a study on singular perturbation correction to bond prices under affine term structure models

7/29/2019 A Study on Singular Perturbation Correction to Bond Prices Under Affine Term Structure Models

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Abstract

The technique of singular perturbation can be applied to fixed income deriva-tive pricing. It provides a convenient and efficient way to account for stochasticinterest rate volatility. We evaluate the yield curve fitting performance of a per-turbation corrected Vasicek model by comparing it to the Fong-Vasicek model.It is found that the accuracies of the perturbation scheme and the exact analyticscheme are comparable, while the former requires much less computational time.We extend this scheme to a perturbation corrected CIR model, in which casethe advantage in speed is diminished due to the need for numerical methods..


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A Study on Singular Perturbation Correction toBond Prices Under Affine Term Structure

Models

Frank FungA REPORT SUBMITTED IN FULFILLMENT

OF THE REQUIREMENT FOR THEINDEPENDENT STUDY

OF THEBERKELEY MFE PROGRAM

15 October 2010


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Contents

1 Introduction 2

2 Literature Review 42.1 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . 42.2 Application of Singular Perturbation in Derivative Pricing . . . . 6

2.2.1 Equity Derivatives . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Fixed Income Derivatives . . . . . . . . . . . . . . . . . . 7

3 Results and Discussion 10

3.1 Performance of Singular Perturbation Under Fong-Vasicek Model 103.1.1 Application of Singular Perturbation Under Fong-Vasicek

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.2 Comparison to Analytic Bond Prices . . . . . . . . . . . . 15

3.2 Singular Perturbation Under CIR Model . . . . . . . . . . . . . . 18

4 Conclusion 22

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Chapter 1

Introduction

Affine term structure models are a family of models that share the same under-lying mathematical structure [13, 12]. One of the advantages of affine models istheir mathematical tractability, namely the zero-coupon bond price can be ex-pressed as an exponential of some affine functions of the underlying processes.This feature of affine models greatly facilitates their calibration to the yieldcurve. Well known single factor affine term structure models include short ratemodels such as Ho-Lee, Vasicek and CIR models. On the other hand, multi-factor affine term structure models allow for more flexibility and diversity. Oneway to utilize the additional degrees of freedom is to make the short rate a sumof more than one processes, as in the case of Longstaff and Schwartz [10]. Dueto the mixing of the two processes, randomness is introduced into the interestrate volatility implicitly. Alternatively, one can explicitly designate a stochas-tic process to represent the dynamics of the short rate variance, as in the caseof Fong-Vasicek model [14]. However, as we will see shortly, the computationalcost of introducing stochastic volatility in such a way is quite high. On the otherhand, empirical evidence shows that interest rates and interest rate volatilityvary on different timescales [9], which is a fact we can exploit, together withthe singular perturbation technique, to correct the zero-coupon bond prices forstochastic volatility. The perturbation correction terms are in different orders ofthe reciprocal of variance mean-reversion rate, hence if the interest rate volatilityis fast mean-reverting the first order correction term would be able to capturemost of the effect of stochastic volatility.

In this report we investigate how the theoretical recipe proposed in [1] be-haves in practice. We are mainly interested in the relative gain in accuracy of

such a perturbation correction scheme compared to its uncorrected single-factorcounterpart, and the relative saving in computational efforts of it compared to afull treatment of stochastic volatility. We choose to conduct our study with theVasicek and CIR models because, first of all, their simplicity allows us to focuson the analysis of interest, and secondly the uncorrected single-factor versionsof them are well known. Nevertheless, we note that the application of the cor-rection scheme is not limited to the two cases presented here, and the extension

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to other models would require further investigation.

This report is organized as follows. Chapter 2 is a literature review. InSection 2.2, we briefly review the mathematical background of singular pertur-bation problems and discuss how it differs from regular perturbation problems.Section 2.2.1 and 2.2.2 are reviews on the application of singular perturbationtechniques in equity derivatives and fixed income instruments, respectively. InChapter 3, we derive the formula for perturbation correction specifically forthe Fong-Vasicek model in Section 3.1.1, which is followed by a yield curve fit-ting comparison to the analytic results in Section 3.1.2. Section 3.2 presentsthe perturbation correction to bond prices under a CIR model and an overallcomparison. We summarize our investigation in Chapter 4.

