amf 241 interest rate models

8/9/2019 AMF 241 Interest Rate Models

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Bonds and Interest Rates


Timothy Robin Teng

Bonds and Interest Rates 1 / 54

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Zero coupon bonds and related interest rates

DefinitionA zero coupon bond (or pure discount bond) with maturity date T ,also called a T -bond, is a contract which guarantees its holder the

payment of one unit of currency at time T , with no intermediatepayments. The contract value at time t < T is denoted by P (t , T ).

The convention that the payment at the maturity date, known as the

principal value or face value, equals one is made for computationalconvenience.Note that P (t , t ) = 1 for all t .


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The bond price P (t , T ) is also a stochastic object with two variablest and T , and for each outcome ω in the underlying sample space, thedependence upon these variables is different.

Fix t ; P (t , T ) is a function of maturity date T

The function provides prices for bonds of all possible maturitiesat a fixed time t . The graph of the function is called “bondprice curve at t ”, or “the term structure of t ”. Typically it willbe a very smooth graph, i.e. for each t , P (t , T ) is differentiablewith respect to T .


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Fix T ; P (t , T ) (as a function t ) will be scalar stochastic processThe process gives the prices, at different times, of the bond with

fixed maturity T , and the trajectory will typically be veryirregular.


B d d I t t R t

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Interest Rates

We may now define a number of interest rates based on the zerocoupon bond, and the basic construction is as follows:

We consider three time instants, namely the time t at which the rateis considered, and two other points in time T and S , such thatt < S < T . The goal is to write a contract at time t which allows usto make an investment of one (dollar) at time S , and to have adeterministic rate of return (determined at the contract time t ) overthe interval [S , T ].



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At time t we sell one S -bond. This will give us P (t , S ) dollars.

We use this income to buy exactly P (t , S )/P (t , T ) T -bonds.Thus our net investment at time t equals zero.

At time S , the S -bond matures, so we are obliged to pay outone dollar.



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At time T , the T -bonds mature at one dollar a piece, so we willreceive the amount P (t , S )/P (t , T ) dollars.

The net effect of all this is that, based on a contract at t , aninvestment of one dollar at time S has yielded P (t , S )/P (t , T )

dollars at T .

Thus, at time t , we have made a contract guaranteeing ariskless rate of interest over the future interval [S , T ]. Such aninterest rate is called a forward rate.



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We now proceed to compute the relevant interest rates implied bythe construction above. The simple forward rate (or LIBOR rate) L,is the solution to equation

1 + (T − S )L = P (t , S )

P (t , T )

whereas the continuously compounded forward rate R is the solutionto the equation

e R (T −S ) = P (t , S )

P (t , T )

The simple rate notation is the one used in the market, whereas thecontinuously compounded notation is used in theoretical contexts.



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The simple forward rate for [S , T ] contracted at t , henceforthreferred to as the LIBOR forward rate, is defined as

L(t ; S , T ) = − 1

T − S

P (t , T ) − P (t , S )

P (t , T )

The simple spot rate for [S , T ], henceforth referred to as theLIBOR spot rate, is defined as

L(S , T ) := L(S ; S , T ) =

− 1

T − S P (S , T ) − 1

P (S , T )



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The continuously compounded forward rate for [S , T ]contracted at t is defined as

R (t ; S , T ) = − log P (t , T ) − log P (t , S )

T − S

The continuously compounded spot forward rate, R (S , T ),for the period [S , T ] is defined as

R (S , T ) := R (S ; S , T ) =

−log P (S , T )

T − S



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The instantaneous forward rate with maturity T , contractedat t , is defined by

f (t , T ) =

−

∂

∂ T

log P (t , T )

The instantaneous short rate at time T is defined by

r t = f (t , t )


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The short rate of interest r t , is the (annualized) interest rate at whichan entity can lend or borrow money over an infinitesimally smallinterval [t , t + dt ]. As opposed to stock prices, short rates tend tostay within a certain range, hence they are often described as mean

reverting processes. From the short rate r t , we can define animportant instrument in the market which is the money marketaccount, representing a (locally) riskless investment where profit isaccrued continuously at rate r t .



