A study on singular perturbation correction to bond prices under affine term structure models
1. 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 stochastic
interest 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 analytic
scheme are comparable, while the former requires much less computational time.
We extend this scheme to a perturbation corrected CIR model, in which case
the advantage in speed is diminished due to the need for numerical methods..
2. A Study on Singular Perturbation Correction to
Bond Prices Under Affine Term Structure
Models
Frank Fung
A REPORT SUBMITTED IN FULFILLMENT
OF THE REQUIREMENT FOR THE
INDEPENDENT STUDY
OF THE
BERKELEY MFE PROGRAM
15 October 2010
4. 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 is
their 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 yield
curve. Well known single factor affine term structure models include short rate
models such as Ho-Lee, Vasicek and CIR models. On the other hand, multi-
factor affine term structure models allow for more flexibility and diversity. One
way to utilize the additional degrees of freedom is to make the short rate a sum
of more than one processes, as in the case of Longstaff and Schwartz [10]. Due
to the mixing of the two processes, randomness is introduced into the interest
rate volatility implicitly. Alternatively, one can explicitly designate a stochas-
tic process to represent the dynamics of the short rate variance, as in the case
of Fong-Vasicek model [14]. However, as we will see shortly, the computational
cost of introducing stochastic volatility in such a way is quite high. On the other
hand, empirical evidence shows that interest rates and interest rate volatility
vary on different timescales [9], which is a fact we can exploit, together with
the singular perturbation technique, to correct the zero-coupon bond prices for
stochastic volatility. The perturbation correction terms are in different orders of
the reciprocal of variance mean-reversion rate, hence if the interest rate volatility
is fast mean-reverting the first order correction term would be able to capture
most 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-factor
counterpart, and the relative saving in computational efforts of it compared to a
full treatment of stochastic volatility. We choose to conduct our study with the
Vasicek and CIR models because, first of all, their simplicity allows us to focus
on the analysis of interest, and secondly the uncorrected single-factor versions
of 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
2
5. to other models would require further investigation.
This report is organized as follows. Chapter 2 is a literature review. In
Section 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 perturbation
techniques in equity derivatives and fixed income instruments, respectively. In
Chapter 3, we derive the formula for perturbation correction specifically for
the 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 presents
the perturbation correction to bond prices under a CIR model and an overall
comparison. We summarize our investigation in Chapter 4.
3
6. Chapter 2
Literature Review
2.1 Mathematical Background
Here we provide a brief review on the topic of perturbation. Readers are referred
to [4, 5, 6, 7] for a more detailed treatment. Perturbation theory is the method
of finding mathematical solutions by treating the full problem as a perturbed
simpler problem. The method has a long history of applications in various
fields, including but not limited to fluid dynamics, celestial mechanics as well
as quantum mechanics.
Consider a trivial example. Suppose we have to solve
x2 − 1 = εx (2.1)
The two quadratic roots are
r
ε ε2
± 1+ (2.2)
2 4
Depending on the desired accuracy, we can now perform a Taylor expansion to
Equation (2.2) to obtain an approximated solution. The leading terms in the
Taylor expansion are
ε ε2 ¡ ¢
x = ±1 + ± + O ε3 (2.3)
2 8
where O (•) is the large O notation. As ε → 0, Equation (2.1) becomes simply
x2 − 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, the
perturbation problem is said to be regular.