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Chapter 2

Literature Review

2.1 Mathematical Background

Here we provide a brief review on the topic of perturbation. Readers are referredto [4, 5, 6, 7] for a more detailed treatment. Perturbation theory is the methodof finding mathematical solutions by treating the full problem as a perturbedsimpler problem. The method has a long history of applications in variousfields, including but not limited to fluid dynamics, celestial mechanics as wellas quantum mechanics.

Consider a trivial example. Suppose we have to solve

x2 1 = x (2.1)

The two quadratic roots are

2

r1 +

2

4(2.2)

Depending on the desired accuracy, we can now perform a Taylor expansion toEquation (2.2) to obtain an approximated solution. The leading terms in theTaylor expansion are

x = 1 +

2

2

8+ O

3

(2.3)

where O () is the large O notation. As 0, Equation (2.1) becomes simplyx2

1 = 0 with solutions x

1, and the series solution given by Equation

(2.3) converges uniformly to these solutions for all x. When this is the case, theperturbation problem is said to be regular.

Alternatively, consider an ODE

dy

dx+ y = cosx (2.4)

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with initial condition y (0) = 0. Naively one may be tempted to substitute a

formal expansion of the form

y (x) =Pn=0

nyn (x) (2.5)

which would produce a series solution

y (x) = cos x + sinx 2 cosx + (2.6)The problem with this candidate is that the initial condition y (0) is not satisfied,and hence the approach that we used in the case of regular perturbation abovefails. In fact the solution to the nonhomogeneous ODE in Equation (2.4) is

y (x) =1

ex/ Zx

0

et/ cos tdt (2.7)

as can be obtained by the method of variation of parameters or undeterminedcoefficients. Partial integration to Equation (2.7) gives

y (x) = cos x ex/ + ex/Zx0

et/ sin tdt (2.8)

Note that unlike the naive attempt before, this solution actually satisfies theinitial condition y (0) = 0. The question is then whether we are able to createout of Equation (2.8) a series solution in powers of . Note that ex/ is aquickly varying function in the vicinity ofx = 0. It can be shown that the series

Sm (x) =

Xn=0

(

1)n n hcosn (x) ex/ sinn (x)i (2.9)

is a legitimate approximation satisfying the initial condition. The convergenceof this series to the exact solution as 0, however, is nonuniform in the sensethat

limx0

lim0

Sm (x)6= lim

0

limx0

Sm (x)

(2.10)

since

limx0

lim0

Sm (x)

= 1 (2.11)

lim0

limx0

Sm

(x) = 0Thus we are faced with a singular, as opposed to regular, perturbation problem.This is observed when the small parameter ( in this case) is multiplied by thehighest-order derivative in the ODE.

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2.2 Application of Singular Perturbation in Deriv-

ative Pricing

2.2.1 Equity Derivatives

In Section 2.1 we see that a singular perturbation problem can arise if we aretrying to expand a solution with respect to a small parameter that is multipliedby the highest -order derivative. Now if we look at the Black-Scholes PDE:

1

22S2

2V

S2+ rS

V

S+

V

t rV = 0 (2.12)

the highest-order derivative is 2SV. Thus the condition for singular perturbationis met if2, the volatility, plays the role of the small parameter in Section 2.1.Typical value of 2 ranges from 0.01 to 0.2, which is indeed small comparedto the option price V. As demonstrated in [8], this fact can be exploited toefficiently price European, American and barrier options.

Another possible application of singular perturbation in the variable 2 isstudied by Fouque et al. [3]. They considered option pricing under the processes

dSt = rStdt + f(Yt)StdW1t (2.13)

dYt =

"1

(m Yt)

v

2 (Yt)

#dt +

v

2dW2t

in the risk-neutral measure, where (Yt) is the combined market price of risk(of both the stock price and the volatility), f is some non-negative functionand is a small parameter, i.e. the volatility is fast mean-reverting (empiricalevidence for and analysis about the mean-reversion of volatility can be found in[2, 9]). Express the option price P as a series

