80-646-08StochasticcalculusIGeneviŁveGauthier Thetermstructureofinterestrates

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Notation Term structure Zero-coupon Forward rate The short rate References

The term structure of interest rates

80-646-08 Stochastic calculus I

Geneviève Gauthier

HEC Montréal

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Notation I

This section draws heavily for inspiration on the book ”La structure à terme des taux d’intérêt" by Christophe Bisière.

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

P(T,T) =1.

Bisière, p.4.

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Notation II

R(t,T) =the rate of returnat time t of a zero-coupon bond maturing at timeT.

It is the interest rate which, being continuously applied to an investment of amountP(t,T)at time t, provides the investor with one unit at time T :

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

We therefore have

R(t,T) = ¹

T tln[P(t,T)]. Bisière, p.5-6.

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Notation III

r(t) =theshort rateat time t.

r(t) =lim_T_#_tR(t,T)

It is the rate yielded at time t by a loan that must be repaid 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 to determine in advance (i.e. at timet) the interest rate of a loan starting at time T1 t and maturing at time

T₂ T₁.

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Notation V

At time t :

sale of a bond maturing at timeT₂ and

purchase ofP(t,T₂)/P(t,T₁)bond maturing atT₁ for a 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 the quantity 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 must therefore pay

P(T₂,T₂) =1.

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Notation VI

Thus, f (t,T₁,T₂)is the interest rate that satis…es the equation

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 T₁ ln P(t,T₂) P(t,T₁) ^.

Bisière, p.10-12.

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Notation VII

WhenT₂ tends toT₁, the forward rate becomes the rate of a loan with a shorter and shorter lifetime.

f (t,T) = theinstantaneous forward rate at timet for the future instant T

f (t,T) = lim_ε_#₀f (t,T,T +ε) It is possible to show that

P(t,T) =exph RT

t f (t,s) dsi f(t,T) = ^∂ln^P(t,u)_∂u

Bisière, p.10-12. u=T

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Notation VIII

Indeed, by de…nition,f (t,T₁,T₂)satis…es the equation f (t,T1,T2) = ¹

T2 T1

ln P(t,T2) P(t,T1) ^. Thus

f (t,T,T +ε) = ¹

T +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) lim

ε#⁰f (t,T,T +ε)

= lim

ε#⁰

ln[P(t,T +ε)] ln[P(t,T)]

ε

= ^∂^ln[P(t,u)]

∂u _u₌_T .

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Notation IX

Hence Z T

t

f (t,s) ds =

Z T

t

∂ln[P(t,s)]

∂s ds

= ln[P(t,T)] +ln[P(t,t)]

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

Which completes the proof since, then exp

Z T

t

f (t,s) ds =P(t,T).

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Notation Term structure

Zero-coupon Forward rate The short rate References

Term structure I

Let’s assume that today corresponds to time t=0. The only known bond prices are of typeP(0,T). Prices of type P(t,T),t >0 cannot be observed yet.

As a consequence, from observed prices, we can deduce the 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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Notation Term structure Zero-coupon

Forward rate The short rate References

Zero-coupon I

Assume that

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

1 k

dW^P(t)

k 1

whereW^P is a multidimensional Brownian motion constructed on the space(Ω,F^,^P).

SinceP(T,T) =1 and since holding a bond one instant before its maturity amounts to invest money in a bank account yielding the riskless rater(T), we have

α(T,T) = r(T) etδ(T,T) = 0.

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Zero-coupon II

Theorem

It is possible to show (the proof is following!) that P(t,T) =E^P_t exp

Z _T

t

α(s,T) ds Bisière, p.52.

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Zero-coupon III

Proof. Let’s set

V(u) =exp Z u

t

α(s,T) ds . Since

dV (u) = α(u,T)V (u) du and

dP(u,T) = α(u,T) P(u,T) du +P(u,T)δ^>(u,T)

1 k

dW^P(u)

k 1

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Zero-coupon IV

then the multiplication rule (Itô’s lemma), yields

dP(u,T)V(u)

= V(u) dP(u,T) +P(u,T) dV(u) +dh^V^,^Piu

= V(u) α(u,T)P(u,T) du+P(u,T) δ^>(u,T)

1 k

dW^P(u)

k 1

!

+P(u,T) ( α(u,T)V(u) du)

= V(u) P(u,T)δ^>(u,T)

1 k

dW^P(u)

k 1

.

Expressed in integral form, we obtain

P(T,T)

| {z }

=1

V(T) P(t,T)V(t)

| {z }

=1

= Z_T

t V(u) P(u,T)δ^>(u,T)

1 k

dW^P(u)

k 1

.