Alternatively, consider an ODE
dy
ε + y = cos x (2.4)
dx
4
7. with initial condition y (0) = 0. Naively one may be tempted to substitute a
formal expansion of the form
P
∞
yε (x) = εn yn (x) (2.5)
n=0
which would produce a series solution
yε (x) = cos x + ε sin x − ε2 cos x + · · · (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 above
fails. In fact the solution to the nonhomogeneous ODE in Equation (2.4) is
Z
1 −x/ε x t/ε
yε (x) = e e cos tdt (2.7)
ε 0
as can be obtained by the method of variation of parameters or undetermined
coefficients. Partial integration to Equation (2.7) gives
Z x
yε (x) = cos x − e−x/ε + e−x/ε et/ε sin tdt (2.8)
0
Note that unlike the naive attempt before, this solution actually satisfies the
initial condition y (0) = 0. The question is then whether we are able to create
out of Equation (2.8) a series solution in powers of ε. Note that e−x/ε is a
quickly varying function in the vicinity of x = 0. It can be shown that the series
∞
X h i
Sm (x) = (−1)n εn cosn (x) − e−x/ε sinn (x) (2.9)
n=0
is a legitimate approximation satisfying the initial condition. The convergence
of this series to the exact solution as ε → 0, however, is nonuniform in the sense
that
³ ´ ³ ´
lim lim Sm (x) 6= lim lim Sm (x) (2.10)
x→0 ε→0 ε→0 x→0
since
³ ´
lim lim Sm (x) = 1 (2.11)
x→0 ε→0
³ ´
lim lim Sm (x) = 0
ε→0 x→0
Thus we are faced with a singular, as opposed to regular, perturbation problem.
This is observed when the small parameter (ε in this case) is multiplied by the
highest-order derivative in the ODE.
5
8. 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 are
trying to expand a solution with respect to a small parameter that is multiplied
by the highest -order derivative. Now if we look at the Black-Scholes PDE:
1 2 2 ∂2V ∂V ∂V
σ S 2
+ rS + − rV = 0 (2.12)
2 ∂S ∂S ∂t
2
the highest-order derivative is ∂S V . Thus the condition for singular perturbation
2
is met if σ , the volatility, plays the role of the small parameter ε in Section 2.1.
Typical value of σ 2 ranges from 0.01 to 0.2, which is indeed small compared
to the option price V . As demonstrated in [8], this fact can be exploited to
efficiently price European, American and barrier options.
Another possible application of singular perturbation in the variable σ 2 is
studied by Fouque et al. [3]. They considered option pricing under the processes
dSt = rSt dt + f (Yt ) St dWt1 (2.13)
" √ # √
1 v 2 v 2
dYt = (m − Yt ) − √ Λ (Yt ) dt + √ dWt2
ε ε ε
in the risk-neutral measure, where Λ (Yt ) is the combined market price of risk
(of both the stock price and the volatility), f is some non-negative function
and ε is a small parameter, i.e. the volatility is fast mean-reverting (empirical
evidence for and analysis about the mean-reversion of volatility can be found in
[2, 9]). Express the option price P ε as a series
√
P ε = P0 + εP1 + εP2 + · · · (2.14)
and define the differential operators
1 1
Lε = L0 + √ L1 + L2 (2.15)
ε ε
where
∂2 ∂
L0 = v2 2
+ (m − y) (2.16)
∂y ∂y
√ ∂2 √ ∂
L1 = 2ρvf (y) − 2vΛ (y)
∂x∂y ∂y
2
µ ¶
∂ 1 2 ∂ 1 2 ∂
L2 = + f (y) + r − f (y) −r
∂t 2 ∂x2 2 ∂x
6
9. It can then be shown, by considering Lε P ε = 0 and matching terms with the
same order in ε, that
µ ¶
√ ∂3 ∂2 ∂
εP1 = − (T − t) V3 3 + (V2 − 3V3 ) 2 + (2V3 − V2 ) P0 (2.17)
∂x ∂x ∂x
where V2 and V3 are group parameters that are averaged over the probability
distribution of y and P0 is the Black-Scholes price with effective variance σ 2 =
2®
f . In other words, the first order correction can be obtained by solving an
ODE that involves the zeroth order price.
