lecture 7 interest rate models i: short rate models€¦ · lecture 7 interest rate models i: short...

LECTURE 7

Interest Rate Models I: Short Rate Models

Spring Term 2012

MSc Financial Engineering

School of Economics, Mathematics and Statistics

Birkbeck College

Lecturer: Adriana Breccia

email: [email protected]

Lecture notes will be posted at

http://www.ems.bbk.ac.uk/for_students/msc_finEng/pricing_emms014p/index_html.

This note is part of a set of lecture notes, written by Raymond Brummelhuis,

PRICING II: MARTINGALE PRICING ( PricingII-Lecture Notes (2007).pdf available at

http://econ109.econ.bbk.ac.uk/fineng/RB_Pricing/.

1

http://www.ems.bbk.ac.uk/for_students/msc_finEng/pricing_emms014p/index_html

http://econ109.econ.bbk.ac.uk/fineng/RB_Pricing/

PRICING II: MARTINGALE PRICING 55

5. Lecture V: Interest Rate Models I: Short Rate Models

The earliest interest rate models took as their starting point a sto-chastic model for the short rate, or instantaneous interest rate, rt de-fined as the rate of interest for the (infinitesimal) interval [t, t+ dt]:

(106) rtdt = total interest gained in [t, t+ dt].

In practice, one takes the yield in a 1 month US Treasury bill, or acomparable sort-maturity bond, as a proxy for the sort rate.

Starting point of these sort rate models is a SDE for rt:

(107) drt = α(rt, t)dt+ σ(rt, t)dWt,

for given coefficients α(r, t) and σ(r, t).

Examples 5.1. Important examples of short rate models are:

(i) The Vasicek model (historically the first):

(108) drt = α(θ − rt)dt+ σdWt.

Observe that the Vasicek-model is mean reverting (since it is simplyan Ornstein-Uhlenbeck process: cf. Math. Methods I), which is eco-nomically reasonable (interest rates should fluctuate along a long termmean equilibrium rate, determined by economic equilibrium betweendemand and supply), and a common feature of all short rate models.A disadvantage of the Vasicek model is that interest rates can becomenegative, which is economically undesirable (why?), although it doesseem to happen every once and then.

(ii) The Cox, Ingersoll and Ross model, or CIR-model:

(109) drt = α(θ − rt)dt+ σ√rt dWt.

The CIR-model has the advantage that the short rate will stay positive(≥ 0), with probability 1.

(iii) The Longstaff model:

(110) drt = α(θ −√rt)dt+ σ

√rt dWt,

a.k.a. the double square root process.

(iv) Hull and White’s generalized Vasicek model, in which one allowsthe coefficients to be (deterministic) functions of time:

(111) drt = α(t)(θ(t)− rt)dt+ σ(t)dWt.

A disadvantage of the earlier models is that they have too little pa-rameters to be able to closely fit the initial yield curve at a given time.Making the coefficients time-dependent (even one of them, usually θ)enables one to have an exact fit. One can similarly introduce time-dependent coefficients in the CIR or in Longstaff’s model.

56 PRICING II: MARTINGALE PRICING

(iv) The lognormal model, (Black, Derman and Toy , Sandmann andSondermann):

(112) d log rt = α(t)(θ(t)− log rt)dt+ σ(t)dWt,

with deterministic time-dependent coefficients α(t), θ(t) and σ(t). In-terest rates clearly stay positive, and are mean-reverting, but prices ofcertain instruments van become infinite, which is a drawback.

(v) The Ahn and Gao model:

(113) drt = α(θ − rt)rt dt+ σr3/2t dWt,

where one could allow for deterministic time-dependent coefficientsalso.

