antonov2.pdf

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66 RkMay 2011

cuttingedge.inteRestRates

underlie the rst stepsof quantitative nance

development. Te main feature of such models consists of postu-lating the short-rate process. Vasicek (1977) and Hull & White(1990) pioneered a Gaussian short-rate model still popular amongpractitioners due to its analytical tractability and transparency.Black & Karasinski (1991) have proposed a lognormal short-ratemodel. Both models share the same Gaussian mean-revertingprocess but with dierent interpretations in terms of the shortrate. Later, there appeared multi-factor generalisations (see, forexample, Hull & White, 1994) as well as functional generalisa-tions (for example, the generalised Black-Karasinski (BK) (our-ruco, Hagan & Schleiniger, 2007) and quadratic short-ratemodels (Piterbarg, 2009)). One can also mention the exactly solv-able Cox-Ingersoll-Ross (CIR) short-rate model (1985).

In this article, we consider a multi-factor mean-reverting Gaus-sian underlying process and view the short rate as being an arbi-trary function of it. We refer to such models as generalised short-rate models with Gaussian underlying (GSRGs). Note that theCIR model remains slightly apart from the GSRG frameworkbecause its short rate cannot be represented as a deterministicfunction of a Gaussian process.

In general, the GSRGs do not permit exact analytical interpre-tation for both zero-coupon bonds and swaption prices1 but haveattractive features of implied volatility skew control. Note that

the classic Hull & White model has a xed normal skew and theBK model possesses the almost-at implied volatility form due toits quasi-lognormality.

Te current post-crisis market favours simple models havingseveral essential elements such as the presence of a multi-factorunderlying (to treat correlations) and the implied volatility skewcontrol. Te usage of the arbitrage-free models was recentlyextended to economic scenario generation, especially in the insur-ance industry. However, the standard models lead to undesirablyextreme rates. Tus, a short-rate model with bounded evolutionof the rates can be attractive for practitioners.

Te GSRGs can satisfy these properties and thus become ade-

quate for the current market environment. What has been lack-ing is analytical tractability. Indeed, in spite of the simplicity ofthe lattice implementation for the GSRG, analytical expressionsfor swaption prices necessary for ecient calibration remainobscure. A step forward in this direction is the analytical expres-sion for zero-coupon bonds for the generalised BK model. our-ruco, Hagan & Schleiniger (2007) have derived the analyticalapproximation for the one-factor case.

In this article, we go further and come up with the multi-dimensional case analytical formulas for both zero-coupon bondand European swaption prices. Dierences from the ourruco,Hagan & Schleiniger approach, such as the underlying processdrift adaptation to the yield curve, wil l be explained.

We il lustrate the method with two examples. Te rst is ourversion of the generalised multi-factor BK model allowing forskew control. Te second is a new bounded short-rate modelwhere the rates evolve between cer tain user-dened limits.

Te article is organised as follows. First, we introduce nota-tions, present swaption pricing logic and derive the main partialdierential equations (PDEs). Ten we present the main results:quadrature formulas for the zero-coupon bond and the Arrow-Debreu (AD) price. We give two examples of the short-rate mod-els: our version of the generalised multi-factor BK model and thebounded short-rate model. Finally, we present a numerical conr-mation of the approximation accuracy, followed by conclusions.

Preliminary remarks

Here, we introduce the main notations, derive the PDEs anddetermine the expansion strategy.

Denote an underlyingF-factor mean-reverting process as:

dxi t( ) = i t( )ai t( )xi t( )( )dt+i t( ) dW t( )for i =1,L,F

(1)

for the vector volatilities li(t), vector uncorrelated Brownian

motions dW(t), mean-reversions ai(t) and drifts f

i(t), serving to

link the short-rate process to a given yield curve. Te dot opera-tion denes the vector inner product.

An equivalent, alternative form of the underlying can containcorrelated scalar Brownian motions dW

i(t) and scalar volatilities

gi(t), that is, dx

i(t) = (f

i(t) a

i(t)x

i(t))dt+ g

i(t)dW

i(t).