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Remark: One should note that in practice, the process r t is notobservable. The shortest maturity rate available is the overnight rate,which is conceptually quite different from an instantaneous spot rate.

Nevertheless, it is quite frequent that one-night and even one-monthor three-month rates are used as proxies for r t in the empirical termstructure literature



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Definition(Money-market account). We define B t to be the value of a bankaccount at time t ∈ R+, and its given by

B t = e t

0 r (u )du

that is, it evolves according to the following initial value problem:

dB t = r t B t dt , B 0 = 1



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As an immediate consequence of the definitions we have the followinguseful formula.

LemmaThe price of the bond can be expressed as

P (t , T ) = exp− T

t f (t , u )du

and for t ≤ s ≤ T , we have

P (t , T ) = P (t , s ) · exp− T

s

f (t , u )du



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Relations between df (t , T ), dP (t , T ) and dr (t )

We will consider dynamics of the following form:Short rate dynamics

dr (t ) = a(t )dt + b (t )dW t (1)

Bond price dynamics

dP (t ,T ) = P (t ,T )m(t ,T )dt + P (t ,T )v (t ,T )dW t (2)

Forward rate dynamics

df (t ,T ) = α(t ,T )dt + σ(t ,T )dW t (3)

The processes a

(t

) and b

(t

) are scalar adapted processes, whereasm(t ,T ), v (t ,T ), α(t ,T ) and σ(t ,T ) are adapted processes paramaterized bytime of maturity T . The interpretation of the bond price equation (2) and theforward rate equation (3) is that these are scalar stochastic differential equations(in the t -variable) for each fixed time of maturity T .



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We will study the formal relations which must hold between bondprices and interest rates, and for this we establish the following

assumptions:1 For each fixed ω, t all the objects m(t , T ), v (t , T ), α(t , T ) and

σ(t , T ) are assumed to be continuously differentiable in theT -variable. This partial T -derivative is sometimes denoted by

mT (t , T ) etc.2 All processes are such that we can differentiate under the

integral sign as well as interchange the order of integration.

The main result is as follows. Note that the results below hold,regardless of the measure under consideration, and in particular, wedo not assume that markets are free of arbitrage.



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Proposition.

1 If P (t , T ) satisfies

dP (t , T ) = P (t , T )m(t , T )dt + P (t , T )v (t , T )dW t

the the forward rate dynamics have

df (t , T ) = α(t , T )dt + σ(t , T )dW t

where α and σ are given by α(t , T ) = v T (t , T ) · v (t , T ) − mT (t , T )σ(t , T ) = −v T (t , T )



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Proposition

2. If f (t , T ) satisfies


then the short rate satisfies

dr (t ) = a(t )dt + b (t )dW t

where a(t ) = f T (t , t ) + α(t , t )b (t ) = σ(t , t )



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Proposition

3. If f (t ,T ) satisfies

df (t ,T ) = α(t ,T )dt + σ(t ,T )dW t

then P (t ,T ) satisfies

dP (t ,T ) = P (t ,T )

r (t ) + A(t ,T ) +

12S

2(t ,T )

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

where

A(t ,T ) = − T

t α(t , s )ds

S (t ,T ) = − T t σ(t , s )ds



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Short rate models

Interest models behave differently from stock prices and require thedevelopment of specific models to account for properties such aspositivity, boundedness and return to equilibrium. Refer to the other

handouts for a list of short rate models

For this section, we consider classical time homogenous short ratemodels, i.e. the assumed short rate dynamics depended only constantcoefficients. The focus would be on the Vasicek (1977), the Dothan

(1978) and the Cox, Ingersoll and Ross (1985) models.



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The Vasicek Model (1977)

Vasicek introduced the first model to capture the mean reversionproperty of interest rates. In the Vasicek model, which is based onthe Ornstein-Uhlenbeck process, the short term interest rate processr t satisfies the SDE

dr t = k (θ − r t ) dt + σdW t

where k , θ and σ are positive constants. The model has the propertyof being statistically stationary in time, i.e. the distribution of r t

−r s

depends only on t − s . However, for each time t , the short rate r t can be negative with positive probability.