The first aim of a short rate model (indeed, of any interest ratemodel) is to price zero-coupon bonds. A zero-coupon bond (also calleda discount bond) is a bond which does not pay any coupons, but whichpays its nominal-, or face-, value at maturity T . The face value willusually be normalized to 1 (of whatever currency we’re working in:Sterling, $, Euro, etc.). We will let

Pt,T ,

be the price at t ≤ T of a 0-coupon with maturity T and face valuePT,T = 1. We will suppose that there exist 0-coupon bonds of all pos-sible maturities T ≥ 0. This is of course not entirely realistic: firstof all, there are only a finitely many maturities traded, and secondly,the longer-maturity bonds will pay coupons; however, these can be de-composed (‘stripped’) into a series of 0-coupon bonds with maturitiescorresponding to the coupon dates. Note, that having 0-coupon bondsof all maturities T ≥ 0 means that we are dealing wit a market withinfinitely many (in practice, a very large number of) assets.

In a second stage, we would also like to price options on bonds, likesimple calls and puts on zero-coupon bonds of a given maturity. Thefact that interest rates are stochastic, and closely correlated with thepay-offs will make the pricing of such options more complicated thanfor options on simple stock.

The short rates themselves are not directly traded, only the (zero-)coupon bonds are. This makes the short rate models effectively in-complete: the basic risk-factor rt is not a tradable, and we can onlytrade rt, which is basically a European derivative on rt, with a pay-offof 1 at maturity. In particular, we cannot hedge Pt,T by buying orselling units of rt, we can only hedge a bond of a given maturity bybuying/selling bonds of another maturity. This will lead to the intro-duction of a market price of risk, which makes up for the difference indrift between the stochastic evolution of the interest rate with respectto the objective (or physical) probability, and the risk-neutral one. Itsorigin is probably most easily explained using the ‘risk-less portfolio’


approach from PDE-style pricing, so we (uncharacteristically, for theselectures) do this approach first.

5.1. Derivation of the Bond price equation. . We suppose thatthe price Pt,T is a function of time t and the short-rate rt at t:

Pt,T = p(rt, t;T ).

(This can be a posteriori justified). An standard application of Ito’slemma then yields that

(114) dPt,T = at,TPt,Tdt+ bt,TPt,TdWt, t < T,

with

(115) bt,T =σ(rt, t)

p(rt, t)

∂p

∂r,

and

(116) at,T =1

p(rt, t)

(∂p

∂t+ α(rt, t)

∂p

∂r+

1

2σ(rt, t)

2∂2p

∂r2

),

all derivatives of p evaluated in (rt, t).

Now consider two zero-coupon bonds having maturities T1 < T2. Wetry to make a locally risk-free portfolio

Vt = ∆1,tPt,T1 −∆2,tPt,T2 .

Computing the change in value between t and t+ dt as

dVt = ∆1,t dPt,T1 −∆2,t dPt,T2 =

(∆1,tat,T1Pt,T1 −∆2,tat,T2Pt,T2) dt+ (∆1,tbt,T1Pt,T1 −∆2,tbt,T2Pt,T2) dWt,

we see that the risk over [t, t+ dt] vanishes if:

∆1,t

∆2,t

=bt,T2Pt,T2

bt,T1Pt,T1

.

We can take, for example,

∆1,t = bt,T2Pt,T2 , ∆2,T = bt,T1Pt,T1 .

Since such a risk-free portfolio can only earn the risk-less rate, rt, over[t, t+ dt], we must have, for these ∆, that

(∆1,tat,T1 −∆2,tat,T2) dt = rt (∆1,tPt,T1 −∆2,tPt,T2) dt,

or

(at,T1bt,T2 − at,T2bt,T1)Pt,T1Pt,T2 = rt (bt,T2 − bt,T1)Pt,T1Pt,T2 .

Re-arranging, this gives:

at,T1 − rt

bt,T1

=at,T2 − rt

bt,T2

.


In other words,

at,T − rt

bt,T=

return on Pt,T − rt

volatility of Pt,T

is independent of the maturity T and can therefore only be a functionof rt and t, say q(rt, t). This function q = q(r, t) is called the marketprice of risk: using the standard deviation as a measure of financialrisk, we have that

• q(r, t) = extra return per unit of risk an investor requires to holdthe bond Pt,T at time t, if rt = r.

(for return on Pt,s = rt + q(rt, t)· (volatility of Pt,s ) .)

