rate Models 2010-2012 c○ Marco Marchioro 

(4) 

(5)


Instantaneous forward rate (1/2) 

Given two future dates d1 and d2, the forward rate was defined as, 

rfwd(d1, d2) = 

� � 

1 D (d1) − D (d2) 

T (d1, d2) D (d2) 

(6) 

We define the instantaneous forward rate f(d1) as the limit, 

f(d1) = lim rfwd(d1, d2) (7) 

d2→d1 



Instantaneous forward rate (2/2) 

Given certain day-conventions, set T = T (d0, d) then after preforming 

a change of variable from d to T we have, 

� � 

1 D(T ) − D(T + ∆t) 

f(T ) = lim 

(8) 

∆t→0 ∆t D(T + ∆t) 

It can be shown that 

f(T ) = − 1 ∂D(T ) 

D(T ) ∂T 

= −∂ log [D(T )] 

∂T 


(9)


Instantaneous forward rate for flat curve 

Consider a continuously-compounded flat-forward curve 

D(d) = e −z T (d0,d) ⇐⇒ D(T ) = e −z T 

with a given zero rate z, then 

∂ log [D(T )] 

f(T ) = − 

∂T 

∂ [−z T ] 

= − 

∂T 

is the instantaneous forward rate 

= z 

� � 

∂ log e−z T 

= − 

∂T 

(10) 



Discount from instantaneous forward rate 

Integrating the expression for the instantaneous forward rate 

� � 

∂ log [D(t)] 

dt = − 

∂T 

f(t)dt ⇐⇒ 

� T 

log [D(T )] = − 

and taking the exponential we obtain 

D(T ) = exp 

� 

− 

� T 

0 f(t)dt 

so that choosing f(t) results in a discount factor 

� 

0 f(t)dt 


Recall 


Forward expectations 

D(T ) = E 

� 

e − � T 

0 r(t)dt� 

= e − � T 

0 f(t)dt 

Similarly in the forward measure (see Brigo Mercurio) 

rfwd(t, T ) = E T 

� � 

1 T 

T − t t r(t′ )dt ′ 

� 

and 

(11) 

(12) 

f(T ) = E T [r(t)dt] (13) 



Piecewise-flat forward curve (1/2) 

Given a number of nodes, T1 < T2 < T3, define the instantaneous 

forward rate as 

until the last node 

f(t) = f1 for t ≤ T1 (14) 

f(t) = f2 for T1 < t ≤ T2 (15) 

f(t) = f3 for T2 < t ≤ T3 (16) 

f(t) = . . . 



Piecewise-flat forward curve (2/2) 

We determine the discount factor D(T ) using equation 

It can be shown that 

Recall that T0 = 0 

D(T ) = exp 

� 

− 

� T 

0 f(t)dt 

D(T ) = 1 · e −f1(T −T0) 

for T ≤ T1 (17) 

D(T ) = D(T1) e −f2(T −T1) 

for T1 < T ≤ T2 (18) 

. . . = . . . (19) 

D(T ) = D(Ti) e −fi+1(T −Ti) 

for Ti < T ≤ Ti+1 (20) 


�


Questions? 



(The art of) choosing the curve nodes 

• Choose d0 the earliest settlement date 

• First few nodes to fit deposit rates (until 1st futures?) 

• Some nodes to fit futures until about 2 years 

• Final nodes to fit swap rates 



Why discard long-maturity deposit rates? 

Compare cash flows of a deposit and a one-year payer swap for a 

notional of 100,000$ 

Date Deposit IRS Fixed Leg IRS Ibor Leg 

Today - 100,000$ 0$ 0$ 

Today + 6m 0$ 0$ 1,200$ 

Today + 12m 102,400$ -2,500$ 1,280 ∗ $ 

For maturities longer than 6 months credit risk is not negligible 

*Estimated by the forward rate 



Talking to the trader: bootstrap 

• Deposit rates are unreliable: quoted rates may not be tradable 

• Libor fixings are better but fixed once a day (great for riskmanagement 

purposes!) 