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Zero-coupon V

Recall:

P(T,T)

| {z }

=1

V(T) P(t,T)V(t)

| {z }

=1

= Z_T

t V(u) P(u,T)δ^>(u,T)

1 k

dW^P(u)

k 1

.

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

E^P_t [V(T)] E^P_t [P(t,T)] =0.

As a consequence,

P(t,T) = E^P_t [P(t,T)] =E^P_t [V(T)]

= E^P_t exp

Z T

t α(s,T) ds .

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Absence of arbitrage I

An argument based on the absence of arbitrage opportunity and the construction of a locally riskless portfolio containing k+1 bonds establishes a link between α,r and δ: there exists a process Λsuch that

α(t,T) r(t) = Λ^>t 1 k

δ(t,T)

k 1

for all 0 t T < ∞. Such as result is important in that it shows there is a link between the drift coe¢ cient and the di¤usion coe¢ cient of the stochastic di¤erential equation modeling the evolution of a bond price.

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Absence of arbitrage II

Idea of the construction. Let’s consider a self-…nancing trading strategy on the time interval [0,T], made up of k+1 bonds with di¤erent maturities (such maturities being after time T).

Let φ₁(t), ..., φ_k₊₁(t)be the quantities of thek+1 bonds held at timet.

The value of the trading strategy is given by the stochastic process V and

V_t =

k+1 j

∑

=1

φ_j(t)P(t,T_j).

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Absence of arbitrage III

Since the strategy is self-…nancing, we have

dVt

=

k+1∑

j=1

φ_j(t) dP t,T_j

=

k+1∑

j=1

φ_j(t) α t,T_j P t,T_j dt+P t,T_j δ^> t,T_j

1 k

dW^P(t)

k 1

!

=

k+1∑

j=1

φ_j(t) P t,T_j α t,T_j dt

+

k+1 j=1∑

φ_j(t) P t,T_j δ^> t,T_j

1 k

dW^P(t)

k 1

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Absence of arbitrage IV

=

k+1 j=1∑

φ_j(t) P t,T_j Vt

| {z }

wj(t)

Vt α t,Tj dt

+

k+1∑

j=1

φ_j(t) P t,Tj

V_t

| {z }

wj(t)

Vt δ^> t,T_j

1 k

dW^P(t)

k 1

=

k+1 j=1∑

wj(t) Vt α(t,Tj) dt+

k+1 j∑=1

wj(t) Vt δ^> t,Tj 1 k

dW^P(t)

k 1

wherew_k(t)represents the portion at timet of the portfolio total value invested in thekth asset.

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Absence of arbitrage V

Idea of the construction (continued) Recall

dVt

=

k+1∑

j=1

wj(t) Vt α t,Tj dt+

k+1∑

j=1

wj(t) Vt δ^> t,Tj 1 k

dW^P(t)

k 1

=

k+1∑

j=1

wj(t) Vt α t,Tj dt+

k+1∑

j=1

wj(t) Vt

∑k i=1

δi t,Tj dWi(t)

=

k+1∑

j=1

w_j(t) Vt α t,T_j dt+

∑k i=1

k+1∑

j=1

w_j(t) Vt δ_i t,T_j

!

dW_i(t)

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Absence of arbitrage VI

w1(t), ..., wk+1(t) could be chosen so that∑^kj=⁺1¹wj(t) Vt δi(t,T_j) =0 for alli 2 f^{1, ...,}^kg

It is a linear system ofk 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:

2 66 64

α(t,T1) r(t) α(t,Tk+1) r(t) δ1(t,T1) δ1(t,T_k₊₁)

.. .

.. . δk(t,T1) δk(t,Tk+1)

3 77 75 2 66 64

w1(t) w2(t)

.. . wk+1(t)

3 77 75=

2 66 64

0 0 .. . 0

3 77 75

For such a system to have a non-trivial solution, i.e.

di¤erent from zero), the determinant of the square matrix of dimensionk+1 must be equal to zero, and its rank must be strictly smaller than k+1.

The rows of that matrix being linearly dependent, there exists a non-zero linear combination of them that is equal to a row-vector ofk+1 zeros.

Bisière, page 55.

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Absence of arbitrage VIII

As such a property is totally independent from the selected securities, the vector of coe¢ cients in the linear

combination does not depend on the maturities T₁, ...,T_k₊₁ selected.

As a consequence, there exists a process Λsuch that α(t,T) r(t) =_Λ^>_t

1 k

δ(t,T)

k 1

for all 0 t T <_∞. Bisière, page 55.

It should however be veri…ed that the selected strategy is indeed self-…nancing...

