Interest rates

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Forward rate

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The term structure of interest rates80-646-08

Stochastic calculus I

Geneviève Gauthier

HEC Montréal

Interest rates

Notation

Termstructure

Zero-coupon

Forward rate

The short rate

References

Notation I

This section draws heavily for inspiration on the book Lastructure à terme des taux dintérêt" by Christophe Bisière.

P (t,T ) = the price at time t of a zero-coupon bondmaturing at time T .

P (T ,T ) = 1.

Bisière, p.4.

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Notation II

R (t,T ) = the rate of return at time t of a zero-couponbond maturing at time T .

It is the interest rate which, being continuously applied toan investment of amount P (t,T ) at time t, provides theinvestor with one unit at time T :

P (t,T ) exp [R (t,T ) (T t)] = 1.

We therefore have

R (t,T ) = 1T t ln [P (t,T )] .

Bisière, p.5-6.

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Notation III

r (t) = the short rate at time t.

r (t) = limT #t R (t,T )

It is the rate yielded at time t by a loan that must berepaid instantaneously!

Bisière, p.8-10.

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Notation IV

f (t,T1,T2) = forward rate

It is possible to construct a bond portfolio that allows todetermine in advance (i.e. at time t) the interest rate of aloan starting at time T1 t and maturing at timeT2 T1.

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Notation VAt time t :

sale of a bond maturing at time T2 andpurchase of P (t,T2) /P (t,T1) bond maturing at T1 fora cost of

P (t,T2)P (t,T2)P (t,T1)

P (t,T1) = 0.

At time T1, we receive the cash ows generated by thequantity of bonds purchased: we therefore receive

P (t,T2)P (t,T1)

P (T1,T1) =P (t,T2)P (t,T1)

.

At time T2, we must pay back the bond sold: we musttherefore pay

P (T2,T2) = 1.

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Notation VI

Thus, f (t,T1,T2) is the interest rate that satises theequation

P (t,T2)P (t,T1)

exp [f (t,T1,T2) (T2 T1)] = 1.

It is possible to show that

f (t,T1,T2) = 1

T2 T1lnP (t,T2)P (t,T1)

.

Bisière, p.10-12.

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Notation VII

When T2 tends to T1, the forward rate becomes the rateof a loan with a shorter and shorter lifetime.

f (t,T ) = the instantaneous forward rate at time t forthe future instant T

f (t,T ) = limε#0 f (t,T ,T + ε)

It is possible to show that

P (t,T ) = exphR Tt f (t, s) ds

if (t,T ) = ∂ lnP (t ,u)

∂u

u=T

Bisière, p.10-12.

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Notation VIIIIndeed, by denition, f (t,T1,T2) satises the equation

f (t,T1,T2) = 1

T2 T1lnP (t,T2)P (t,T1)

.

Thus

f (t,T ,T + ε) = 1T + ε T ln

P (t,T + ε)

P (t,T )

= ln [P (t,T + ε)] ln [P (t,T )]

ε.

As a consequence,

f (t,T ) limε#0f (t,T ,T + ε)

= limε#0 ln [P (t,T + ε)] ln [P (t,T )]

ε

=

∂ ln [P (t, u)]∂u

u=T

.

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Notation IX

HenceZ T

tf (t, s) ds =

Z T

t

∂ ln [P (t, s)]∂s

ds

= ln [P (t,T )] + ln [P (t, t)]= ln [P (t,T )] since P (t, t) = 1.

Which completes the proof since, then

expZ T

tf (t, s) ds

= P (t,T ) .

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Term structure I

Lets assume that today corresponds to time t = 0. Theonly known bond prices are of type P(0,T ). Prices oftype P(t,T ), t > 0 cannot be observed yet.

As a consequence, from observed prices, we can deducethe forward rates

f (0,T1,T2) , 0 T1 T2

but the forward rates of type

f (t,T1,T2) , 0 < t T1 T2

are still unknown.

So they must be modeled!

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Zero-coupon I

Assume that

dP (t,T ) = α (t,T ) P (t,T ) dt+P (t,T ) δ> (t,T )1k

dWP (t)k1

where WP is a multidimensional Brownian motion constructedon the space (Ω,F ,P) .Since P (T ,T ) = 1 and since holding a bond one instantbefore its maturity amounts to invest money in a bank accountyielding the riskless rate r (T ), we have

α (T ,T ) = r (T )

et δ (T ,T ) = 0.