Inserting formulas (115) and (116) into the relation

at,T − rt

bt,T= q(r, t),

we find:

Proposition 5.2. The bond pricing function p(r, t;T ) satisfies the fol-lowing PDE:

(117)∂p

∂t+

1

2σ(r, t)2∂

2p

∂r2+ (α(r, t)− q(r, t)σ(r, t))

∂p

∂r= rtp, t < T,

with final value

(118) p(r, T ;T ) = 1.

So we have derived a boundary value problem for the bond-price,and all that is left is to solve it for the various models proposed. Thiscan be done either by attacking the PDE directly, or by using Feynmanand Kac. As an example we look at the extended Vasicek model of Hulland White.

5.2. Solving the Vasicek equation: analytic approach. We takea(r, t) = α(θ − r) and σ(r, t) = σ, with α. θ and σ constants, so that

drt = α(θ − rt)dt+ σdWt.

For the market price of risk we simply chooses a constant: q(r, t) = q aconstant. Note that in that case, α(r, t)−q(r, t)σ(r, t) = α(θ−r)−qσ =α(θ∗ − r), with θ∗ = θ − qσ/α. The bond price equation becomes:

(119)∂P

∂t+ α (θ∗ − r)

∂P

∂r+σ2

2

∂2P

∂r2= rP, on t < T,

which we have to solve with boundary condition (118). To do thisanalytically, we look for solutions having the special form:

(120) eA(t;T )−rB(t;T ).

To satisfy the boundary condition, we have to have

(121) A(T ;T ) = 0, B(T ;T ) = 0.


Substituting (120) into (119) gives:

(122)∂A

∂t− αθ∗B +

σ2

2B2 = r ·

(∂B

∂t− αB + 1

)Note that, for fixed maturity T , A and B are functions of t only. De-riving both sides with respect to r gives:

(123)∂B

∂t− αB + 1 = 0,

and, upon substituting this again in (122):

(124)∂A

∂t− αθ∗B +

σ2

2B2 = 0.

General solution of (123):

Ceαt +1

α.

By the second equation of (121), this has to be 0 for t = s. It followsthat C = e−αs/α, and therefore:

(125) B(t, T ) =1

α

(1− eα(t−T )

).

¿From this and (124) and (122), A can be found by integration:

(126) A(t;T ) = −αθ∗∫ T

t

B(τ, T )dτ +σ2

2

∫ T

t

B(τ, T )2dτ

Observe that these integrals are automatically 0 if t = T ; the extraminus-sign is explained by the fact that differentiating an integral w.r.t.the lower bound of the integration domain gives minus the integrand.)

After some calculations one finds that A(t;T ) can be written as:

A(t;T ) =

(θ∗ − σ2

2α2

)(t−T )+

1

α

(1− eα(t−T )

)(θ∗ − σ2

2α2

)− σ2

4α3

(1− eα(t−T )

)2.

Remembering (120) and (125), and introducing the quantity:

R∞ = θ∗ − σ2

2α2,

we find that:

pVasicek(r, t;T ) =(127)

exp(R∞(t− s) + 1

α

(1− eα(t−T )

)(R∞ − r)− σ2

4α3

(1− eα(t−T )

)2).


5.3. Solving the Vasicek equation: probabilistic approach. Weknow, by the Feynman-Kac theorem (see Math. Methods I), that thesolution to (119) with boundary value p(r, T ;T ) = F (r) is given by:

(128) p(r, t, ;T ) = E(

exp

(−∫ T

t

rτdτ

)F (rT , T )|rt = r

),

where the hatted, ‘risk-adjusted’ process follows the SDE:

dtrt = α(θ∗ − r)dt+ σdWt.

This is again an Ornstein-Ulenbeck SDE, whose solution we know isfor times between t and T , with initial value r at t, is given by:

(129) rτ = θ(1− e−α(τ−t)

)+ re−α(τ−t) + σ

∫ τ

t

eα(s−τ)dWs.

In particular, we know that rτ is Gaussian, whose mean and variancewe computed in Math. Methods I. Applying (128) with F (r) = 1 gives

p(r, t;T ) = E(

exp

(−∫ T

t

rτdτ

)|rt = r

),

and to be able to proceed, we have to analyze the integrated process

IT =

∫ T

t

rτdτ.