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Alternative approach I

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

This is equivalent to saying that, under a risk-neutral measure, the instantaneous return of the bond is the riskless rate:

dP(t,T)

= α(t,T) P(t,T) dt+P(t,T)δ^>(t,T)

1 k

dW^P(t)

k 1

= α(t,T) P(t,T) dt+P(t,T)

∑

k i=1

δi(t,T) dW_i(t)

= α(t,T)

∑

k i=1

δi(t,T)γ_i(t)

!

P(t,T) dt +P(t,T)

∑

k i=1

δi(t,T) d Wi(t) +

Z t

0 γ_i(s)ds

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Alternative approach II

dP(t,T) = α(t,T)

∑

k i=1

δi(t,T)γ_i(t)

!

P(t,T) dt +P(t,T)

∑

k i=1

δi(t,T) dWf_i(t) where

f

W_i(t) =W_i(t) +

Z t

0 γ_i(s)ds.

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Alternative approach III

So let’s set

α(t,T)

∑

k i=1

δi(t,T)γ_i(t) =r(t) which can be rewritten in matrix form as follows:

α(t,T) r(t) = _Λ^>

1 k δ(t,T).

It should then be veri…ed that the process Λ satis…es the Novikov condition, in order to ensure that W^Q is a Q Brownian motion.

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We also obtain the bond price:

P(t,T)

= E^P_t exp Z_T

t α(s,T) ds

= E^P_t exp Z_T

t r(s) ds 1 2

Z_T

t Λ^>_sΛs ds Z_T

t Λ^>_s dW^P_s

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

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Interest rates

Proof. Let’s set

V(u) =exp 0 BB

@ Z _u

t

r(s) ds 1 2

Z _u

t Λ^>s Λs ds Z _u

t Λ^>s dW_s^P

| {z }

Yu

1 CC A.

ConsideringV as a function of u andY, Itô’s lemma can be used to establish the following:

dV (u) = ^∂V

∂udu+ ^∂V

∂ydY_u+¹ 2

∂²V

∂y²dh^Yiu

= r(u) ¹

2Λ^>uΛu V(u)

| {z }

∂V

∂u

du

V (u)

| {z }

∂V

∂y

Λ^>u dW^P_u

| {z }

dYu

+¹ 2V (u)

| {z }

∂2V

∂y2

Λu^>Λu du

| {z }

dh^Yiu

= r(u)V (u) du V(u)Λ^>u dW^P_u

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Recall that

dV (u) = r(u)V(u) du V (u)Λ^>u dW^P_u anddP(u,T) = α(u,T) P(u,T) du

+P(u,T)δ^>(u,T)

1 k

dW^P(u)

k 1

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The multiplication rule (Itô’s lemma), yields

dP(u,T)V(u)

= V(u) dP(u,T) +P(u,T) dV(u) +dh^V^,^Piu

= V(u) α(u,T)P(u,T) du+P(u,T) δ^>(u,T) dW^P(u) +P(u,T) r(u)V(u) du V(u)Λ^>_u dW^P_u

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

1 k

dW^P(u)

k 1

.

= V(u) P(u,T) δ^>(u,T) Λ^>_u

1 k

dW^P(u)

k 1

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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

t V(u) P(u,T) δ^>(u,T) Λ^>u 1 k

dW^P(u)

k 1

By taking the conditional expectation on both sides, we get E^P_t [V(T)] P(t,T) =0

hence

P(t,T) =E^P_t exp Z _T

t r(s) ds 1 2

Z_T

t Λ^>sΛs ds Z _T

t Λ^>s dW^P_s .

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Lastly I

P(t,T)

= E^P_t exp Z T

t rs ds 1

2 ZT

t Λ^>_sΛs ds Z T

t Λ_s^>dW^P_s

= E^Q_t exp Z_T

t rs ds

where dP

dQ =exp 1 2

Z T

0 Λ^>s Λs ds Z T

0 Λ^>s dW^P_s Using Girsanov theorem, we can state that

W_t^Q=W^P_t +

Z t

0 Λsds :t 0 is a multidimensional Q Brownian motion.

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Lastly II

dP(t,T)

1 k

dW^P(t)

k 1

1 k

d W^Q_t

k 1 Z_t

0 Λs k 1

ds

!

= α(t,T) δ^>(t,T)

1 k

Λt k 1

!

P(t,T) dt+P(t,T)δ^>(t,T)

1 k

dW^Q(t)

k 1

= rt P(t,T) dt+P(t,T)δ^>(t,T)

1 k

dW^Q(t)

k 1

sinceα(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, under the empirical measure P, by using the right rate of returnα, or under the risk-neutral measure Q, by working with the

instantaneous short rater.

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Notation Term structure Zero-coupon Forward rate

The short rate References

Forward rate I

Theorem Assume that

df (t,T) =η(t,T) dt+ θ^>(t,T)

1 k

dW^P(t)

k 1

.