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Zero-coupon II

TheoremIt is possible to show (the proof is following!) that

P (t,T ) = EPt

exp

Z T

tα (s,T ) ds

Bisière, p.52.

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Forward rate

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Zero-coupon III

Proof. Lets set

V (u) = expZ u

tα (s,T ) ds

.

SincedV (u) = α (u,T )V (u) du

and

dP (u,T ) = α (u,T ) P (u,T ) du

+P (u,T ) δ> (u,T )1k

dWP (u)k1

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Zero-coupon IV

then the multiplication rule (Itôs lemma), yields

dP (u,T )V (u)

= V (u) dP (u,T ) + P (u,T ) dV (u) + d hV ,Piu

= V (u)

α (u,T )P (u,T ) du + P (u,T ) δ> (u,T )

1kdWP (u)

k1

!+P (u,T ) (α (u,T )V (u) du)

= V (u) P (u,T ) δ> (u,T )1k

dWP (u)k1

.

Expressed in integral form, we obtain

P (T ,T )| z =1

V (T ) P (t,T )V (t)| z =1

=Z T

tV (u) P (u,T ) δ> (u,T )

1kdWP (u)

k1.

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Forward rate

The short rate

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Zero-coupon V

Recall:

P (T ,T )| z =1

V (T ) P (t,T )V (t)| z =1

=Z T

tV (u) P (u,T ) δ> (u,T )

1kdWP (u)

k1.

By taking the conditional expectation on both sides of theabove equality, we get

EPt [V (T )] EP

t [P (t,T )] = 0.

As a consequence,

P (t,T ) = EPt [P (t,T )] = EP

t [V (T )]

= EPt

exp

Z T

tα (s,T ) ds

.

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Absence of arbitrage I

An argument based on the absence of arbitrageopportunity and the construction of a locally risklessportfolio containing k + 1 bonds establishes a link betweenα, r and δ : there exists a process Λ such that

α (t,T ) r (t) = Λ>t

1kδ (t,T )k1

for all 0 t T < ∞.

Such as result is important in that it shows there is a linkbetween the drift coe¢ cient and the di¤usion coe¢ cientof the stochastic di¤erential equation modeling theevolution of a bond price.

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Absence of arbitrage II

Idea of the construction. Lets consider a self-nancingtrading strategy on the time interval [0,T ], made up ofk + 1 bonds with di¤erent maturities (such maturitiesbeing after time T ).

Let φ1 (t), ..., φk+1 (t) be the quantities of the k + 1bonds held at time t.

The value of the trading strategy is given by the stochasticprocess V and

Vt =k+1

∑j=1

φj (t)P (t,Tj ) .

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Forward rate

The short rate

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Absence of arbitrage III

Since the strategy is self-nancing, we have

dVt

=k+1

∑j=1

φj (t) dPt,Tj

=

k+1

∑j=1

φj (t)

αt,Tj

Pt,Tj

dt + P

t,Tj

δ>t,Tj

1k

dWP (t)k1

!

=k+1

∑j=1

φj (t) Pt,Tj

αt,Tj

dt

+k+1

∑j=1

φj (t) Pt,Tj

δ>t,Tj

1k

dWP (t)k1

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Absence of arbitrage IV

=k+1

∑j=1

φj (t) Pt,Tj

Vt| z

wj (t)

Vt αt,Tj

dt

+k+1

∑j=1

φj (t) Pt,Tj

Vt| z

wj (t)

Vt δ>t,Tj

1k

dWP (t)k1

=k+1

∑j=1

wj (t) Vt α (t,Tj) dt +k+1

∑j=1

wj (t) Vt δ>t,Tj

1k

dWP (t)k1

where wk (t) represents the portion at time t of the portfoliototal value invested in the kth asset.

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Absence of arbitrage V

Idea of the construction (continued)Recall

dVt

=k+1

∑j=1

wj (t) Vt αt,Tj

dt +

k+1

∑j=1

wj (t) Vt δ>t,Tj

1k

dWP (t)k1

=k+1

∑j=1

wj (t) Vt αt,Tj

dt +

k+1

∑j=1

wj (t) Vtk

∑i=1

δit,Tj

dWi (t)

=k+1

∑j=1

wj (t) Vt αt,Tj

dt +

k

∑i=1

k+1

∑j=1

wj (t) Vt δit,Tj

!dWi (t)

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Forward rate

The short rate

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Absence of arbitrage VI

w1 (t), ..., wk+1 (t) could be chosen so that ∑k+1j=1 wj (t)

Vt δi (t,Tj ) = 0 for all i 2 f1, ..., kgIt is a linear system of k equations and k + 1 unknowns.