Inserting (129), and changing order of integration, we find that

IT =∫ T

t

θ(1− e−α(τ−t)

)dτ +

∫ T

t

re−α(τ−t)dτ + σ

∫ T

t

∫ τ

t

eα(s−τ)dWsdτ

= θ(T − t)− θ

α(1− e−α(T−t)) +

r

α(1− e−α(T−t)) + σ

∫ T

t

dWs

∫ T

s

eα(s−τ)dτ

= θ(T − t)− θ

α(1− e−α(T−t)) +

r

α(1− e−α(T−t)) +

σ

α

∫ T

t

(1− eα(s−T ))dWs.

It follows from the theory of Ito integrals with deterministic integrandsthat IT is again Gaussian, with mean

µI = θ(T − t)− θ

α(1− e−α(T−t)) +

r

α(1− e−α(T−t)),

and variance

σ2I =

σ2

α2E

((∫ T

t

(1− eα(s−T ))dWs

)2)

=σ2

α2

∫ T

t

(1− 2eα(s−T ) + e2α(s−T )) ds

=σ2

α2(T − t)− 2

σ2

α3(1− e−α(T−t) +

σ2

2α3(1− e−2α(T−t)).


Now we know from Math Methods I (and this is also easily verifieddirectly), that if I ∼ N(µI , σ

2I ), then

E(e−I)

= e−µI+σ2I/2.

Substituting the expressions we obtained for µI and σ2I , and re-arranging,

we find once more formula (127) for p = pVasicek = E(exp(IT )).

The Vasicek model as three parameters, α, θ∗ and σ with which tofit bond prices of all maturities, which there are in principle an infinitenumber, and one doesn’t always get a good fit. This difficulty can becircumvented by letting the coefficients depend deterministically on t:this will allow us to fit at least any initial yield curve:

5.4. Hull and White’s extended Vasicek model. We look at thesimplest variant, with α and σ constant, and θ∗ = θ∗(t) a deterministicfunction of t. Solving the new bond price equation,

∂p

∂t+ α (θ∗(t)− r)

∂p

∂r+σ2

2

∂2p

∂r2= rp, on t < T,

using for example the direct, analytical method, and looking for solu-tions of the same form (120), 121) as before, we find that A(t;T ) andB(t;T ) have to satisfy the same equations (123), (124), but the latterof course with θ∗(t). B(t, T ) will be given by the same function (125)as before, but (126) will have to be changed to:

(130) AHW (t;T ) = −∫ T

t

θ∗(τ)B(τ, T )dτ +σ2

2

∫ T

t

B(τ ;T )2dτ

= −∫ T

t

θ∗(τ)B(τ, T )dτ+σ2

2α2

((t− T )− 2

αeα(t−T ) +

1

2αe2α(t−T ) +

3

2α

).

Hence,

(131) pHW(r, t;T ) = exp(AHW(t;T )− rB(t;T ))

where the suffix ”HW” stands for ”Hull and White”. Making θ∗ time-dependent may look like introducing an additional complication, butin fact gives an opportunity to do:

Exact Yield curve fitting: We suppose that σ and α are alreadydetermined from examining time series data for short term interestrates (a typical econometrical problem). Then we find θ∗ by fittingat date t = 0 (‘today’) the theoretical prices pHW(r0, 0, ;T ), where r0is today’s observed short rate, to the observed market prices P0,T =PMarket

0,T for all maturities T > 0. Taking logarithms, we find that θ∗(τ)has to satisfy the integral equation

−∫ T

0θ∗(τ)B(τ ;T )dτ =

logP0,T + rB(0;T )− 12σ2∫ T

0B(τ, T )2dτ =

logP0,T + r0

α(1− e−αT )− σ2

2α2

(T + 2

αe−αT − 1

2αe−2αT − 3

2α

).(132)


This equation can be solved for θ∗(T ) by differentiating twice withrespect to the maturity T :

∂

∂T

∫ T

0

θ∗(τ)B(τ ;T )dτ = θ∗(T )B(T ;T ) +

∫ T

0

θ∗(τ)∂B(τ ;T )

∂T

=

∫ T

0

θ∗(τ)eα(τ−T )dτ

= e−αT

∫ T

0

θ∗(τ)eατdτ,

whence

∂2

∂T 2

∫ T

0

θ∗(τ)B(τ ;T )dτ = θ∗(T )− αe−αT

∫ T

0

θ∗(τ)eατdτ.