We postulate the short-rate process as an arbitrary function ofthe underlying processes:

r t( ) = f t, x( ) (2)

Having sucient freedom, we choose the zero initial values xi(0)

= 0. Now we dene the numeraire process, a savings account,Nt=

et0dsr(s). A zero-coupon bond with maturityTequals:

P t,T( ) = NtE1

NT

Ft

= E e

d s r s( )t

T

Ft

In other words, the zero-coupon bond is a function of the under-lying process. Here, and throughout this article, we refer to theoperatorE[] as the expectation in the risk-neutral measure.

General short-rate analyticsAlexandre AntonovandMichael Spectorpresent

an analytical approximation o zero-coupon bonds

and swaption prices or general short-rate models.

The approximation is based on regular and singular

expansions with respect to low volatility and contains

a low-dimensional integration. The model in hand

assumes the short rate is an arbitrary unction o

a multi-dimensional Gaussian underlying process.

The high approximation accuracy is confrmed bynumerical experiments

Short-rate models

1 Except the Hull-White model and the Piterbarg quadratic model (Piterbarg, 2009)


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risk.net/risk-magazine 67

Te discount factor D(T) P(0, T) is a zero-coupon bondmaturing at Tas seen from the origin. Denote todays forwardrate asR(t):

D T( ) = e d s R s( )

0

T

Ten, without loss of generality, we can represent 2 the short ratemoving around todays forward rate:

f t,0( ) = R t( )

(3)

Te discount factor is considered as a model input and the driftparameters f

i(t) should be chosen to t it:

D T( ) = E e ds f s,x s( )( )

0

T

(4)

Note that the drifts will have the second order in volatilities dueto the condition (3).

It is convenient to remove the mean-reverting term from theunderlyingxs using an appropriate multiplier:

yi t( ) = xi t( )Ai t( ) for Ai t( ) = e dsai s( )0t

We introduce normal processe syisatisfying the stochastic dier-

ential equation (SDE):

dyi t( ) = i t( )dt+ i t( ) dW t( )

(5)

where qi(t) = f

i(t)A

i(t) and s

i(t) = l

i(t)A

i(t). Obviously, the proc-

esses start with zero values,yi(0) = 0.

Te short rate (2) in terms of the Gaussian ys will look asfollows:

r t( ) = r t,y( ) f t, Ai t( )

1yi t( ){ }( )

(6)

o calculate a generalised option price with payout at time T:

n

P T,Tn( )

n

+

one should be able to evaluate the expectation:

V = E

n P T,Tn( )n( )+

NT

Note that the payment coecients pn

could have opposite signsfor non-trivial optionality.

After some experiments with the measure choices, we have con-cluded that the most ecient way is to proceed in the risk-neutralmeasure. We use the AD price q(t, u):

q t,u( ) E y t( ) u( )e

ds r s( )0

t

= E e ds r s( )

0

t

y t( ) = u

E y t( ) u( )

where u is a vector and the multi-dimensional delta function isdened by the product d(y(t) u) = PF

i=1d(y

i(t) u

i). It permits us

to represent the option price as a multi-dimensional integral:

V= du nP T,Tn;u( )n

+

q T,u( )

(7)

Tus, our goal is to approximate the zero-coupon bond priceP(t, T; u) and the AD price q(t, u) as functions of arbitraryu.Ten, the integration (7) will be performed numerically.

At the end of this section, we present dierential equations forboth components of the interest. Te zero-coupon bond price corre-

sponds to a backward PDE, (t+L(t, u))P(t, T; u) = 0, with operator:

L t,u( ) = r t,u( )+ ii

t( )i +1

2Cij t( )ij

i, j

(8)

and the nal conditions P(T, T; u) = 1. We have denoted spacederivatives as

iui and the instantaneous covariance matrix as

Cij(t) =s

i(t) s

j(t). Te backward PDE can be easily derived by Its

formula requiring zero drift of the martingale et0dsr(s)P(t, T;y(t)).Te AD price, q(t, u), satises the conjugate forward PDE,

(tL*(t, u))q(t, u) = 0, for the operator:

L*

t,u( ) = r t,u( ) i t( )ii

+1

2Ci,j t( )ij

i,j

(9)

with the initial condition q(0, u) = id(u

iy

i(0)). Indeed, the inte-

gral P(0,T) = duP(t, T; u)q(t, u) is independent of time t. After its

dierentiation over time, we obtain the desired forward PDE.o derive our approximation, we will apply the perturbation

technique for small volatilities to the PDEs above. With the yieldcurve t (4) the leading order of the drifts q

iwill be s2, because of

the invariance s s. Tus, we will consider the transforma-tion qeq and s2es2 in the backward and forward PDEs.