2.2.2 Fixed Income Derivatives
In this section we discuss the procedure for applying singular perturbation to
fixed income derivative pricing. Once again this is motivated by the fact that
interest rate volatility varies on a different (shorter) timescale from the interest
rate itself. Since the detailed derivation for the specific case of Fong-Vasicek
would be provided in Section 3.1.1, we only outline the method here. Consider
the processes
drt = a(r∞ − rt )dt + f (Yt )dWt1 (2.18)
dYt = α(m − Yt )dt + βdWt2
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 can
be shown that the lowest order correction term satisfies
LLHS P1 = 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 have
estimated V2 and V3 from the historical stock price time series. On the other
hand for fixed income derivatives pricing, since we have an observable yield
curve, we can calibrate our model using the stochastic volatility corrected bond
prices 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 dealing
with affine models, in which case the bond price correction term P1 could have
closed-form solution, as we shall see later. Secondly, the method introduced
here is flexible enough to be applied to different affine short rate models. With
these two observations we move on to the next chapter, where we compare the
application of singular perturbation to yield curve fitting under different models
(Vasicek and CIR) and during different periods. We will also compare how the
7
10. 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 perturbation
corrected curves are shown. In Figure 2.1 the uncorrected and perturbation
corrected yield curves have sum of squared errors of 4.5409 × 10−5 and 1.2793 ×
10−5 , respectively. In Figure 2.2 the uncorrected and perturbation corrected
yield curves have sum of squared errors of 9.4260 × 10−6 and 6.2520 × 10−7 ,
respectively.
0.06
0.059
0.058
0.057
Yield
0.056
0.055
0.054
0.053
0 5 10 15 20 25 30
Maturity (year)
Figure 2.1: Fitted yield curves to swap rates observed on 09/07/1998. The
crosses are the raw data points, the red curve is fitted with Vasicek bond prices
and the blue curve is fitted with perturbation corrected Vasicek bond prices.
8
11. 0.062
0.061
0.06
Yield
0.059
0.058
0.057
0.056
0 5 10 15 20 25 30
Maturity (year)
Figure 2.2: Fitted yield curves to swap rates observed on 08/07/1998. The
crosses are the raw data points, the red curve is fitted with Vasicek bond prices
and the blue curve is fitted with perturbation corrected Vasicek bond prices.
9
12. 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 order
correction 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 capturing
the effect of volatility. Ideally, we would want to study a model having the
same 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 a
model. Alternatively, we try to shed light on the comparison by investigate the
Fong-Vasicek model. We derive in details the bond price obtained by singular
perturbation 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:
p
drt = a(r∞ − rt )dt + Yt dWt1 (3.1)
p
dYt = κ(θ − Yt )dt + σ Y Yt dWt2
where hdWt1 dWt2 i = ρdt. Comparing to the system of stochastic differential
equations in [1], which we reproduce here,
drt = a(r∞ − rt )dt + f (Yt )dWt1 (3.2)
dYt = α(m − Yt )dt + βdWt2
10
13. we see that the diffusion of Yt 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
dξ
z(Y ) = √ (3.3)
ξ
It can be easily shown that the appropriate transformation is z(Y ) = Y 1/2 (up
to a factor). Using Ito’s lemma, the dynamics of z is
∙ ¸
1 ¡ 2
¢ 1 2 1 1
dz = κ θ−z − σY dt + σ Y dWt2 (3.4)
2z 8 z 2
and we succeeded in making the diffusion coefficient constant. Recall that the
order matching scheme described in Section 2.2.1 requires the rate of mean
reversion of interest rate volatility to be high, which in turn dictates the drift
coefficient in the volatility process (i.e. α in Equation (2.18)) be large. On the
other hand, in Equation (3.4) we can no longer be sure about the rate of mean
reversion due to the dependence of the drift coefficient on z. Hence we have to
go back to the original Fong-Vasicek SDE’s as a starting point.