It follows that

θ∗(T ) =∂2

∂T 2

∫ T

0

θ∗(τ)B(τ ;T )dτ − α∂

∂T

∫ T

0

θ∗(τ)B(τ ;T )dτ,

and substituting the right hand side of (132) one finds, after somealgebra, that

(133) θ∗(T ) = − ∂2

∂T 2logP0,T − α

∂

∂TlogP0,T +

σ2

2α

(1− e−2αT

).

To use formula (133) in practice, we need to construct a smooth (thatis, twice-differentiable) curve P0,T out of the finitely many bonds whichare in fact quoted (corresponding to some finite sequence of maturitiesT1, · · · , TN). This is usually done by spline interpolation.

Finally we mention a problem with yield curve fitting. Suppose wecalibrate with today’s yield curve, and find a function θ∗now(t). If wecalibrate again in one week’s time, say, we would in general find adifferent function θ∗then(t), whereas this function should be the same, atleast if the theory consistently describes reality. A slightly non-realisticaspect of the Vasicek model, and other one-factor models, is of coursethat there is only one risk-factor: Wt, driving the sort rate, for thewole range of maturities. One could reasonably argue that at the longend of the yield curve (that is, for big T ), other risks will come intoplay. This has led to the introduction of multi-factor models,: for areview of the literature, see Musiela and Rutkowski, section 12.4. TheHJM-models which we will study in the next lecture are typically alsomulti-factor.

5.5. Martingale approach to short rate models. Introduce, asbefore, the savings account Bt

(134) Bt = e∫ t0 rsds.

This is simply the amount of money in a bank account to which an ini-tial deposit of 1, continuously re-invested at the short rate rt, as grown


at time t (to see this, somply observe that dBt = rtBtdt, and integrate).We are then dealing with a market consisting of the savings bond Bt,and infinitely many 0-coupons Pt,T , T > t. Extrapolating from themulti-asset markets treated in Lecture IV (1 risk factor, and infinitelymany assets), we expect that there will be absence of arbitrage iff thereis an E(quivalent) M(artingale) M(easure), Q for the discounted assetsPt,T/Bt. That is, if t < u ≤ T ,

(135) B−1t Pt,T = EQ

(B−1

u Pu,T |Ft

).

In particular, taking u = T , and using PT,T = 1, and (134), we find thepricing equation:

(136) Pt,T = EQ

(e−

∫ Tt rsds|Ft

).

Observe that we obtain a similar equation (but with rt replaced byrt) by applying the Feynman-Kac formula to the bond price equation(117): changing from rt to rt can also be understood as replacing theSDE for rt, drt = αtdt + σtdWt (where we have written αt = α(rt, t),σt = σ(rt, t)) by

drt = (αt − qtσt) dt+ σt)dWt, qt = q(rt, t),

with dWt = q(rt, t)dt + σ(rt, t)dWt being a Brownian motion with re-spect to some new probability measure Q, obtained from Girsanov’stheorem.

Conversely, it can be proved that on the Brownian FWt , any EMM

is of the Girsanov form:

dQ = exp

(∫ t

0

−γsdWs −1

2

∫ t

0

γ2sds

),

for some adapted process γt, with Wt, defined by dWt = γtdt+ dWt, aBrownian motion with respect to Q. With respect to Q, rt follows theSDE

drt = (αt − γtσt) dt+ σdWt,

so that γt is the price of risk.In the case of for example the Vasicek model, formula (136), together

withdrt = α(θ∗ − rt)dt+ σdWt,

Wt a Q-Brownian motion, can be used to directly compute the 0-couponprices, by the same arguments as in 5.3 above.