Main results

In this section, we present the main results: quadrature formulasfor the zero-coupon bond and the AD price. We use two types ofexpansion technique: regular for the zero-coupon bonds and sin-gular for the AD price.n Zero-coupon bond. Introducing e in the backward para-bolic operator:

L t,u( ) = r t,u( )+ i t( )i + 12i

Cij t( )iji,j

(10)

we denote the unperturbed operator as L0(t) and a perturbation

asL1(t):

L0 t( ) = r t,u( )

(11)

L1 t( ) = i t( )i +1

2i Cij t( )ij

i,j

(12)

Te solution of the unperturbed equation (t+L

0(t))P

0(t, T, u)

= 0 for the unit nal value, P0(T, T, u) = 1, is very simple:

P0 t,T,u( ) = exp dt

T

r ,u( )( )

(13)

Te rst and second corrections can be obtained using the reg-ular perturbation technique3:

P1 t,T,u( ) = P0 t,T,u( ) ds i s( )gi s,T,u( )i

t

T

+1

2Cij s( ) gi s,T,u( )gj s,T,u( ) gij s,T,u( )( )

i,j

(14)

with:

2One can use a proper shift,xi(t) x

i(t) +

t

0dsm

i(s), for some functionsm

i(s)

3 More details can be found in Antonov & Spector (2010)


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gi s,T,u( ) = dir ,u( )sT

and

gij s,T,u( ) = dijrsT

,u( )

(15)

Let us stress that the short rate r(t, u) and its derivatives are func-

tions of the normal underlyings y; in order to handle the initialdependence of the underlying mean-reverting process x oneshould use formula (6).

Note that ourruco, Hagan and Schleiniger (2007) have pre-sented the zero-coupon bond in exponential form:

P t,T;u( ) = P0 t,T,u( )exp 1 t,T;u( )+22 t,T;u( )+O

3( ){ }

which can sometimes deliver explosive va lues, especial ly for largematurities and volatilities.

o nalise the zero-coupon bond approximation, we should xthe drifts, unknown so far, to reproduce the yield curve or dis-count factor D(T). Equivalently, we can calculate the averagesE[y

i(t)] = m

i(t) =t

0dsq

i(s), requiring the t:

D T

( )= P

0 0,T,0( )

+P1 0,

T,0( )

to the leading order in the volatilities. Noticing that:

P0 0,T,0( ) = exp d r ,0( )0T

( ) = exp dR0T

( ) = P 0,T( )we conclude that, in order to t the yield curve, we should choosethe averages m

i(t) such that the second-order correction equals

zero, that is, P1(0, T, 0) = 0 or:

ds i s( ) gi s,T,0( )i

0

T

=

1

2ds Cij s( ) gi s,T,0( ) gj s,T,0( ) gij s,T,0( )( )

i,j

0

T

Dierentiating in T, we obtain:

ir T,0( ) mi T( )i

= ir T,0( ) djr ,0( )Vij ( )0T

i,j

1

2ijr T,0( )Vij T( )

i,j

(16)

where Vij(T) is theys covariance:

E yi t( ) mi t( )( ) yj t( ) mj t( )( ) =Vij t( ) = ds Cij s( )0t

Either solution of equation (16) for individual averages mi(T) is

suitable for the approximation. Note that ourruco, Hagan &Schleiniger (2007) have matched the yield curve by a rough mul-tiplicative adjustment that does not x the underlying drifts.