P = P0 +P1 + P2 + (2.14)

and define the differential operators

L =1

L0 +

1L1 + L2 (2.15)

where

L0 = v2 2

y2+ (m y)

y(2.16)

L1 =

2vf(y)2

xy

2v (y)

y

L2 =

t+

1

2f(y)

2 2

x2+

r 1

2f(y)

2

x r

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It can then be shown, by considering LP = 0 and matching terms with the

same order in , that

P1 = (T t)

V3

3

x3+ (V2 3V3)

2

x2+ (2V3 V2)

x

P0 (2.17)

where V2 and V3 are group parameters that are averaged over the probabilitydistribution ofy and P0 is the Black-Scholes price with effective variance

2 =-f2

. In other words, the first order correction can be obtained by solving anODE that involves the zeroth order price.

2.2.2 Fixed Income Derivatives

In this section we discuss the procedure for applying singular perturbation tofixed income derivative pricing. Once again this is motivated by the fact thatinterest rate volatility varies on a different (shorter) timescale from the interestrate itself. Since the detailed derivation for the specific case of Fong-Vasicekwould be provided in Section 3.1.1, we only outline the method here. Considerthe processes

drt = a(r rt)dt + f(Yt)dW1t (2.18)dYt = (m Yt)dt + dW2t

where both the interest rate and the interest rate volatility are mean-reverting,and f is some non-negative function. Expanding the bond price as a series and

defining a set of differential operators properly as we did in Section 2.2.1, it canbe shown that the lowest order correction term satisfies

LLHSP1 = LRHS (V1, V2, V3)P0 (2.19)

for some differential operators LLHS and LRHS.Although the procedure here is formally identical to what we have seen in

Section 2.2.1, there are two subtle differences. First, in Section 2.2.1, Equation(2.17) suggests that we can calculate the option price correction after we haveestimated V2 and V3 from the historical stock price time series. On the otherhand for fixed income derivatives pricing, since we have an observable yieldcurve, we can calibrate our model using the stochastic volatility corrected bondprices of zero-coupon bonds. That is, the group parameters V1, V2 and V3 can be

backed out from the yield curve. This is particularly practical if we are dealingwith affine models, in which case the bond price correction term P1 could haveclosed-form solution, as we shall see later. Secondly, the method introducedhere is flexible enough to be applied to different affine short rate models. Withthese two observations we move on to the next chapter, where we compare theapplication of singular perturbation to yield curve fitting under different models(Vasicek and CIR) and during different periods. We will also compare how the

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approximation performs alongside an exact treatment of stochastic volatility

under Fong-Vasicek.Figure 2.1 and 2.2 are attempts to reproduce the results obtained in [1].The Vasicek model is considered and both the uncorrected and perturbationcorrected curves are shown. In Figure 2.1 the uncorrected and perturbationcorrected yield curves have sum of squared errors of 4.5409105 and 1.2793105, respectively. In Figure 2.2 the uncorrected and perturbation correctedyield curves have sum of squared errors of 9.4260 106 and 6.2520 107,respectively.

0 5 10 15 20 25 30

0.053

0.054

0.055

0.056

0.057

0.058

0.059

0.06

Maturity (year)

Yield

Figure 2.1: Fitted yield curves to swap rates observed on 09/07/1998. Thecrosses are the raw data points, the red curve is fitted with Vasicek bond pricesand the blue curve is fitted with perturbation corrected Vasicek bond prices.

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0 5 10 15 20 25 300.056

0.057

0.058

0.059

0.06

0.061

0.062

Maturity (year)

Yield

Figure 2.2: Fitted yield curves to swap rates observed on 08/07/1998. Thecrosses are the raw data points, the red curve is fitted with Vasicek bond pricesand the blue curve is fitted with perturbation corrected Vasicek bond prices.

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Chapter 3

Results and Discussion

3.1 Performance of Singular Perturbation Un-

der Fong-Vasicek Model

3.1.1 Application of Singular Perturbation Under Fong-

Vasicek Model

In Section 2.2.2 we see how singular perturbation method produces a first ordercorrection to the zero-coupon bond price in the presence of stochastic volatility.One question that is of interest is the performance of this scheme in capturingthe effect of volatility. Ideally, we would want to study a model having thesame form as Equation (2.18), with a specified functional form for f and with

analytic zero-coupon bond price. Unfortunately, we are not aware of such amodel. Alternatively, we try to shed light on the comparison by investigate theFong-Vasicek model. We derive in details the bond price obtained by singularperturbation correction and compare it to the analytic solution [11], which re-quires numerical integration. The derivation presented here follows [1] closely,with the missing steps filled in.