Returning to Equation (3.1), the differential equations in risk-neutral mea-
sure becomes
h p i p
drt = a(r∞ − rt ) − λ(Yt ) Yt dt + Yt dWt1 (3.5)
h p i p
dYt = κ(θ − Yt ) − σ Y Yt Λ(Yt ) dt + σ Y Yt dWt2
p
where λ and γ are the two market prices of risk and Λ ≡ ρλ + 1 − ρ2 γ. The
corresponding Feynman-Kac PDE is
1 2 √
∂t P + y∂x P + [a(r∞ − x) − λ(y) y] ∂x P − xP (3.6)
2
1 √
+ρσY y∂xy P + σ 2 y∂y P + [κ(θ − y) − σ Y yΛ(y)] ∂y P = 0
2
Y
2
2
To save space we write it as
Lε P = 0 (3.7)
where the subscript ε is a reminder that the differential operator Lε depends on
ε. We then rearrange the differential operators into three groups,
1 2 2
L0 = σ y∂ + κ(θ − y)∂y (3.8)
2 Y y
2 √
L1 = ρσ Y y∂xy − σ Y yΛ(y)∂y
1 2 √
L2 = ∂t + y∂x + [a(r∞ − x) − λ(y) y] ∂x − x
2
11
14. With
1
ε =
κr
2
σY = v
ε
the differential operators, expressed explicitly in powers of ε, are defines as
L0 = v 2 y∂y − (θ − y) ∂y
2
(3.9)
√ 2
√ √
L1 = 2vρy∂xy − 2v yΛ (y) ∂y
1 2 √
L2 = ∂t + y∂x + [a(r∞ − x) − λ(y) y] ∂x − x
2
so that
1 1
Lε = L0 + √ L1 + L2 (3.10)
ε ε
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 ε (t, x, y; T ) = P0 (t, x, y; T ) + εP1 (t, x, y; T ) + εP2 (t, x, y; T ) + · · · (3.11)
The O(ε−1 ) equation is
L0 P0 = 0 (3.12)
which indicates that P0 (t, x; T ) is independent of y.
The O(ε−1/2 ) equation is
L0 P1 + L1 P0 = 0 (3.13)
Since P0 = P0 (t, x; T ), L1 P0 = 0 and we conclude that P1 (t, x; T ) is also inde-
pendent of y. The O(1) equation is
L2 P0 + L1 P1 + L0 P2 = L2 P0 + L0 P2 = 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 L2 P0 |x=x0 can be thought of as a function of y
which we denote as h(y). Then, holding x constant,
(L0 P2 + h(y)) |x=x0 = 0 (3.15)
is a Poisson equation for P2 in the variable y and it can be shown that a solution
P2 (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 = hL2 P0 i = 0, where the average
12
15. 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 hL2 i P0 = 0. The operator L2 is
formally the same as the differential operator of the Feynman-Kac PDE if we
are considering a Vasicek model with constant volatility:
1 2 √
hL2 i = ∂t + hyi ∂x + [a(r∞ − x) − hλ(y) yi] ∂x − x (3.16)
2
1
≡ ∂t + σ 2 ∂x + [a(r∗ − x)] ∂x − x
2
2
√ ®
where σ2 ≡ hyi and r∗ = r∞ − λ(y) y /a. Therefore the solution to hL2 i P0 =
0 is
P0 (t, x; T ) = A(T − t)e−B(T −t)x (3.17)
where A(T − t) and B(T − t) are given by
1 − e−a(τ )
B(τ ) = (3.18)
a
½ ∙
σ2 ³ ´2 ¸¾
A(τ ) = exp − Rτ − RB(τ ) + 1 − e−a(τ )
4a3
where τ = T − t and R ≡ r∗ − σ 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 σ2 .
With the centering condition, we can write
L2 P0 = L2 P0 − hL2 i P0 = (L2 − hL2 i) P0 (3.19)
Substituting back into the Poisson Equation (3.15) would give
L0 P2 = − (L2 − hL2 i) P0 (3.20)
Or equivalently
P2 = −L−1 (L2 − hL2 i) P0 + k (x, t)
0 (3.21)
where the function√ arises as an integration constant (in the variable y).
k
Finally, the O ( ε) equation is
L2 P1 + L1 P2 + L0 P3 = 0 (3.22)
This is a Poisson equation of P3 , and a solution exists only if the centering
condition hL2 P1 + L1 P2 i = 0 is fulfilled. Once again, P1 is y-independent and
hence hL2 P1 i = hL2 i P1 . Equation (3.22), together with the centering condition,
gives
13
16. hL2 i P1 = − hL1 P2 i (3.23)
®
= L1 L−1 (L2 − hL2 i) P0 + L1 k (x, t)
0
®
= L1 L−1 (L2 − hL2 i) P0
0
The second line makes use of the fact that k(x, t) does not depend on y. As P0
is already found, solving Equation (3.23) would give us an expression for P1 .