If that is the case, the return must be the riskless rate:

k+1

∑j=1

wj (t) Vt α (t,Tj ) dt = r (t) Vt dt

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Absence of arbitrage VII

The linear system to be solved is written as follows:

26664α (t ,T1) r (t) α (t ,Tk+1) r (t)

δ1 (t ,T1) δ1 (t ,Tk+1)...

...δk (t ,T1) δk (t ,Tk+1)

3777526664

w1 (t)w2 (t)...

wk+1 (t)

37775 =2666400...0

37775

For such a system to have a non-trivial solution, i.e.di¤erent from zero), the determinant of the square matrixof dimension k + 1 must be equal to zero, and its rankmust be strictly smaller than k + 1.

The rows of that matrix being linearly dependent, thereexists a non-zero linear combination of them that is equalto a row-vector of k + 1 zeros.

Bisière, page 55.

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Absence of arbitrage VIII

As such a property is totally independent from the selectedsecurities, the vector of coe¢ cients in the linearcombination does not depend on the maturitiesT1, ...,Tk+1 selected.

As a consequence, there exists a process Λ such that

α (t,T ) r (t) = Λ>t

1kδ (t,T )k1

for all 0 t T < ∞.

Bisière, page 55.

It should however be veried that the selected strategy isindeed self-nancing...

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Forward rate

The short rate

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Alternative approach I

Under a risk-neutral measure, the present value of a bondmust be a martingale.

This is equivalent to saying that, under a risk-neutralmeasure, the instantaneous return of the bond is theriskless rate:

dP (t,T )

= α (t,T ) P (t,T ) dt + P (t,T ) δ> (t,T )1k

dWP (t)k1

= α (t,T ) P (t,T ) dt + P (t,T )k

∑i=1

δi (t,T ) dWi (t)

=

α (t,T )

k

∑i=1

δi (t,T ) γi (t)

!P (t,T ) dt

+P (t,T )k

∑i=1

δi (t,T ) dWi (t) +

Z t

0γi (s) ds

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Alternative approach II

dP (t,T ) =

α (t,T )

k

∑i=1

δi (t,T ) γi (t)

!P (t,T ) dt

+P (t,T )k

∑i=1

δi (t,T ) dfWi (t)

where fWi (t) = Wi (t) +Z t

0γi (s) ds.

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Alternative approach III

So lets set

α (t,T )k

∑i=1

δi (t,T ) γi (t) = r (t)

which can be rewritten in matrix form as follows:

α (t,T ) r (t) = Λ>1k

δ (t,T ) .

It should then be veried that the process Λ satises theNovikov condition, in order to ensure that WQ is aQBrownian motion.

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We also obtain the bond price:

P (t,T )

= EPt

exp

Z T

tα (s ,T ) ds

= EP

t

exp

Z T

tr (s) ds 1

2

Z T

tΛ>s Λs ds

Z T

tΛ>s dW

Ps

(The proof is following) Bisière, p.55-58.

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Proof. Lets set

V (u) = exp

0BB@ Z u

tr (s) ds 1

2

Z u

tΛ>s Λs ds

Z u

tΛ>s dW

Ps| z

Yu

1CCA .Considering V as a function of u and Y , Itôs lemma can beused to establish the following:

dV (u) =∂V∂udu +

∂V∂ydYu +

12

∂2V∂y2

d hY iu

=

r (u) 1

2Λ>u Λu

V (u)| z

∂V∂u

du

V (u)| z ∂V∂y

Λ>u dW

Pu| z

dYu

+12V (u)| z

∂2V∂y2

Λ>u Λu du| z d hY iu

= r (u)V (u) du V (u)Λ>u dW

Pu

Interest rates

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Forward rate

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Recall that

dV (u) = r (u)V (u) du V (u)Λ>u dW

Pu

and dP (u,T ) = α (u,T ) P (u,T ) du

+P (u,T ) δ> (u,T )1k

dWP (u)k1

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The multiplication rule (Itôs lemma), yields

dP (u,T )V (u)

= V (u) dP (u,T ) + P (u,T ) dV (u) + d hV ,Piu= V (u)

α (u,T )P (u,T ) du + P (u,T ) δ> (u,T ) dWP (u)

+P (u,T )

r (u)V (u) du V (u)Λ>u dW

Pu

V (u)P (u,T ) Λ>u δ (u,T ) du

= V (u) P (u,T )

α (u,T ) r (u)Λ>u δ (u,T )

| z =0

du

+V (u) P (u,T )

δ> (u,T )Λ>u

1kdWP (u)

k1.