The dynamics of the short rate and the interest rate volatility, respectively,follow the system of equations:

drt = a(r rt)dt +pYtdW

1t (3.1)

dYt = (

Yt)dt + YpYtdW2t

where hdW1t dW2t i = dt. Comparing to the system of stochastic differential

equations in [1], which we reproduce here,

drt = a(r rt)dt + f(Yt)dW1t (3.2)dYt = (m Yt)dt + dW2t

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we see that the diffusion ofYt here is a constant, while the Fong-Vasicek variance

follows a CIR process. Inspired by [15], we seek a transformation z such that

z(Y) =

ZY d

(3.3)

It can be easily shown that the appropriate transformation is z(Y) = Y1/2 (upto a factor). Using Itos lemma, the dynamics of z is

dz =1

2z

z2

1

82Y

1

z

dt +

1

2YdW

2t (3.4)

and we succeeded in making the diffusion coefficient constant. Recall that theorder matching scheme described in Section 2.2.1 requires the rate of meanreversion of interest rate volatility to be high, which in turn dictates the drift

coefficient in the volatility process (i.e. in Equation (2.18)) be large. On theother hand, in Equation (3.4) we can no longer be sure about the rate of meanreversion due to the dependence of the drift coefficient on z. Hence we have togo back to the original Fong-Vasicek SDEs as a starting point.

Returning to Equation (3.1), the differential equations in risk-neutral mea-sure becomes

drt =ha(r rt) (Yt)

pYtidt +

pYtdW

1t (3.5)

dYt =h( Yt) Y

pYt(Yt)

idt + Y

pYtdW

2t

where and are the two market prices of risk and

+ p1 2. Thecorresponding Feynman-Kac PDE istP+

1

2y2xP+ [a(r x) (y)

y]xP xP (3.6)

+Yy2xyP+

1

22Yy

2yP+ [( y) Y

y(y)]yP = 0

To save space we write it as

LP = 0 (3.7)

where the subscript is a reminder that the differential operator L depends on

. We then rearrange the diff

erential operators into three groups,

L0 =1

22Yy

2y + ( y)y (3.8)

L1 = Yy2xy Y

y(y)y

L2 = t +1

2y2x + [a(r x) (y)

y] x x

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With

=1

Y = v

r2

the differential operators, expressed explicitly in powers of , are defines as

L0 = v2y2y ( y) y (3.9)

L1 =

2vy2xy

2vy (y) y

L2

= t

+1

2y2

x+ [a(r

x)(y)

y]

x x

so that

L =1

L0 +

1L1 + L2 (3.10)

Note that L0 contains y differential operators only and L2 contains no y differ-ential operator. We seek an asymptotic solution of the form

P(t,x,y;T) = P0(t,x,y;T) +P1(t,x,y;T) + P2(t,x,y;T) + (3.11)

The O(1) equation is

L0P0 = 0 (3.12)

which indicates that P0(t, x;T) is independent ofy.The O(1/2) equation is

L0P1 + L1P0 = 0 (3.13)

Since P0 = P0(t, x;T), L1P0 = 0 and we conclude that P1(t, x;T) is also inde-pendent ofy. The O(1) equation is

L2P0 + L1P1 + L0P2 = L2P0 + L0P2 = 0 (3.14)

Noting that L2 contains no y derivative and P0 does not depend on y, by fixing

a particular x = x0 the term L2P0|x=x0 can be thought of as a function of ywhich we denote as h(y). Then, holding x constant,

(L0P2 + h(y)) |x=x0 = 0 (3.15)

is a Poisson equation for P2 in the variable y and it can be shown that a solutionP2(t,x,y;T)|x=x0 exists only if the so called centering condition is fulfilled for y[2]. The centering condition is expressed as hhi = hL2P0i = 0, where the average

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is taken with respect to the probability distribution of Yt over the whole range