Te second-order correction was calculated in Antonov & Spec-tor (2010). Although, according to our experiments, the second-

order approximation (precisionO(e2))

does not deliver a reasonablequality increase but can substantial ly slow down the procedure.nAD price. In the case of the AD price, we deal with a singularperturbation technique for the parabolic operator:

L*

t,u( ) = r t,u( ) i t( )i +1

2 Cij t( )ij

i,j

i

(17)

Te AD price satises the forward PDE (tL* (t, u))q(t, u) = 0

with the init ial condition q(0, u) = Pid(u

iy

i(0)).

Applying the Dyson technique would require careful calcu la-tion of all terms in e due to the singularity of the initial value.Instead, we use the Heat-Kernel singular expansion method.

It is easy to check that the Gaussian density:

q0 t,u( ) =exp 1

21

u m t( )( )T

V1

t( ) u m t( )( )( )2( )

n2 V t( )

12

(18)

where m(t) is they(t) centre and V(t) is the covariance in vector/matrix notations, satises the PDE without the rate term:

tq0 t,u( ) = i t( )iq0 t,u( )+1

2 Cij t( )ijq0 t,u( )i,j

i

(19)

Now, following the Heat-Kernel expansion logic, we look forthe solution as q(t, u) = q

0(t, u)(t, u), where (t, u) is a regular

function ofe with the unit initial condition (0, u) = 1. Aftersubstitution into the initial PDE, we get the solution for the ADprice up to the rst order in low volatilities4:

q t,u( ) =exp 1

21 u m t( )( )

TV1 t( ) u m t( )( )( )

2( )n2 V t( )

12

exp d s r s,V s( )V1 t( )u( )0t

1+ d T ( ) mT ( )V1 ( )C ( )( )0

t{

V1 ( ) h

1( ) ;t,u( )1

2 d trV1

0

t ( )C ( )V1 ( )

h1( ) ;t,u( ) h

1( ) ;t,u( ) h2( ) ;t,u( )

}

(20)

where vector:

hn

1( ) ;t,u( ) = dsVnk s( ) %k0

j

r s,V s( )V1 t( )u( )

(21)

and matrix:

hnl2( ) ;t,u( ) = dsVnk s( ) %k%mr s,V s( )V1 t( )u( )Vml s( )0

km

(22)

Note that the main formula contains vector/matrix notationsincluding standard linear algebra products. For example, outerproduct ab of two vectors means a matrix with elements a

ibj.

Here, the derivative ~

kdenotes the short-rate derivative over the

space argument:

%ir s,z u( )( ) =

r s,z u( )( )zi u( )

Model examples

Here, we give two GSRG examples. One is our version of thegeneralised multi-factor BK model and the other is the bounded

short-rate model, which can be suitable for scenario generationwhere the rates evolve inside given boundaries.We will use a symmetric form of the GSRGs:

r t( ) = f t, xii

As mentioned above, we can always represent the short-rate func-tionf(t,x) as being equal to the forward rate for zero values of theunderlying processx. Similar scaling arguments can be applied toset the space derivatives of the short rate to one for x = 0. Tesetwo properties,f(t, 0) =R(t) and

if(t, 0) = 1, are convenient for

4More details can also be found in Antonov & Spector (2010)


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analysing the calibration results. Indeed, typical underlyingxparameters correspond to the normal case, that is, volatilities have11.5% order of magnitude for any short rate.nGeneralised multi-factor BK model. We propose a version ofthe generalised multi-factor BK model with many attractive fea-

tures, including swaption skew control and multiple factors per-mitting decorrelation of the rates. Te short rate is presented inthe form:

r t( ) = f t,x( ) = R t( )+e

S t( ) xii 1

S t( )

(23)

whereR(t) is todays forward rate and positive function S(t) is ashift parameter controlling the swaption skew. For a special caseofS(t) =R1(t), the model degenerates to the standard multi-factorBK model, r(t) = R(t)eR

1(t)ixi, whereas when the shift parametertends to zero, we obtain the Hull-White model, r(t) =R(t) +

ix

i.

Note that typical underlyingx parameters correspond to thenormal case, that is, volatilities have 11.5% order of magni-tude for any shift. Tis is another attractive feature of the pro-

posed model.Te analytical expressions can be calculated using the general

formulas for the zero-coupon bond (13) and (14) and the ADprice (20).