= V (u) P (u,T )

δ> (u,T )Λ>u

1kdWP (u)

k1

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The short rate

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Written in its integral form, we obtain

P (T ,T )| z =1

V (T ) P (t,T )V (t)| z =1

=Z T

tV (u) P (u,T )

δ> (u,T )Λ>u

1k

dWP (u)k1

By taking the conditional expectation on both sides, we get

EPt [V (T )] P (t,T ) = 0

hence

P (t,T ) = EPt

exp

Z T

tr (s) ds 1

2

Z T

tΛ>s Λs ds

Z T

tΛ>s dW

Ps

.

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Forward rate

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Lastly I

P (t,T )

= EPt

exp

Z T

trs ds

12

Z T

tΛ>s Λs ds

Z T

tΛ>s dW

Ps

= EQ

t

exp

Z T

trs ds

where

dP

dQ= exp

12

Z T

0Λ>s Λs ds

Z T

0Λ>s dW

Ps

Using Girsanov theorem, we can state that

WQt =W

Pt +

Z t

0Λsds : t 0

is a multidimensional QBrownian motion.

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Forward rate

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Lastly II

dP (t,T )

= α (t,T ) P (t,T ) dt + P (t,T ) δ> (t,T )1k

dWP (t)k1

= α (t,T ) P (t,T ) dt + P (t,T ) δ> (t,T )1k

d

WQt

k1Z t

0Λsk1

ds

!

=

α (t,T ) δ> (t,T )

1kΛtk1

!P (t,T ) dt + P (t,T ) δ> (t,T )

1kdWQ (t)

k1

= rt P (t,T ) dt + P (t,T ) δ> (t,T )1k

dWQ (t)k1

since α (t,T ) r (t) = Λ>t δ (t,T )

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Lastly III

The morale of that story. We can price bonds directly, underthe empirical measure P, by using the right rate of return α, orunder the risk-neutral measure Q, by working with theinstantaneous short rate r .

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Forward rate I

TheoremAssume that

df (t,T ) = η (t,T ) dt + θ> (t,T )1k

dWP (t)k1

.

Since f (t,T ) = ∂ ln P (t ,u)∂u

u=T

then we apply Itôs lemma to

lnP (t, u) in order to determine coe¢ cients η and θ. We obtain

η (t,T ) = δ>T (t,T )1k

δ (t,T )k1

αT (t,T )

and θ> (t,T ) = δ>T (t,T )

Bisière, p.59-61.

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Forward rate II

Proof. To be shown: if

dP (t,T ) = α (t,T ) P (t,T ) dt + P (t,T ) δ> (t,T )1k

dWP (t)k1

and df (t,T ) = η (t,T ) dt + θ> (t,T )1k

dWP (t)k1

.

then

η (t,T ) = δ>T (t,T )1k

δ (t,T )k1

αT (t,T )

and θ> (t,T ) = δ>T (t,T )

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Forward rate III

Recall that

dP (t,T ) = α (t,T ) P (t,T ) dt+P (t,T ) δ> (t,T )1k

dWP (t)k1

and

f (t,T ) = ∂ lnP (t, u)∂u

u=T

.

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The short rate

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Forward rate IV

Itôs lemma, applied to lnP (t,T ), yields

d lnP (t,T )

=1

P (t,T )dP (t,T ) 1

2

1

P (t,T )

2d hPit

=

α (t,T ) dt + δ> (t,T )

1kdWP (t)

k1

! 12

δ> (t,T )1k

δ (t,T )k1

dt

In integral form, we obtain

lnP (t,T ) lnP (0,T )

=Z t

0

α (s ,T ) 1

2δ> (s ,T )

1kδ (s ,T )k1

!ds +

Z t

0δ> (s ,T )

1kdWP (s)

k1.