(,). But we have already shown that P0(t, x;T) is independent of y,hence the centering condition simplifies to hL2iP0 = 0. The operator L2 isformally the same as the differential operator of the Feynman-Kac PDE if weare considering a Vasicek model with constant volatility:

hL2i = t +1

2hyi 2x + [a(r x) h(y)

yi]x x (3.16)

t +1

222x + [a(r

x)]x x

where 2 hyi and r = r-(y)

y/a. Therefore the solution to hL2iP0 =

0 is

P0 (t, x;T) = A(T t)eB(Tt)x (3.17)where A(T t) and B(T t) are given by

B() =1 ea()

a(3.18)

A() = exp

RRB() +

2

4a3

1 ea()

2

where = T t and R r 2/2a2, which is consistent with the y-independence requirement. Note that Equation (3.17) is nothing but the zero-coupon bond price for the one-factor Vasicek model with effective volatility 2.

With the centering condition, we can write

L2P0 = L2P0 hL2iP0 = (L2 hL2i)P0 (3.19)Substituting back into the Poisson Equation (3.15) would give

L0P2 = (L2 hL2i)P0 (3.20)Or equivalently

P2 = L10 (L2 hL2i)P0 + k (x, t) (3.21)where the function k arises as an integration constant (in the variable y).

Finally, the O () equation is

L2P1 + L1P2 + L0P3 = 0 (3.22)

This is a Poisson equation of P3, and a solution exists only if the centeringcondition hL2P1 + L1P2i = 0 is fulfilled. Once again, P1 is y-independent andhence hL2P1i = hL2iP1. Equation (3.22), together with the centering condition,gives

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hL2iP1 = hL1P2i (3.23)=

-L1L

10 (L2 hL2i)P0 + L1k (x, t)

=

-L1L

10 (L2 hL2i)

P0

The second line makes use of the fact that k(x, t) does not depend on y. As P0is already found, solving Equation (3.23) would give us an expression for P1.The operator L2 hL2i is

1

2

y 2

2x (

y hi) x (3.24)

Since L0 contains y differential operators only,

L10 (L2 hL2i) =1

2

L10

y 2

2x

L10 (

y hi)

x (3.25)

Introduce two functions, and , with the following properties:

L0 = y 2 (3.26)L0 =

y hi

Equation (3.23) then becomes

hL2iP

1= L

11

22

x

xP

0(3.27)

=

2vy2xy

2vy (y) y

122x x

P0

Using the relations

hy (nx )i =

-0nx (3.28)

hy (nx )i =

-0nx

for some y-independent function (x), and defining

V1 = 12v

-y0

(3.29)

V2 = 1

2v-y0

2v-

y 0

V3 =

2v-

y0x

we can rewrite Equation (3.27) as

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t + 1

222x + [a(r

x)]x xP1 =

V1

3x + V2

2x + V3x

P0 (3.30)

Due to the centering condition, all y-dependences are contained within the groupparameters V1,2,3, which allows us to solve a PDE in x and t only. SubstituteP0 = A()e

B()x into Equation (3.30) and after some algebra we find that

P1 =1

222xP1 + [a(r

x)]xP1 xP1 +V1B

3 V2B2 + V3BP0 (3.31)

with the initial condition P1 (t = T, x) = 0. It is trivial to check that the solutionto the PDE is

P1 (t, x) = D ()A()eB()x (3.32)

where

D () =V3a3

B () 1

2aB ()2 1

3a2B ()3

(3.33)

V2a2

B () 1

2aB ()2

+

V1a

(B ())

The corrected zero-coupon bond price up to O1/2

is therefore

P' 1 +D ()A()eB()x (3.34)

In summary we showed that compared to the canonical Vasicek model stud-ied in [1], although the volatility dynamics of Fong-Vasicek model has a diffusionterm proportional to a function ofYt, the first order correction to the bond price(i.e. Equation (3.34)) remains the same form except that V1,2,3 are defined dif-ferently.