Explicit formulas for the main derivatives (15) are given by:

gi s,T,u( ) = dAi ( )1

s

T e

S Ai ( )1u

i

and:

gij s,T,u( ) = dAisT

( )1Aj ( )

1Se

S Ai ( )1u

i

Te conditions for the drifts simplify also, and one can choose asymmetric form:

mi T( ) = dVij ( )Aj1

( )1

2ST

0

T

j Vij T( )Aj

1

T( )j

nBounded short-rate model. Sometimes practitioners, espe-cially from the insurance industry, are interested in a boundedrates evolution for scenario generation. Te current short-ratemodels do not meet that bounded requirement. Tere exists aprobability such that rates can exceed an a priori-given (posi-tive) value.

We propose the following bounded short-rate model:

r t( ) = f t,x( ) =U t( )e

t( ) xii+L t( ) t( )

e t( ) xii

+ t( )

(24)

where time-dependent functions L(t) and U(t) are the lower and

upper boundaries, respectively. As usual, the forward rate isdenoted asR(t). o ensure the attractive properties off(t, 0) =R(t)andf(t, 0) = 1, we dene the functions a(t) and b(t) as:

t( ) =U t( ) R t( )R t( ) L t( )

and:

t( ) =U t( ) L t( )

R t( ) L t( )( ) U t( ) R t( )( )We see that the function (24) i s a monotone function ofX=

ix

iand its inverse has a simple form:

X= lnU t( ) R t( )

U t( ) FF L t( )

R t( ) L t( )

1t( )

for F=f(t, X). Obviously, the short rate has two desired limits:

limXf(t,X) =L(t) and limX+f(t,X) = U(t).A special case of the bounded short-rate model U(t) + rep-

resents the generalised multi-factor BK model (23) for the skewparameter S(t) = (R(t) L(t))1. We see that, for this special case,the skewness of the implied volatility depends on the bounds.Tis property holds also in the general bounded model set-up. overify it explicitly, consider the short-rate SDE:

dr t( ) =Ldt+

r t( ) L t( )( ) U t( ) r t( )( )R t( ) L t( )( ) U t( ) R t( )( )

dxi t( )i

Te diusion coecient:

D t,r( ) ~r L t( )( ) U t( ) r( )

R t( ) L t( )( ) U t( ) R t( )( )dxi t( )

i

or local volatility, can give an approximate form of the implied

volatility smile a s a function of strike K:

I

K( ) ~D t,

R t( )+K2( )

R t( )+K2

We see that the implied volatility form depends on the bounda-ries. Tus, if one xes the boundaries, one loses the skew controland the model can be calibrated only to one point of the impliedvolatility for a given swaption (say at-the-money). On the otherhand, xing, say, the lower boundary, one can still calibrate to theoption skew using a residual freedom in the upper boundary.

Numerical experiments

Here, we present numerical results for swaption prices compar-ing our analytical approximation (Approx) with the exactanswer (Exact) obtained in a high-quality least-squares MonteCarlo simulation.

Te approximation is performed in the rst order in variances:

P t,T;u( ) = P0 t,T;u( )+P1 t,T;u( )

from (13) and (14) for the zero-coupon bond and q(t, u) AD pricedened in (20). Note that, according to our experiments, the second-order approximation does not deliver a reasonable quality increasebut can substantially slow down the procedure. Te integration:

V= du nP T,Tn;u( )

n

+

q T,u( )

(25)

is taken numerically using, for example, an adaptive grid.Note that the forwards duP(T, T

n; u) q(T, u) are not consistent

with the theoretica l ones, the discount factorsD(T

n), due toapproximation errors. Tus it can be useful to perform a multipli-

cative adjustment for the underlying products, P(T, Tn; u) q(T, u),

to match the forwards exactly.nPerformance. Te numerical integration (25) includes calcula-tion of integrals over time in the zero-coupon bond and the ADprice for any xed space point u. Consider, for example, the zero-bonds. Te unperturbed zero-coupon bond integral (13) can beeciently taken using the Simpson method or using functionalproxies for few integration points. Te rst-order correction (14)accuracy is less important for the convergence due to its smallvalue. Moreover, the underlying two-dimensional integrals con-


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70 RkMay 2011


tain ordered arguments that lead to linear integration algorithmsin the number of time-discretisation points. Note also that cer-tain values, such as the short rate, its derivatives and their inte-

grals, can be pre-calculated from the smallest exercise date untilthe largest zero-coupon bond maturity and reused in the underly-ing expressions (13), (14) and (15) for all exercise dates and allzero-coupon bond maturities.