The operator L2 − hL2 i is
1¡ ¢ 2 √
y − σ 2 ∂x − (λ y − hλi σ) ∂x (3.24)
2
Since L0 contains y differential operators only,
1 £ −1 ¡ ¢¤ 2 £ √ ¤
L−1 (L2 − hL2 i) =
0 L0 y − σ 2 ∂x − L−1 (λ y − hλi σ) ∂x
0 (3.25)
2
Introduce two functions, φ and ψ, with the following properties:
L0 φ = y − σ 2 (3.26)
√
L0 ψ = λ y − hλi σ
Equation (3.23) then becomes
¿ µ ¶À
1 2
hL2 i P1 = L1 φ∂x − ψ∂x P0 (3.27)
2
¿³ ´ µ1 ¶À
√ 2
√ √ 2
= 2vρy∂xy − 2v yΛ (y) ∂y φ∂x − ψ∂x P0
2
Using the relations
n
0® n
h∂y (φ∂x )i Γ = φ ∂x Γ (3.28)
n
0® n
h∂y (ψ∂x )i Γ = ψ ∂x Γ
for some y-independent function Γ(x), and defining
1 ®
V1 = √ vρ yφ0 (3.29)
2
1 ® √ √ ®
V2 = − √ v yΛφ0 − 2vρ yψ 0
2
√ √ ®
V3 = 2v yΛψ 0 ∂x
we can rewrite Equation (3.27) as
14
17. ∙ ¸
1 £ ¤
∂t + σ 2 ∂x + [a(r∗ − x)] ∂x − x P1 = V1 ∂x + V2 ∂x + V3 ∂x P0
2 3 2
(3.30)
2
Due to the centering condition, all y-dependences are contained within the group
parameters V1,2,3 , which allows us to solve a PDE in x and t only. Substitute
P0 = A(τ )e−B(τ )x into Equation (3.30) and after some algebra we find that
1 2 2 £ ¤
∂τ P1 = σ ∂x P1 + [a(r∗ − x)] ∂x P1 − xP1 + V1 B 3 − V2 B 2 + V3 B P0 (3.31)
2
with the initial condition P1 (t = T, x) = 0. It is trivial to check that the solution
to the PDE is
P1 (t, x) = D (τ ) A(τ )e−B(τ )x (3.32)
where
µ ¶
V3 1 1
D (τ ) = τ − B (τ ) − aB (τ )2 − a2 B (τ )3 (3.33)
a3 2 3
µ ¶
V2 1 V1
− 2 τ − B (τ ) − aB (τ )2 + (τ − B (τ ))
a 2 a
¡ ¢
The corrected zero-coupon bond price up to O ε1/2 is therefore
¡ √ ¢
P ' 1 + εD (τ ) A(τ )e−B(τ )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 diffusion
term proportional to a function of Yt , 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 an
analytic bond price [11]. The zero-coupon bond price under Fong-Vasicek model
is P (τ , r) = A (τ ) e−B(τ )r−C(τ )y , with A (τ ), B (τ ) and C (τ ) give by the system
of ordinary differential equations
dA
= −A (ar∗ B + κθC) (3.35)
dτ
dB
= −aB + 1
dτ
dC B2 σ2 C 2
= −λB − κC − γσ Y C − − Y − σ Y ρBC
dτ 2 2
15
18. 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 the
A and B equations are
1 − e−aτ
B (τ ) = (3.36)
µa Z ¶
τ
A (τ ) = exp −r∗ τ + r∗ B (τ ) − κθ C (s) ds
0
Before going into yield curve fitting results, we shall inspect the connection
between the Fong-Vasicek analytic bond price and perturbation correction term.