5.6. Exercises to Lecture VI.

Exercise 5.3. Write out the details of the computation leading to(114), (115), (116).

Exercise 5.4. Verify equation (133), and find a formula for Pt,T interms of today’s market prices, P0,T .

Exercise 5.5. A first order ODE of the form

y′ = p(x)y2 + q(x)y + r(x),

for an unknown function y = y(x), with given functions p(x), q(x) andr(x), is called a Riccati equation. As usual, y′ = dy/dx. Suppose wedispose of a particular solution y0 = y0(x). Then we can transform theoriginal equation into a first order linear equation, which (being firstorder) is explicitly solvable. Indeed, let u = u(x) be defined by

y = y0 +1

u,

or, equivalently,

u(x) =1

y(x)− y0(x).

(Let us not worry about the possible singularity in points x for whichy(x) = y0(x): this would have to be studied in any particular case towhich we would apply this method).

a) Show that u = u(x) satisfies the ODE

−u′ = p(x) + 2p(x)y0(x)u+ q(x)u.

b) Consider from now the special case in which p(x), q(x) and r(x) areconstants, p, q and r, respectively. Show that y0(x) = v± are specialsolutions of

(137) y′ = py2 + qy + r,

if v± are the zeros of

(138) pv2 + qv + r = 0,

that is,

v± =−q ±

√q2 − 4pr

2p.

c) Take y0 = v− as special solution. Show that u = u(x) satisfies

u′ = −p+ du,

where d :=√q2 − 4pr is the discriminant of the quadratic equation

(138). Solve this equation by the usual method (”general solution =particular solution + general solution homogeneous equation”) to find

u(x) =p

d+ Cedx,

with C an arbitrary constant. d) Conclude that the general solution of


(137) is given by

(139) y(x) =(pv− + d) + C ′edx

p+ C ′edx,

where C ′ is some constant (C ′ = dv−C if C is the constant from partc) ).

e) Determine C ′ such that y(0) = 0, and show that the correspondingy is given by

(140) y(x) =v+(1− edx)

1− v+edx,

where you will have to use that

pv− + d = pv+.

(verify this!).

f) Show that, if y(x) is given by (140),

(141)

∫ x

0

y(s)ds = x−(v+ − 1

v+d

)log(1− v+e

dx).

(Hint: Observe that

1− edx

1− v+edx= 1+(v+−1)

edx

1− v+edx= 1−

(v+ − 1

v+d

)(log(1− v+e

dx))′. )

Exercise 5.6. We can use the results of the previous exercise to findthe price of a zero coupon bond in the CIR model:

drt = α(θ − rt)dt+ σ√rtdWt.

a) Let Pt,T = pT (r, t) be the price of a pure discount bond of maturityT . Show that, under the assumption that the market price of risk isa constant times

√r (Corrected!), p = pT solves the boundary value

problem

∂p

∂t+ α (θ∗ − r)

∂p

∂r+

1

2σ2r

∂2p

∂r2= rp, t < T,(142)

p(r, T ) = 1,

where θ∗ is the risk-adjusted mean rate.

b) Making the ”Ansatz”12

(143) p(r, t) = eA(τ)−rB(τ),

12meaning trial solution


where τ := T − t, and A(τ) and B(τ) are functions of τ only, show thatthe latter ave to satisfy the following system of differential equations:

dA

dτ= −rB,

dB

dτ= −σ

2

2B2 − αB + 1,

with initial conditions A(0) = B(0) = 1.(N.B. The reason for introducing the new time variable τ is to haveinitial conditions at τ = 0, instead of final conditions at t = T , simplybecause this is computationally more convenient).

c) You should recognize the differential equation for B(τ) as a Riccatiequation. Use the results of the previous exercise on such equationsto find A(τ) and B(τ), and to derive an explicit formula for the bondprice in the CIR model. Ceck your answer against formulas which canbe found in the literature.

lecture 7 interest rate models i: short rate models€¦ · lecture 7 interest rate models i: short...

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