Bootstrapping the interest-rate term structure

Marco Marchioro www.marchioro.org

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

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)

Major

liquid

quoted interest-rate derivatives

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

• Deposit rates

• Interest-rate futures (FRA not reliable!)

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

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 r_fix(1) = D(T1) = e

−r T_{1 (2)}

and for the second deposit rate 1

1 + T₁ r_fix(2) = D(T2) = e

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

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

Instantaneous forward rate (1/2)

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

r_fwd(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) = lim

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 he−z Ti ∂T = −∂ [−z T] ∂T = z

Discount from instantaneous forward rate

Integrating the expression for the instantaneous forward rate

Z ∂ log [D(t)] ∂T dt = − Z f(t)dt ⇐⇒ log [D(T)] = − Z T 0 f (t)dt

and taking the exponential we obtain

D(T) = exp ( − Z T 0 f (t)dt )

Forward expectations

Similarly in the forward measure (see Brigo Mercurio)

r_fwd(t, T) = ET " 1 T − t Z T t r(t0)dt0 # (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) = . . .

Piecewise-flat forward curve (2/2)

We determine the discount factor D(T) using equation

D(T) = exp ( − Z T 0 f (t)dt )

It can be shown that

D(T) = 1 · e−f1(T−T0) _{for T ≤ T} 1 (17) D(T) = D(T₁) e−f2(T−T1) _{for T} 1 < T ≤ T2 (18) . . . = . . . (19) D(T) = D(T_i) e−fi+1(T−Ti) _{for T} i < T ≤ Ti+1 (20) Recall that T₀ = 0

(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

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∗$

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!)

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 rates (%) Depo1Y + Swaps Depo6m + Swaps Depo3m + Swaps Depo3m + Futs + Swaps Depo2m + 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

Spread over risk free (%)

time to maturity

Boostrap of the USD curve using different helper lists

Depo1Y + Swaps Depo6m + Swaps Depo3m + Swaps Depo3m + Futs + Swaps

Discount interpolation

Taking the logarithm in the piecewise-flat forward curve log [D(T)] = log

D(T_i−1)

− (T − T_i)f_i+1 (21) discount factors are interpolated log linearly

• Other interpolations are possible and give slightly different results between nodes (see QuantLib for a list)

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−f1T1 ₌ 1

1 + T₁ r_fix(1) (22)

This equation can be solved for f₁ to give,

f₁ = 1

T₁ log

1 + T₁ r_fix(1) (23)

Bootstrapping the first node (2/2)

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−f1T1 _e−f2(T2−T1) ₌ 1

1 + T₂ r_fix(2) (24)

from which we obtain

f₂ =

log1 + T₂ r_fix(2) − f₁ T₁

T₂ − T₁ (25)

Bootstrapping the second node (2/2)

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

Bootstrapping from quoted futures (2/2)

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

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

QuantLib: forward curve

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

QuantLib: rate helpers

Containers with the logic and data needed for bootstrapping

• Function qlDepositRateHelper for deposit rates

• Function qlFuturesRateHelper for futures quotes

QuantLib: bootstrapped curve

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

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

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

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 X_e$

PVe

X_e$ = D

e_(d) Ne

NPV of an FX forward

The net present value of the forward contract in the $ currency is NPV$_fx−fwd = PV e X_e$ − PV $ = De(d) N e X_e$ − D $₍ d) N$ (32)

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

Arbitrage-free forward FX rate

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

N$ = D

e_(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)

De(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.

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:

DV01_P = P(r₁ + ∆r, r₂ + ∆r, . . .) − P(r₁, r₂, . . .) (35) with ∆r=0.01%

Using a Taylor approximation

DV01_P ' ∂P

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

Managing interest-rate risk (2/2)

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

H = −DV01P

DV01_S (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

Advanced interest-rate risk management (1/2)

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

CV_P ' ∂ 2_P

∂r2 ∆r (40)

For portfolio with highly varying cash flows compute as many DV 01 as the number of maturities. E.g. DV012Y , DV013Y , . . .

Advanced interest-rate risk management (2/2)

Build the hedging book as

B = P + H2Y S2Y + H3Y S3Y + . . . (42) with H2Y = −DV01 2Y P DV012_SY , H 3Y _{= −}DV013PY DV013_SY , . . . (43)

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

B(r + ∆r) − B(r) ' DV01_P2Y ∆r + H2Y DV012_SY ∆r + (44)

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)