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Solving the above equation will give us,

r t = θ + e −kt

r 0 − θ +

t

0

σe ku dW u

= r 0e −kt + θ 1 − e −kt + σ t

0

e −k (t −u )dW u

and for s ≤ t

r t = r s e −k (t −s ) + θ 1 − e −k (t −s ) + σ t

s




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Hence, r t conditional on F s is normally distributed with mean and

variance given respectively by

E [ r t | F s ] = E

r s e −k (t −s ) + θ

1 − e

−k (t −s )

+ σ

t

s

e −k (t −u )

dW u

F s

= r

s e −k (t −s )

+ θ

1 −e −k (t −s )

+ E

σ t s

e −k (t −u )

dW u F s

= r s e −k (t −s ) + θ

1 − e

−k (t −s )

+ E

σ

t

s

e −k (t −u )

dW u

= r s e −k (t −s ) + θ 1

−e −k (t −s )



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Var [ r t | F s ] = E (r t − E [ r t | F s ])2

F s = E

σ2

t

s


2F s

= σ2E t

s

e −k (t −u )dW u 2

= σ2E

t

s

e −2k (t −u )du

= σ2

E t

s e −2k (t −u )

du

= σ2

2k

1 − e −2k (t −s )



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Based on the conditional mean and variance, the short rate r t ismean reverting, since the expected rate approaches the value θ as t

goes to infinity. The fact that θ can be regarded as a long term

average rate could be inferred from the dynamics of the SDE itself.Notice that the drift of the process r t is positive whenever the shortrate is below θ and negative otherwise, so that r is pushed, at everytime t , closer towards the average level θ.



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The Dothan Model (1978)

The Dothan model basically follows the GBM dynamics for the shortrate

dr t = ar t dt + σr t dW t

in which case

r t = r s exp

a − 1

2σ2

(t − s ) + σ (W t −W s )

for s

≤t .



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Hence, r t conditional on F s is lognormally distributed with mean and

variance given by

E [ r t | F s ] = E

r s exp

a − 1

2σ2

(t − s ) + σ (W t −W s )

F s

= r s e (a− 1

2σ2)(t −s )

E

e σ(W t −W s )

|F s = r s e (a−

12σ2)(t −s )E

e σ(W t −W s )

W t −W s ∼ N (0, t − s )

= r s e (a−12σ2)(t −s )e

12σ2(t −s )

= r s e a(t −s )



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Var [ r t | F s ] = E

r 2t |F s − (E [r t |F s ])2

= E

r 2s e (2a−σ2)(t −s )+2σ(W t −W s )|F s − r 2s e 2a(t −s )

= r 2s e (2a−σ2)(t −s )E e 2σ(W t −W s )

|F s − r 2s e 2a(t −s )

= r 2s e (2a−σ2)(t −s )E

e 2σ(W t −W s )− r 2s e 2a(t −s )

= r 2s e (2a−σ2)(t −s )e 12

(2σ)2(t −s ) − r 2s e 2a(t −s )

= r 2s e 2a(t −s ) e σ2(t −s )

−1



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The lognormal distribution implies that r t is always positive for eacht , so that the main drawback of Vasicek is addressed here. However,the process r t is mean-reverting if and only if a < 0, with thereversion level that must be necessarily equal to 0.

Remark: An alternative to this, which is also a lognormal short ratemodel, is the Exponential Vasicek model, where the short rate isalways mean reverting (refer to handout for the formulation andproperties of the model)



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Cox-Ingersoll-Ross (1985)

The general equilibrium approach developed by Cox, Ingersoll andRoss (1985) led to the introduction of a “square root” term in thediffusion coefficient of the instantaneous short rate dynamics

proposed by Vasicek. The resulting model, which is also meanreverting, has been a benchmark for many years because of itsanalytical tractability and the fact that the short rate is alwayspositive.