3.1.2 Comparison to Analytic Bond Prices

We compare the performance of singular perturbation on Fong-Vasicek to ananalytic bond price [11]. The zero-coupon bond price under Fong-Vasicek modelis P(, r) = A () eB()rC()y, with A (), B () and C() give by the systemof ordinary differential equations

dA

d= A (arB + C) (3.35)

dB

d= aB + 1

dC

d= B C YC

B2

2

2YC

2

2 YBC

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Figure 3.1: Comparison of the two yield curve fitting schemes.

with initial conditions A (0) = 1, B (0) = 0 and C(0) = 0. The solutions to theA and B equations are

B () =1 ea

a(3.36)

A () = exp

r+ rB ()

Z0

C(s) ds

Before going into yield curve fitting results, we shall inspect the connectionbetween the Fong-Vasicek analytic bond price and perturbation correction term.If the approximation scheme is correct, we should have

exp(

r+ rB ()

B () r)exp Z

0

C(s) ds exp(C() y) (3.37)=

1 +

D ()

exp

RRB() +

2

4a3

1 ea()

2+ B () r

+ O ()

As a reminder, = 1/. The reason singular perturbation is required is nowapparent. Although the correction to the bond price is small under our assump-tions, it cannot be easily obtained by performing a series expansion directly onthe Fong-Vasicek analytic price since the parameter appearing in the exponen-tial, , is not small.

To fit the yield curve with the analytic Fong-Vasicek bond price, we approx-imate C() using Euler method starting with the initial condition C(0) = 0.The integral

RC(s) ds within A () is approximated using a Riemann sum. The

singular perturbation results are identical to those from Section 2.2.2 because,

as we have shown in the previous section, the perturbation term is not alteredby a change in the variance SDE diffusion coefficient (it is absorbed in the groupparameters). Figure 3.1 shows the details of the two schemes using nonlinearleast square fitting in Matlab.

Figure 3.2 shows the fitted yield curves.Judging from Figure 3.2 and also from the fitting errors reported in Figure

3.1, we see that the singular perturbation scheme performs better in terms of

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0 5 10 15 20 25 300.053

0.054

0.055

0.056

0.057

0.058

0.059

0.06

Maturity (years)

Yield

Figure 3.2: Fitted yield curves to swap rates observed on 09/07/1998. Thecrosses are the raw data points, the red curve is fitted with analytic Fong-Vasicek bond prices and the blue curve is fitted with perturbation correctedVasicek bond prices.

computational speed but does not do as well in terms of fitness to data. This isto be expected since the perturbation scheme is accurate only up to O (

). An-

other observation is that the optimized parameters for the analytic Fong-Vasicekscheme are the variables that enter into the SDE of Equation (3.1), while thereis no intuitive interpretation for the optimized parameters of the perturbationscheme. We would like to point out two issues in the implementation of theyield curve fitting schemes. First, the longer CPU time in the analytic schemeis to a large extent due to looping when implementing the Euler scheme andthe numerical integration for C(). The relative disadvantage in speed of theanalytic scheme may be reduced under a different implementation. Second, theanalytic bond price depends on y, which is an unobservable quantity. We proxyy with the historical variance of the one-week yield for the analytic scheme.Figure 3.3 shows a yield curve that is less demanding for the Vasicek model,in which case the Fong-Vasicek and the perturbation prices are very close (sum

of squared error of 5.2853 10

7 for the former versus 4.5923 10

7 for thelatter).

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0 5 10 15 20 25 300.056

0.057

0.058

0.059

0.06

0.061

0.062

Yield

Maturity (year)

Figure 3.3: Fitted yield curves to swap rates observed on 08/07/1998. Thecrosses are the raw data points, the red curve is fitted with analytic Fong-Vasicek bond prices and the blue curve is fitted with perturbation correctedVasicek bond prices. The two curves are indistinguishable under this resolution.