If the approximation scheme is correct, we should have
µ Z τ ¶
∗ ∗
exp (−r τ + r B (τ ) − B (τ ) r) exp −κθ C (s) ds exp (−C (τ ) y) (3.37)
0
½ ∙ ¸¾
¡ √ ¢ σ2 ³ ´2
= 1 + εD (τ ) exp − Rτ − RB(τ ) + 3 1 − e−a(τ ) + B (τ ) r + O (ε)
4a
As a reminder, ε = 1/κ. The reason singular perturbation is required is now
apparent. Although the correction to the bond price is small under our assump-
tions, it cannot be easily obtained by performing a series expansion directly on
the Fong-Vasicek analytic price since the parameter appearing in the exponen-
tial, κ, is not small.
To fit the yield curve with the analytic Fong-Vasicek bond price, we approx-
imate C (τ ) using Euler method starting with the initial condition C (0) = 0.
R
The integral C (s) ds within A (τ ) 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 altered
by a change in the variance SDE diffusion coefficient (it is absorbed in the group
parameters). Figure 3.1 shows the details of the two schemes using nonlinear
least 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
16
19. 0.06
0.059
0.058
0.057
Yield
0.056
0.055
0.054
0.053
0 5 10 15 20 25 30
Maturity (years)
Figure 3.2: Fitted yield curves to swap rates observed on 09/07/1998. The
crosses are the raw data points, the red curve is fitted with analytic Fong-
Vasicek bond prices and the blue curve is fitted with perturbation corrected
Vasicek bond prices.
computational speed but does not do as well in terms of fitness to data. This is
√
to be expected since the perturbation scheme is accurate only up to O ( ε). An-
other observation is that the optimized parameters for the analytic Fong-Vasicek
scheme are the variables that enter into the SDE of Equation (3.1), while there
is no intuitive interpretation for the optimized parameters of the perturbation
scheme. We would like to point out two issues in the implementation of the
yield curve fitting schemes. First, the longer CPU time in the analytic scheme
is to a large extent due to looping when implementing the Euler scheme and
the numerical integration for C (τ ). The relative disadvantage in speed of the
analytic scheme may be reduced under a different implementation. Second, the
analytic bond price depends on y, which is an unobservable quantity. We proxy
y 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 the
latter).
17
20. 0.062
0.061
0.06
Yield
0.059
0.058
0.057
0.056
0 5 10 15 20 25 30
Maturity (year)
Figure 3.3: Fitted yield curves to swap rates observed on 08/07/1998. The
crosses are the raw data points, the red curve is fitted with analytic Fong-
Vasicek bond prices and the blue curve is fitted with perturbation corrected
Vasicek 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. We
first briefly review the partial results presented in [1]. The dynamics of the
two-factor CIR model we study is described by the processes
√
drt = a (r∗ − rt ) dt + f (Yt ) rt dWt1 (3.38)
√ ¡ ¢
dYt = αrt (m − Yt ) dt + β rt ρdWt1 + ρ0 dWt2
where dWt1 and dWt2 are Wiener processes under the risk-neutral measure. The
differential operators in this case are
¡ ¢
L0 = x v 2 ∂y + (m − y) ∂y
2
(3.39)
√ 2
L1 = 2vρxf (y) ∂xy
1
L2 = ∂t + f (y)2 x∂x + a(r∗ − x)∂x − x
2
2
Note that unlike when we studied the Vasicek model in Section 3.1.1 where
a change of measure is performed, here we start working in the risk-neutral
measure directly. Expressing the zero-coupon bond price as
18
21. √
P ε (t, x, y) = P0 (t, x) + εP1 (t, x) + O (ε) (3.40)
it can be shown that
P0 (t, x) = A (τ ) e−B(τ )x (3.41)
where
µ ¶2ar∗ /σ2
2θe(θ+a)τ /2
A (τ ) = (3.42)
(θ + a) (eθτ − 1) + 2θ
¡ ¢
2 eθτ − 1
B (τ ) =
(θ + a) (eθτ − 1) + 2θ
p
θ = a2 + 2σ 2
The correction term then satisfies
3
hL2 i P1 = V3 x∂x P0 (3.43)
P1 (T, x) = 0
which has a solution
P1 (t, x) = (D1 (τ ) x + D2 (τ )) A (τ ) e−B(τ )x (3.44)
where D1 and D2 satisfy the ODE’s
∂D1 ¡ ¢
= V3 B 3 − σ 2 B + a D1 (3.45)
∂τ
∂D2
= ar∗ D1
∂τ
D1 (0) = D2 (0) = 0
Unlike in the case of singular perturbation for the Vasicek model where the
correction depends on one function D that can be expressed in terms of B (τ ),
here we are faced with two functions D1 and D2 that are related through a
system of coupled nonhomogeneous differential equations.