• FRA quotes are even more unreliable than deposit rates 


Zero rates (%) 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 


Boostrap of the USD curve using different helper lists 

Depo2m + Futs + Swaps 

0.1 

0 0.5 1 1.5 2 2.5 3 3.5 

time to maturity 

Depo1Y + Swaps 

Depo6m + Swaps 




Spread over risk free (%) 

0.4 

0.35 

0.3 

0.25 

0.2 

0.15 

0.1 

0.05 

0 


Boostrap of the USD curve using different helper lists 

-0.05 

0 0.5 1 1.5 2 2.5 3 3.5 

time to maturity 

Depo1Y + Swaps 






Discount interpolation 

Taking the logarithm in the piecewise-flat forward curve 

log [D(T )] = log � D(Ti−1) � − (T − Ti)f i+1 

discount factors are interpolated log linearly 

(21) 

• Other interpolations are possible and give slightly different results 

between nodes (see QuantLib for a list) 

• Important: use the same type of interpolation for all curves! 



Bootstrapping the first node (1/2) 

Set the first node to the maturity of the first depo rate. 

Recalling equation (2) for f1 = r, 

D(T1) = e −f1 T1 = 

This equation can be solved for f1 to give, 

f1 = 1 

we obtain the value of f1. 

T1 

1 

1 + T1 r fix(1) 

log � 

� 

1 + T1 rfix(1) (22) 

(23) 


6.0% 

5.0% 

4.0% 

3.0% 

2.0% 

1.0% 

✻ 

• 

f1 

• 


Bootstrapping the first node (2/2) 

3m 6m 1y 2y 3y 4y 5y 7y 10y 


✲


Bootstrapping the second node (1/2) 

Set the second node to the maturity of the second depo rate. 

The equivalent equation for the second node gives, 

from which we obtain 

D(T2) = e −f1 T1 e −f2 (T2−T1) = 

f2 = 

� 

� 

log 1 + T2 rfix(2) T2 − T1 

1 

1 + T2 r fix(2) 

− f1 T1 

Continue for all deposit rates to be included in the term structure 

(24) 

(25) 


6.0% 

5.0% 

4.0% 

3.0% 

2.0% 

1.0% 

✻ 

• 

f1 

• 


Bootstrapping the second node (2/2) 

f2 

• 

3m 6m 1y 2y 3y 4y 5y 7y 10y 


✲


Bootstrapping from quoted futures (1/2) 

For each futures included in the term structure 

• Add the futures maturity + tenor date to the node list 

• Solve for the appropriate forward rates that reprice the futures 

Note: futures are great hedging tools 


6.0% 

5.0% 

4.0% 

3.0% 

2.0% 

1.0% 

✻ 

• 

f1 


Bootstrapping from quoted futures (2/2) 

• 

f2 

• 

f3 

• 

f4 

• 

3m 6m 1y 2y 3y 4y 5y 7y 10y 


✲


Bootstrapping from quoted swap rates 

For each interest-rate swap to be included in the term structure 

• Add the swap maturity date to the node list 

• Solve for the appropriate forward rate that give null NPV to the 

given swap 


6.0% 

5.0% 

4.0% 

3.0% 

2.0% 

1.0% 

✻ 

• 

f1 

• 


Final piecewise-flat forward curve 

f2 

• 

f3 

• 

f4 

f5 

f6 

f7 

f8 

f9 

• 

• • • • 

• 

3m 6m 1y 2y 3y 4y 5y 7y 10y 


✲


Extrapolation 

Sometimes we need to compute the discount factor beyond the last 

quoted node 

We assume the last forward rate to extend beyond the last maturity 

D(T ) = D(Tn) e −fn(T −Tn) 

for T > Tn 

(26) 



Questions? 



QuantLib: forward curve 

The curve defined in equations (17)-(20) is available in QuantLib as 

qlForwardCurve 



QuantLib: rate helpers 

Containers with the logic and data needed for bootstrapping 

• Function qlDepositRateHelper for deposit rates 

• Function qlFuturesRateHelper for futures quotes 

• Function qlSwapRateHelper2 for swap fair rates 



QuantLib: bootstrapped curve 

• qlPiecewiseYieldCurve: a curve that fits a series of market quotes 

• qlRateHelperSelection: a helper-class useful to pick rate helpers 



Questions? 



Foreign-exchange rates 

Very often derivatives are used in order to hedge against future changes 

in foreign exchange rates. 

We extend the approach of the previous sections to contracts that 

involve two different currencies. 