3.2 Singular Perturbation Under CIR Model

Here we extend the singular perturbation correction to the CIR model. Wefirst briefly review the partial results presented in [1]. The dynamics of thetwo-factor CIR model we study is described by the processes

drt = a (r rt) dt + f(Yt)

rtdW

1t (3.38)

dYt = rt (m Yt) dt + rtdW1t +

0dW2t

where dW1t and dW2t are Wiener processes under the risk-neutral measure. The

differential operators in this case are

L0 = x v22y + (m y) y (3.39)L1 =

2vxf(y) 2xy

L2 = t +1

2f(y)2 x2x + a(r

x)x x

Note that unlike when we studied the Vasicek model in Section 3.1.1 wherea change of measure is performed, here we start working in the risk-neutralmeasure directly. Expressing the zero-coupon bond price as

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P

(t,x,y) = P0 (t, x) +P1 (t, x) + O () (3.40)it can be shown that

P0(t, x) = A () eB()x (3.41)

where

A () =

2e(+a)/2

( + a) (e 1) + 2

2ar/2(3.42)

B () =2e 1

( + a) (e 1) + 2

=pa2 + 22

The correction term then satisfies

hL2iP1 = V3x3xP0 (3.43)

P1 (T, x) = 0

which has a solution

P1 (t, x) = (D1 ()x + D2 ())A () eB()x (3.44)

where D1 and D2 satisfy the ODEs

D1

= V3B3

2B + a

D1 (3.45)

D2

= arD1

D1 (0) = D2 (0) = 0

Unlike in the case of singular perturbation for the Vasicek model where thecorrection depends on one function D that can be expressed in terms of B (),here we are faced with two functions D1 and D2 that are related through asystem of coupled nonhomogeneous differential equations.

Rearranging the terms in Equation (3.45), we have

+2B + a

D1 = V3B

3 (3.46)

This is a linear first-order ODE that can be solved by integrating factor. Tofind the integrating factor, we set the RHS to be zero and solveZ

du

u=

Z2B () + a

d (3.47)

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0 5 10 15 20 25 300.053

0.054

0.055

0.056

0.057

0.058

0.059

0.06

Maturity (year)

Yield

Figure 3.4: Fitted yield curves to swap rates observed on 09/07/1998. Thecrosses are the raw data points, the green curve is fitted with CIR bond pricesand the blue curve is fitted with perturbation corrected CIR bond prices.

The integrating factor u is

u () = c exp

"22

( + a) 2 log ( + a) e 1 + 22 a2

a# (3.48)Interested readers are referred to Appendix A for details. With the integratingfactor given by Equation (3.48), the full solution to the nonhomogeneous ODEis

D1 =

Ru ()V3B ()

3 d+ c

u ()(3.49)

where c can be determined using the initial condition. The initial conditionD1 (0) = 0 requires that c = 0. Figure 3.4 shows the fitted yield curves of CIR(with a sum of squared error 3.9777 105) and perturbation corrected CIR

models (with a sum of squared error 3.0147106). Although the p erturbationcorrected CIR model has only one group parameter (V3) available for fitting, itsperformance is comparable to that of the perturbation corrected Vasicek model,which has three group parameter, V1, V2 and V3 (see Section 3.1.2).

Figure 3.5 summarizes all the yield curve fitting results performed on the09/07/1998 data. Our results suggest a number of cautions for yield curvefitting in practice. First, a larger number of free parameters to be optimized

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Figure 3.5: A summary of all the yield curve fitting schemes investigated. Allfittings are done on the 09/07/1998 data. Vas is the uncorrected Vasicek model,Vas is the perturbation corrected Vasicek model, F-V is the Fong-Vasicekmodel, CIR is the uncorrected CIR model and CIR is the perturbation cor-rected CIR model.

does not necessarily guarantee a better performance. The performance is alsoheavily affected by the model specification and the implementation, especiallyif numerical methods are required to compute the zero-coupon bond price. Sec-ondly, the optimized parameters in the Fong-Vasicek model are the drift anddiffusion coefficients of the underlying processes, while the optimized parame-ters in the perturbation schemes (i.e. the group parameters V1, V2 and V3 inVas and V3 in CIR) have no intuitive interpretation. The question of whetherit is possible to back out the SDE parameters from the group parameters wouldcall for further study, but assuming it is possible it would certainly require nu-merical integration. Hence, if we are interested in finding the dynamics impliedby the yield curve under a certain model, the singular perturbation scheme isnot feasible; on the other hand if the main objective is to price fixed incomederivatives efficiently, the interpretations of the optimized parameters is not so

important and the feasibility of the perturbation correction may be justified.

