Marco Marchioro · Bootstrapping the interest-rate term structure Marco Marchioro October 20th,...

Bootstrapping the interest-rate term structure

Marco Marchioro

www.marchioro.org

October 20th, 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 (1)

Matching exactly the implied discount for the first deposit rate

1

1 + T1 rfix(1)= D(T1) = e−r T1 (2)

and for the second deposit rate

1

1 + T1 rfix(2)= D(T2) = e−r T2 (3)



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

Yielding

r =1

T1log

(1 + T1 rfix(1)

)(4)

and

r =1

T2log

(1 + T2 rfix(2)

)(5)

which would imply two values for the same r. Hence,

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



Instantaneous forward rate (1/2)

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

rfwd(d1, d2) =1

T (d1, d2)

[D (d1)−D (d2)

D (d2)

](6)

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

f(d1) = limd2→d1

rfwd(d1, d2) (7)



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,

f(T ) = lim∆t→0

1

∆t

[D(T )−D(T + ∆t)

D(T + ∆t)

](8)

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 (10)

with a given zero rate z, then

f(T ) = −∂ log [D(T )]

∂T= −

∂ log[e−z T

]∂T

= −∂ [−z T ]

∂T= z

is the instantaneous forward rate



Discount from instantaneous forward rate

Integrating the expression for the instantaneous forward rate∫∂ log [D(t)]

∂Tdt = −

∫f(t)dt ⇐⇒ log [D(T )] = −

∫ T

0f(t)dt

and taking the exponential we obtain

D(T ) = exp

{−∫ T

0f(t)dt

}

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



Forward expectations

Recall

D(T ) = E

[e−∫ T

0 r(t)dt]

= e−∫ T

0 f(t)dt (11)

Similarly in the forward measure (see Brigo Mercurio)

rfwd(t, T ) = ET

[1

T − t

∫ T

tr(t′)dt′

](12)

and

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



Piecewise-flat forward curve (1/2)

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

forward rate as

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) = . . .

until the last node



Piecewise-flat forward curve (2/2)

We determine the discount factor D(T ) using equation

D(T ) = exp

{−∫ T

0f(t)dt

}It can be shown that

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)

Recall that T0 = 0



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 LegToday - 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 risk-

management purposes!)

• FRA quotes are even more unreliable than deposit rates



0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.5 1 1.5 2 2.5 3 3.5

Zero

rate

s (%

)

time to maturity

Boostrap of the USD curve using different helper lists

Depo1Y + Swaps

Depo6m + Swaps

Depo3m + Swaps

Depo3m + Futs + Swaps




-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.5 1 1.5 2 2.5 3 3.5

Spre

ad o

ver r

isk

free

(%)

time to maturity

Boostrap of the USD curve using different helper lists

Depo1Y + Swaps

Depo6m + Swaps

Depo3m + Swaps




Discount interpolation

Taking the logarithm in the piecewise-flat forward curve

log [D(T )] = log[D(Ti−1)

]− (T − Ti)fi+1 (21)

discount factors are interpolated log linearly

• 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 =1

1 + T1 rfix(1)(22)

This equation can be solved for f1 to give,

f1 =1

T1log

(1 + T1 rfix(1)

)(23)

we obtain the value of f1.



Bootstrapping the first node (2/2)

-

6

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

6.0%5.0%4.0%3.0%2.0%1.0%

••f1



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,

D(T2) = e−f1 T1 e−f2 (T2−T1) =1

1 + T2 rfix(2)(24)

from which we obtain

f2 =log

(1 + T2 rfix(2)

)− f1 T1

T2 − T1(25)

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



Bootstrapping the second node (2/2)

-

6

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

6.0%5.0%4.0%3.0%2.0%1.0%

••f1

•f2



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



Bootstrapping from quoted futures (2/2)

-

6

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

6.0%5.0%4.0%3.0%2.0%1.0%

••f1

•f2 •f3 •f4



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



Final piecewise-flat forward curve

-

6

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

6.0%5.0%4.0%3.0%2.0%1.0%

••f1

•f2 •f3 •f4 •f5 •f6 •f7 •f8 •f9



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 Xe$,

1 $ =1e

Xe$(27)



Foreign-exchange forward contract

Given a certain notional amount Ne 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 Ne.

Pay/Receive (at d) = Ne −N$ (28)

Bootstrap the risk-free discount curve De(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 Ne in the home currency is given by

PVe = De(d)Ne (29)

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

PV$ = D$(d)N$ (30)

Dividing the first expression by Xe$

PVe

Xe$= De(d)

Ne

Xe$. (31)



NPV of an FX forward

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

NPV$fx−fwd =

PVe

Xe$− PV$

= De(d)Ne

Xe$−D$(d)N$ (32)

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

NPVefx−fwd = De(d)Ne −Xe$D$(d)N$ (33)



Arbitrage-free forward FX rate

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

N$ =De(d)

Xe$ D$(d)Ne .

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

Xe$(d) = Xe$D$(d)

De(d). (34)

• The exchange rate Xe$(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 port-

folio P is the variation incurred in the portfolio when interest rates

move up one basis point:

DV01P = P (r1 + ∆r, r2 + ∆r, . . .)− P (r1, r2, . . .) (35)

with ∆r=0.01%

Using a Taylor approximation

DV01P '∂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 (DV01P ) and Swap (DV01S)



Managing interest-rate risk (2/2)

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

H = −DV01P

DV01S(37)

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

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

where r is the vector of all interest rates

B(r + ∆r)−B(r) ' DV01P ∆r + H DV01S ∆r ' 0 (39)



Advanced interest-rate risk management (1/2)

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

hedging)

CVP '∂2P

∂r2∆r (40)

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

as the number of maturities. E.g. DV012Y , DV013Y , . . .

DV011YP = P (r1, . . . , r2Y + ∆r, r3Y , . . .)− P (r) (41)



Advanced interest-rate risk management (2/2)

Build the hedging book as

B = P + H2Y S2Y + H3Y S3Y + . . . (42)

with

H2Y = −DV012Y

P

DV012YS

, H3Y = −DV013Y

P

DV013YS

, . . . (43)

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

B(r + ∆r)−B(r) ' DV012YP ∆r + H2Y DV012Y

S ∆r + (44)

+DV013YP ∆r + H2Y DV013Y

S ∆r + . . . ' 0



Questions?



References

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

(from fourth edition)

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

curio, Springer Finance (from first edition)


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