Te same arguments apply to the AD price integrals.Te option price approximation (25) containing eectively a

space-time integration is denitely more attractive than othernumerical methods, including lattices and Monte Carlo simula-tions especially, for multi-dimensional problems. Depending onthe numerical method eciency, the speed gain can be two tofour times for one-dimensional short-rate models and ve to 10times for multi-dimensional cases.

nGeneralised multi-factor BK model. We consider a two-fac-tor case of the generalised multi-factor BK model (23):

r t( ) = R t( )+e

S t( ) x1+x2( )1

S t( )

with the underlying processes:

dxi t( ) = i t( ) ai t( )xi t( )( )dt+ i t( )dWi t( )for i =1,2 and xi 0( ) = 0

with a certa in correlation between Brownian motions, r(t) =E[dW

1(t)dW

2(t)].

We use standard data, summarised in table A.We have chosen particular time-independent parameters for

transparency. However, the proposed approximation works forgeneral time-independent parameters. Note that the shift param-eter corresponds to a skew situated between normal and lognor-

18

19

20

21

22

23

24

25

Strike (%)

Impliedvolatility(%)

ApproxExact

2 4 6 8 10

1comproofhmplvollfor10-yr/

10-yreropwpo(rlwo-for

BKmol)

0

5

10

15

20

25

30

35

Strike (%)

Impliedvolatility(%)

Approx

Exact

0 5 10 15 20 25

2comproofhmplvollfor10-yr/

10-yreropwpo(bohor-rmol)

a.grlwo-forBKp

R R(t) 5.00%

corrlo r(t) 90.00%

shf S(t) 10

Volly1 g1(t) 1.00%

Volly2 g2(t) 1.75%

Rvro1 a1(t) 50.00%

Rvro2 a2(t) 5.00%

B.grlwo-forBKrl

exr

(yr)

Lh

(yr)

srk(%) implvolly(%) error(bp)

approx ex

5 10 3.28 25.77 25.71 5

5 10 3.67 25.04 24.99 6

5 10 4.10 24.34 24.31 3

5 10 4.59 23.69 23.67 2

5 10 5.13 23.09 23.09 0

5 10 5.73 22.53 22.54 1

5 10 6.41 22.01 22.01 0

5 10 7.17 21.52 21.50 2

5 10 8.02 21.10 21.06 4

10 10 2.72 24.39 24.26 13

10 10 3.19 23.36 23.27 9

10 10 3.74 22.42 22.34 8

10 10 4.38 21.54 21.47 7

10 10 5.13 20.75 20.67 8

10 10 6.01 20.03 19.96 7

10 10 7.03 19.39 19.32 7

10 10 8.24 18.82 18.75 6

10 10 9.65 18.30 18.25 5

20 10 2.10 21.59 21.24 35

20 10 2.62 20.20 19.89 31

20 10 3.28 18.94 18.68 26

20 10 4.10 17.87 17.62 26

20 10 5.13 16.91 16.68 23

20 10 6.41 16.05 15.86 19

20 10 8.02 15.35 15.15 20

20 10 10.03 14.72 14.57 15

20 10 12.54 14.19 14.09 11

c.Boo-forhor-rmolp

R R(t) 5.00%

Volly g(t) 1.5%

Rvro a(t) 5.00%

Lowrbo L(t) 0%

upprbo U(t) 20.00%


6/6

mal ones. Recall that the normal model is described with the zeroshift parameter and the lognormal model with the shift equal tothe inverse rate, S(t) =R1(t) = 20.