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The model formulation in this case is given by

dr t = k (θ − r t )dt + σ√

r t dW t

where k , θ and σ are positive constants. The condition 2k θ > σ

2

hasto be imposed to ensure that the origin is inaccessible to the processdescribed above, so that we can grant that r t remains positive.



Here rt follows a noncentral chi-squared distribution. Denoting pY

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Here r t follows a noncentral chi squared distribution. Denoting p Y the density function of the random variable Y ,

p r t (x ) = p χ2(v ,λt )/c t (x ) = c t p χ2(v ,λt )(c t x )

c t = 4k

σ2 (1 − exp(−kt )) v =

4k θ

σ2 λt = c t r 0 exp(−kt )

where the noncentral chi-squared distribution function χ2(·, v , λ) with

v degrees of freedom and non-centrality parameter λ has density

p χ2(v ,λ)(z ) =∞i =0

e −λ/2 (λ/2)i

i ! p Γ(i +v /2,1/2)(z )

p Γ(i +v /2,1/2)(z ) = (1/2)i +v /2

Γ(i + v /2)z i −1+v /2e −z /2 = p χ2(v +2i )(z )

with p χ2(v +2i )(z ) denoting the density of a central chi-squareddistribution with v + 2i degrees of freedom.



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The mean and variance of r t conditional on F s are given by

E [ r t | F s ] = r s e −k (t −s ) + θ(1 − e −k (t −s ))

Var [ r t | F s ] = r s σ2

k

e −k (t −s )

− e −2k (t −s

+ θ

σ2

2k (1 − e −k (t −s )

)2



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We remark that while the above time homogenous models allow forthe analytical pricing of bond and bond options, these models don’t

tend to fit well with initial observed term structure in the market,regardless of how the parameters are chosen. Consequently, somepractitioners are reluctant to apply such kinds of models. Hence, wemay consider time dependent extensions of the above models. For

example, the Hull White model

dr t = k (θt − r t ) dt + σdW t

is a time dependent extension of the Vasicek model. The CIR also

admits a similar time-dependent extension (refer to handout for moremodels).



Bond price and risk neutral measure

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Bond price and risk neutral measure

We highlight the relationship between the bond price P (t , T ) and theshort rate process {r t }t ∈R+ under the absence of arbitrage conditionby considering the following scenarios:

The short rate is a deterministic constant r > 0. In this case,P (t , T ) should satisfy

e r (T −t )P (t , T ) = P (T , T ) = 1, t ∈ [0, T ]

thereforeP (t , T ) = e −r (T −t ), t ∈ [0, T ]



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The short rate is time dependent and deterministic function r t .In this case, it can be shown that

P (t , T ) = e − T t r u du , t ∈ [0, T ] (4)



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The short rate is a stochastic process r t . We remark that (4)does not make sense since the price P (t , T ) being set at time t ,can depend only on information known up to time t . In fact, theprice for this case would be given by

P (t , T ) = E P e −

T

t r u du F t

, t

∈[0, T ]

where the expectation is under the risk neutral measure P. Itgives us the “best possible estimate” of the future quantity

e − T

t r u du given information known up to time t . Furthermore, as

a conditional expectation with respect to F t , the bond priceP (t , T ) is F t -measurable.



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We remark that the risk neutral measure P is equivalent to P, underwhich for all s ∈ [0, T ], the process {P (t , s )}0≤s ≤T defined by

P (t , s ) = e − t

0 r u du P (t , s ) =

P (t , s )

B t

is a P-martingale.



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For the succeeding discussion on bond pricing, we will assume thatthe short rate process is under the measure P, and that the filtration{F t } is generated by the P-Brownian motion {W t }. Indeed, the termstructure, as well as prices of all other interest rate derivatives, arecompletely determined by specifying the dynamics of the short rateunder P, and the objective probability measure P can just be ignored.


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The arbitrage price of the bond is given by

P (t .T ) = E P

e −

T

t r s ds

F t = e A(t ,T )−B (t ,T )r t

where

B (t , T ) = 1b

1 − e −b (T −t )

and

A(t , T ) = 2ab

−σ2

2b 2 (B (t , T ) − T + t ) − σ2

4b B 2(t , T )


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Affine Term Structure Model

Affine term structure models are interest rate models where thecontinuously compounded spot rate R (t ,T ) is an affine function in theshort rate r t , that is

R (t ,T ) = α(t ,T ) + β (t ,T )r t

where α and β are deterministic functions of time. If this happens, themodel is said to possess an affine term structure.