Since f (t,T) = ^∂^ln^P_∂u⁽^t,u⁾

u=T then we apply Itô’s lemma to lnP(t,u)in order to determine coe¢ cientsη andθ.We obtain

η(t,T) = δ^>_T (t,T)

1 k

δ(t,T)

k 1

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)

1 k

dW^P(t)

k 1

anddf(t,T) = η(t,T) dt+ θ^>(t,T)

1 k

dW^P(t)

k 1

.

then

η(t,T) = δ^>_T (t,T)

1 k

δ(t,T)

k 1

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)

1 k

dW^P(t)

k 1

and

f (t,T) = ^∂^ln^P(t,u)

∂u _u₌_T .

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Forward rate IV

Itô’s lemma, applied to lnP(t,T), yields dlnP(t,T)

= ¹

P(t,T)^dP(t,T) ¹ 2

1 P(t,T)

2

dh^Pit

= α(t,T) dt+ δ^>(t,T)

1 k

dW^P(t)

k 1

! 1

2δ^>(t,T)

1 k

δ(t,T)

k 1

dt

In integral form, we obtain

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

= Zt

0 α(s,T) ¹

2δ^>(s,T)

1 k

δ(s,T)

k 1

! ds+

Z t

0 δ^>(s,T)

1 k

dW^P(s)

k 1

.

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Forward rate V

Since

f (t,T) = ^∂^ln^P(t,u)

∂u _u₌_T , then

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

= ^∂^ln^P(t,T)

∂T +^∂^ln^P(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 Zt

0 α(s,T) ¹

2δ^>(s,T)

1 k

δ(s,T)

k 1

! ds+

Zt

0 δ^>(s,T)

1 k

dW^P(s)

k 1

!

= Zt

0 δ^>T(s,T)

1 k

δ(s,T)

k 1

αT(s,T)

! ds

Zt

0 δ^>T(s,T)

1 k

dW^P(s)

k 1

sinceδ^>(s,T)δ(s,T) =_∑^k_i₌₁δ²_i (s,T)implies that

∂

∂Tδ^>(s,T)δ(s,T) = ^∂

∂T

∑k i=1

δ²_i (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 established the following relation:

α(t,T) r(t) = δ^>(t,T)

1 k

Λt k 1

and we have constructed the measure Q and the Q Brownian motionW^Q.

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) dW^P(t)

= θ^>(t,T)

Z T

t θ(t,s)ds+_Λt dt+ θ^>(t,T) dW^P_t since θ^>(t,T) = δ^>_T (t,T)

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Forward rate IX

Let’s now determine the stochastic di¤erential equation f ( ,T)under the risk-neutral measure Q :

df(t,T)

= θ^>(t,T) Z_T

t θ(t,s)ds+Λt dt+ θ^>(t,T) dW^P_t

= θ^>(t,T) Z_T

t θ(t,s)ds+Λt dt+ θ^>(t,T) d W^Q_t

k 1 Z_t

0 Λs k 1

ds

!

= θ^>(t,T) Z _T

t θ(t,s)ds+_Λ_t θ^>(t,T) Λt dt +θ^>(t,T) dW^Q_t

= θ^>(t,T) ZT

t θ(t,s)ds dt+ θ^>(t,T) dW^Q_t

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Short rate I

Recall that

df(t,T) = θ^>(t,T) Z_T

t θ(t,s)ds+_Λ_t dt+ θ^>(t,T) dW^P_t df(t,T) = θ^>(t,T)

Z_T

t θ(t,s)ds dt+ θ^>(t,T) dW^Q_t .

In integral form, we obtain f (t,T) f (0,T) =

Z _t

0 θ^>(u,T)

Z _T

u θ(u,s)ds+_Λu du +

Z t

0 θ^>(u,T) dW^P_u f (t,T) f (0,T) =

Z t

0 θ^>(u,T)

Z T

u θ(u,s)ds du +

Z t

0 θ^>(u,T) dW^Q_u.

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Short rate II

So, the short rate is r(t)

= f (t,t)

= f (0,t) +

Z t

0 θ^>(u,t)

Z t

u θ(u,s)ds+Λu du +

Z _t

0 θ^>(u,t) dW^P_u

= f (0,t) +

Z t

0 θ^>(u,t)

Z t

u θ(u,s)ds du +

Z t

0 θ^>(u,t) dW^Q_u. Bisière, p.59-61.

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References

Martin Baxter and Andrew Rennie (1996). Financial Calculus, an introduction to derivative pricing, Cambridge university press.

Christophe Bisière (1997). La structure par terme des taux d’intérêt, Presses universitaires de France.

Damien Lamberton and Bernard Lapeyre (1991).

Introduction au calcul stochastique appliqué à la …nance, Ellipses.