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Chapter 4

Conclusion

We reviewed the mathematical background of singular perturbation techniqueand the application of it in equity, as well as in fixed income, derivatives pricing.Singular perturbation arises from the lack of uniform convergence, which is ob-served when a small parameter is multiplied by the highest order derivative in adifferential equation. This is exactly the case in the Black-Scholes PDE, giventhat the variance is small. This fact can be exploited to introduce stochasticvolatility corrections. We demonstrated in both equity and fixed income deriv-atives pricing how singular perturbation allows us to expand the full solutionas a series with terms of different orders in

, where is the reciprocal of the

variance mean-reversion rate.To evaluate the performance of the perturbation scheme under investigation,

we proceeded to fit the yield curve using two methods: fitting with Fong-Vasicekbond prices, and fitting with first-order perturbation corrected Vasicek bondprices. We presented a detailed derivation that gives us the expression forthe correction term P1. Since the Fong-Vasicek prices are exact in accountingfor stochastic interest rate volatility, it serves as an analytic benchmark forthe perturbation scheme. The fitting test is conducted on an S-shaped and amonotonically increasing yield curves. We found that in the former case, whilethe perturbation scheme slightly underperforms relative to the analytic schemein terms of minimizing the sum of squared error, it is about 200 times faster.In the latter case, the difference in error minimization is minimal.

Finally, we studied the application of singular perturbation technique to theCIR model. Unlike the perturbation corrected Vasicek model under which thecorrection term has a closed form, here the correction can only be calculated

using numerical integration. The computational time advantage of the pertur-bation scheme is greatly reduced, but we also found that the fitness of the cor-rected CIR model is very good despite its smaller number of group parameters.Directions for further investigations include backing out the underlying processcoefficients from the group parameters, as well as extending the technique tonon-affine interest rate models.

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Appendix A: Integrating Factor forD1 Under CIR

ModelThe full nonhomogeneous ODE is

+2B + a

D1 = V3B

3 (4.1)

The general solution u should solveZdu

u=

Z2B () + a

d+ c (4.2)

Evaluating the LHS and substituting Equation (3.42) into the RHS, we have

logu = 2 Z 2 e 1( + a) (e 1) + 2 d a (4.3)

With the change of variable

d de 1

= ed (4.4)

= logu = 2Z

2

( + a) + 2

d

( + 1) a

it can be shown that the general solution to Equation (4.1), which is also theintegrating factor for solving the original nonhomogeneous ODE, is

u () = exp

"22

( + a) 2 log

( + a)e 1

+ 2

2 a2

a#

(4.5)

which can be readily verified by taking derivative with respect to .

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Bibliography

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[2] J. -P. Fouque, G. Papanicolaou and R. Sincar, 2000, Derivatives in Finan-cial Markets with Stochastic Volatility, Cambridge University Press

[3] J. -P. Fouque, G. Papanicolaou, R. Sincar and K. Solna, 2003, "SingularPerturbations in Option Pricing", SIAM J. Appl. Math., Vol. 63, No. 5,pp. 1648-1665

[4] R. S. Johnson, 2005, Singular Perturbation Theory, Springer

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[7] J. Kevorkian and J. D. Cole, 1996, Multiple Scale and Singular PerturbationMethods, Springer

[8] M. Widdicks, P. W. Duck, A. D. Andricopoulos and D. P. Newton, 2005,"The Black-Scholes Equation Revisited: Asymptotic Expansions and Sin-gular Perturbations", Math. Finance, Vol. 15, No. 2, pp. 373-391

[9] J. -P. Fouque, G. Papanicolaou and R. Sincar, 2000, "Mean-Reverting Sto-chastic Volatility", Int. J. Theoretical Appl. Finance, Vol. 3, No. 1, pp.101-142

[10] F. A. Longstaffand E. S. Schwartz, 1992, "Interest Rate Volatility and theTerm Structure: A Two-Factor General Equilibrium Model", J. Finance,Vol. 47, No. 4, pp. 1259-1282

[11] B. Stehlikova, 2007, "Averaged Bond Prices for Fong-Vasicek and the Gen-eralized Vasicek Interest Rates Models", conference proceedings, Mathe-matical Methods in Economics and Industry

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a study on singular perturbation correction to bond prices under affine term structure models

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