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This relationship is always satisfied when the zero-coupon bond pricecan be written in the form

P (t , T ) = C (t , T )e −B (t ,T )r t

since then clearly it suffices to set

α(t , T ) = − ln C (t , T )/(T − t ) β (t , T ) = B (t , T )/(T − t )

Both Vasicek and CIR models seen earlier are affine models, sincetheir corresponding bond prices has the above form. The Dothanmodel is not an affine model.



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Now assume that we have the risk neutral dynamics for the short rate

given bydr t = b (t , r t )dt + σ(t , r t )dW t

The conditions for which b and σ admits an affine term structure isgiven by

b (t , x ) = λ(t )x + η(t )

σ(t , x ) =

γ (t )x + δ (t )

that is, b and σ2

are also affine functions themselves.



Forward Rate and the HJM condition

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We recall that instantaneous forward rate with maturity T ,contracted at t , is defined by

f (t , T ) = − ∂

∂ T log P (t , T )

If the short rate folows the Vasicek stochastic interest rate model,that is

dr t = (a − br t )dt + σdW t

then

f (t , T ) = r t e −b (T −t ) + a

b

1 − e −b (T −t )

− σ2

2b 2

1 − e −b (T −t )

2



Now suppose we determine the dynamics of the process f (t T ) of

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Now suppose we determine the dynamics of the process f (t , T ) of forward rates in the Vasicek model

df (t , T ) = e −b (T −t )dr t + be −b (T −t )r t dt +

be −b (T −t )dt

dr t

−ae −b (T −t )dt + σ2

b

1 − e −b (T −t )

e −b (T −t )dt

= e −b (T −t ) ((a

−br t )dt + σdW t ) + be −b (T −t )r t dt

−ae −b (T −t )dt + σ2

b

1 − e −b (T −t )

e −b (T −t )dt

= σ2

b 1 − e −b (T −t )

e −b (T −t )dt + σe −b (T −t )dW t

= σ2e −b (T −t ) T

t

e −b (s −t )ds

dt + σe −b (T −t )dW t



Hence df (t , T ) can be written as

( ) ( ) ( )

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with

σ(t , T ) = σe −b

(T −t

)

and

α(t , T ) = σ2e −b (T −t ) T

t

e −b (s −t )ds = σe −b (T −t )

T

t

σe −b (s −t )ds

= σ(t , T ) T

t

σ(t , s )ds

We note that the above relation between σ and α is a not acoincidence, but rather a general consequence of the absence of arbitrage hypothesis on the dynamics of forward rates. This will beelaborated in succeeding section.



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Heath-Jarrow-Morton (HJM) condition

In the HJM model, the dynamics for the instantaneous forward rate isgiven by


where the date T is fixed. At this point, our objective is to determinethe conditions in which the above equation will make sense in thefinancial context (that is, satisfying the absence of arbitragecondition)



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To determine this, we remark that at this point we have two differentformulas for bond prices

P (t , T ) = e − T

t f (t ,u )du and P (t , T ) = E

P

e −

T

t r u du

F t Both of these formulas should hold simultaneously so that there willbe absence of arbitrage.

This will ultimately yield the consistency relation between α and σ,given by

α(t , T ) = σ(t , T ) T

t

σ(t , s )ds



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References

1 Bjork, Tomas. Arbitrage Theory in Continuous Time, 2nd Edition. Oxford University Press, 2004.

2 Privault, Nicolas. An Elementary Introduction to Stochastic Interest Rate Modelling . World Scientific Publishing Co. Pte.

Ltd, 20083 Brigo, Damiano and Mercurio, Fabio. Interest Rate Models -

Theory and Practice (with Smile, Inflation and Credit). SpringerFinance. Springer-Verlag, Berlin, 2006.


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amf 241 interest rate models

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