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Forward rate V

Since

f (t,T ) = ∂ lnP (t, u)∂u

u=T

,

then

f (t,T ) f (0,T )

= ∂ lnP (t,T )∂T

+∂ lnP (0,T )

∂T

= ∂

∂T(lnP (t,T ) lnP (0,T ))

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Forward rate VI

Therefore

f (t ,T ) f (0,T )

= ∂

∂T

Z t

0

α (s ,T ) 1

2δ> (s ,T )

1kδ (s ,T )k1

!ds +

Z t

0δ> (s ,T )

1kdWP (s)

k1

!

=Z t

0

δ>T (s ,T )

1kδ (s ,T )k1

αT (s ,T )

!ds

Z t

0δ>T (s ,T )

1kdWP (s)

k1

since δ> (s,T ) δ (s,T ) = ∑ki=1 δ2i (s,T ) implies that

∂

∂Tδ> (s ,T ) δ (s ,T ) =

∂

∂T

k

∑i=1

δ2i (s ,T ) =k

∑i=1

2δi (s ,T )∂

∂Tδi (s ,T ) .

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Forward rate VII

Under the absence of arbitrage assumption, we have establishedthe following relation:

α (t,T ) r (t) = δ> (t,T )1k

Λtk1

and we have constructed the measure Q and the QBrownianmotion WQ.As a consequence,

η (t,T ) = δ>T (t,T ) δ (t,T ) αT (t,T )

= δ>T (t,T ) δ (t,T ) δ>T (t,T )Λt

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Forward rate VIII

and

df (t,T )

=

δ>T (t,T ) δ (t,T ) δ>T (t,T )Λt

dt

+θ> (t,T ) dWP (t)

= θ> (t,T )Z T

tθ (t, s) ds +Λt

dt + θ> (t,T ) dWP

t

since θ> (t,T ) = δ>T (t,T )

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Forward rate IX

Lets now determine the stochastic di¤erential equationf (,T ) under the risk-neutral measure Q :

df (t,T )

= θ> (t,T )Z T

tθ (t, s) ds +Λt

dt + θ> (t,T ) dWP

t

= θ> (t,T )Z T

tθ (t, s) ds +Λt

dt + θ> (t,T ) d

WQt

k1Z t

0Λsk1

ds

!

=

θ> (t,T )

Z T

tθ (t, s) ds +Λt

θ> (t,T ) Λt

dt

+ θ> (t,T ) dWQt

= θ> (t,T )Z T

tθ (t, s) ds

dt + θ> (t,T ) dWQ

t

Interest rates

Notation

Termstructure

Zero-coupon

Forward rate

The short rate

References

Short rate IRecall that

df (t,T ) = θ> (t,T )Z T

tθ (t, s) ds +Λt

dt + θ> (t,T ) dWP

t

df (t,T ) = θ> (t,T )Z T

tθ (t, s) ds

dt + θ> (t,T ) dWQ

t .

In integral form, we obtain

f (t,T ) f (0,T ) =Z t

0θ> (u,T )

Z T

uθ (u, s) ds +Λu

du

+Z t

0θ> (u,T ) dWP

u

f (t,T ) f (0,T ) =Z t

0θ> (u,T )

Z T

uθ (u, s) ds

du

+Z t

0θ> (u,T ) dWQ

u .

Interest rates

Notation

Termstructure

Zero-coupon

Forward rate

The short rate

References

Short rate II

So, the short rate is

r (t)

= f (t, t)

= f (0, t) +Z t

0θ> (u, t)

Z t

uθ (u, s) ds +Λu

du

+Z t

0θ> (u, t) dWP

u

= f (0, t) +Z t

0θ> (u, t)

Z t

uθ (u, s) ds

du

+Z t

0θ> (u, t) dWQ

u .

Bisière, p.59-61.

Interest rates

Notation

Termstructure

Zero-coupon

Forward rate

The short rate

References

References

Martin Baxter and Andrew Rennie (1996). FinancialCalculus, an introduction to derivative pricing, Cambridgeuniversity press.

Christophe Bisière (1997). La structure par terme des tauxdintérêt, Presses universitaires de France.

Damien Lamberton and Bernard Lapeyre (1991).Introduction au calcul stochastique appliqué à la nance,Ellipses.