We have tested European swaptions with an annual period,10-year length, multiple strikes and maturities of ve, 10 and 20years. Te results for Black implied volatilities are shown in tableB and gure 1. Te at-the-money swaptions are marked in bold.n

Bounded short-rate model. We consider a one-factor case ofthe bounded short-rate model (24):

r t( ) = f t,x( ) =U t( )e

t( )x+L t( ) t( )

e t( )x

+ t( )

with the underlying process:

dx t( ) = t( ) a t( )x t( )( )dt+ t( )dW t( )

starting from zero,x(0) = 0.We use standard data, summarised in table C.We have chosen particular time-independent parameters for

transparency. However, the proposed approximation works forgeneral time-independent ones.

We have tested European swaptions with an annual period,10-year length, multiple strikes and 10-year maturity. Te resultsfor Black implied volatilities are presented in gure 2 and table D.

One can observe zero implied volatilities for a strike of 21%. Tisproperty is also valid for higher strikes due to bounds of ratesevolution. Namely, interest rates are highly correlated with theshort rate (bounded by 20% in our set-up). Tus we expect theoption underlying constant maturity swap 10-year rate to bebounded by a maximum of around 20% of value (around 21%according to the numerical results). Tis leads to a zero value of acall option for strikes higher than the underlying rate bound.

Conclusion

We have analysed GSRGs and derived an ecient swaption priceapproximation. Te result is based on a regular and singularexpansion with respect to low volatility and contains a low-dimensional integration.

We also presented two special cases of the GSRG model. Te

rst is the generalised multi-factor BK model widely used bypractitioners. Te second, the bounded short-rate model, is new.Its rates evolve inside arbitrary thresholds. Tis model is espe-cially attractive for scenario generation. Te implied volatilityskew for the bounded short-rate model is convex for small/moder-ate strikes and concave for bigger ones.

We have conrmed the analytical approximation quality bycomparison with purely numerical results. We have detected anexcellent analytics-numerics t for short/medium maturities, anda good one for large maturities. We have observed a slight qualitydegeneration for extreme strikes. n

alxraoovorv-profqvrrh

Mhlsporrorofqvrrhnmrx.thyr

bosrMhkovforomrlmplmo

hlpwllohrollnmrx,pllyogroryWh

srikovforpporhworkPHrrforh

xll.alxraoovrfloVlmrPrbr

MhlKokovformlo.eml:[email protected],

[email protected]

Antonov A and M Spector, 2010Analytical approximations for short

rate models

SSRN preprint, available at http://ssrn.

com/abstract=1692624

Black F and P Karasinski, 1991

Bond and option pricing when short

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Financial Analysts Journal 47, pages5259

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Econometrica 53, pages 385407

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Pricing interest-rate derivative

securities

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573592

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implementing term structure models

II: two-factor models

Journal of Derivatives 2, pages 3747

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Rates squared

RiskJanuary, pages 100105

Tourruco F, P Hagan and

G Schleiniger, 2007Approximate formulas for zero-

coupon bonds

Applied Mathematical Finance 14(3),

pages 207226

Vasicek O, 1977

An equilibrium characterization of the

term structure

Journal of Financial Economics 5(2),

pages 177188

Rfr


d.Boo-forhor-rmolrl

srk(%) implvolly(%) error(bp)

approx ex

0.01 29.72 32.52 280

0.5 19.98 20.54 56

1 19.01 18.34 67

1.5 18.39 17.79 60

2 17.92 17.49 43

2.5 17.52 17.19 33

3 17.17 16.92 25

3.5 16.85 16.69 16

4 16.56 16.48 8

4.5 16.29 16.27 2

5 16.04 16.07 3

5.5 15.8 15.87 7

6 15.57 15.67 10

6.5 15.35 15.49 14

7 15.14 15.3 16

7.5 14.93 15.11 18

8 14.74 14.93 19

9 14.34 14.55 21

10 13.96 14.17 21

11 13.58 13.79 21

12 13.19 13.39 20

13 12.79 12.96 17

14 12.37 12.52 15

15 11.92 12.11 19

16 11.44 11.72 28

17 10.92 11.32 40

18 10.33 10.93 60

19 9.654 9.972 31.8

20 8.824 8.254 57

20.2 8.632 5.852 278

20.4 8.425 4.662 376.3

20.5 8.315 4.63 368.5

21 0 3.99 399

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