Consider a home currency (e.g. e), a foreign currency (e.g. $), and 

their current currency-exchange rate so that X e$ , 

1 $ = 1e 

X e$ 

(27) 



Foreign-exchange forward contract 

Given a certain notional amount N e in the home currency and a 

notional amount N $ in the foreign currency, consider the contract 

that allows, at a certain future date d, to pay N $ and to receive N e . 

Pay/Receive (at d) = N e − N $ 

(28) 

Bootstrap the risk-free discount curve D e (d) using the appropriate 

quoted instruments in the e currency, and the risk-free discount curve 

D $ (d) similarly. 



Present value of notionals 

The present value of N e in the home currency is given by 

PV e = D e (d) N e 

the present value of N $ in the foreign currency can be written as 

PV $ = D $ (d) N $ 

Dividing the first expression by X e$ 

PV e 

X e$ 

(29) 

(30) 

= D e N e 

(d) . (31) 

Xe$ Advanced Derivatives, Interest Rate Models 2010-2012 c○ Marco Marchioro


NPV of an FX forward 

The net present value of the forward contract in the $ currency is 

NPV $ fx−fwd 

PVe 

= − PV 

Xe$ $ 

= D e N e 

(d) − D 

Xe$ $ (d) N $ 

The same amount can be expressed in the foreign currency as, 

NPV e fx−fwd = De (d)N e − X e$ D $ (d) N $ 

(32) 

(33) 



Arbitrage-free forward FX rate 

The contract is usually struck so the its NPV=0, from equation (32) 

N $ = De (d) 

X e$ D $ (d) N e . 

Comparing with (27), we define the forward exchange rate X e$ (d) 

X e$ (d) = X e$ 

D $ (d) 

D e (d) 

. (34) 

• The exchange rate X e$ (d) is the fair value of an FX rate at d. 

• According to (34) the forward FX rate is highly dependent on the 

discount curves in each respective currency. 



Questions? 



Interest-rate sensitivities 

In order to hedge our interest-rate portfolio we compute the interest 

rate sensitivities 



Dollar Value of 1 basis point 

The Dollar Value of 1 basis point, or DV01, of an interest-rate portfolio 

P is the variation incurred in the portfolio when interest rates 

move up one basis point: 

with ∆r=0.01% 

DV01 P = P (r1 + ∆r, r2 + ∆r, . . .) − P (r1, r2, . . .) (35) 

Using a Taylor approximation 

DV01 P � ∂P 

∂r 

∆r (36) 



Managing interest-rate risk (1/2) 

• Consider an interest-rate portfolio P with a certain maturity T 

• Look for a swap S with the same maturity 

• Compute DV01 for both portfolio (DV01 P ) and Swap (DV01 S) 



Managing interest-rate risk (2/2) 

Buy an amount H, the hedge ratio, of the given swap, 

H = − DV01 P 

DV01 S 

The book composed by the portfolio and the swap is delta hedged 

where r is the vector of all interest rates 

(37) 

B(r) = P (r) + H S(r) (38) 

B(r + ∆r) − B(r) � DV01 P ∆r + H DV01 S ∆r � 0 (39) 



Advanced interest-rate risk management (1/2) 

For highly volatile interest rates use higher-order derivatives (gamma 

hedging) 

CVP � ∂2P ∆r (40) 

∂r2 For portfolio with highly varying cash flows compute as many DV 01 

as the number of maturities. E.g. DV01 2Y , DV01 3Y , . . . 

DV01 1Y 

P = P (r1, . . . , r 2Y + ∆r, r 3Y , . . .) − P (r) (41) 



Advanced interest-rate risk management (2/2) 

Build the hedging book as 

with 

B = P + H 2Y S 2Y + H 3Y S 3Y + . . . (42) 

H 2Y = − DV012Y P 

DV012Y S 

, H 3Y = − DV013Y P 

DV013Y S 

The book is delta hedge with respect to all swap rates: 

B(r + ∆r) − B(r) � DV01 2Y 

P ∆r + H2Y DV01 2Y 

S 

+DV01 3Y 

P ∆r + H2Y DV01 3Y 

S 

, . . . (43) 

∆r + . . . � 0 

∆r + (44) 



Questions? 



References 

• Options, future, & other derivatives, John C. Hull, Prentice Hall 

(from fourth edition) 

• Interest rate models: theory and practice, D. Brigo and F. Mercurio, 

Springer Finance (from first edition)

Bootstrapping the interest-rate term structure - Marco Marchioro

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