Rearranging the terms in Equation (3.45), we have
µ ¶
∂ ¡ ¢
+ σ 2 B + a D1 = V3 B 3 (3.46)
∂τ
This is a linear first-order ODE that can be solved by integrating factor. To
find the integrating factor, we set the RHS to be zero and solve
Z Z
du ¡ 2 ¢
=− σ B (τ ) + a dτ (3.47)
u
19
22. 0.06
0.059
0.058
0.057
Yield
0.056
0.055
0.054
0.053
0 5 10 15 20 25 30
Maturity (year)
Figure 3.4: Fitted yield curves to swap rates observed on 09/07/1998. The
crosses are the raw data points, the green curve is fitted with CIR bond prices
and the blue curve is fitted with perturbation corrected CIR bond prices.
The integrating factor u is
" ¡ ¡ ¢ ¢ #
2 (θ + a) θτ − 2θ log (θ + a) eθτ − 1 + 2θ
u (τ ) = c exp 2σ ¡ ¢ − aτ (3.48)
θ θ2 − a2
Interested readers are referred to Appendix A for details. With the integrating
factor given by Equation (3.48), the full solution to the nonhomogeneous ODE
is
R 3
u (τ ) V3 B (τ ) dτ + c
D1 = (3.49)
u (τ )
where c can be determined using the initial condition. The initial condition
D1 (0) = 0 requires that c = 0. Figure 3.4 shows the fitted yield curves of CIR
(with a sum of squared error 3.9777 × 10−5 ) and perturbation corrected CIR
models (with a sum of squared error 3.0147 × 10−6 ). Although the perturbation
corrected CIR model has only one group parameter (V3 ) available for fitting, its
performance 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 the
09/07/1998 data. Our results suggest a number of cautions for yield curve
fitting in practice. First, a larger number of free parameters to be optimized
20
23. Figure 3.5: A summary of all the yield curve fitting schemes investigated. All
fittings are done on the 09/07/1998 data. Vas is the uncorrected Vasicek model,
Vasε is the perturbation corrected Vasicek model, F-V is the Fong-Vasicek
model, CIR is the uncorrected CIR model and CIRε is the perturbation cor-
rected CIR model.
does not necessarily guarantee a better performance. The performance is also
heavily affected by the model specification and the implementation, especially
if numerical methods are required to compute the zero-coupon bond price. Sec-
ondly, the optimized parameters in the Fong-Vasicek model are the drift and
diffusion coefficients of the underlying processes, while the optimized parame-
ters in the perturbation schemes (i.e. the group parameters V1 , V2 and V3 in
Vasε and V3 in CIRε ) have no intuitive interpretation. The question of whether
it is possible to back out the SDE parameters from the group parameters would
call for further study, but assuming it is possible it would certainly require nu-
merical integration. Hence, if we are interested in finding the dynamics implied
by the yield curve under a certain model, the singular perturbation scheme is
not feasible; on the other hand if the main objective is to price fixed income
derivatives efficiently, the interpretations of the optimized parameters is not so
important and the feasibility of the perturbation correction may be justified.
21
24. Chapter 4
Conclusion
We reviewed the mathematical background of singular perturbation technique
and 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 a
differential equation. This is exactly the case in the Black-Scholes PDE, given
that the variance is small. This fact can be exploited to introduce stochastic
volatility corrections. We demonstrated in both equity and fixed income deriv-
atives pricing how singular perturbation allows us to expand the full solution
√
as a series with terms of different orders in ε, where ε 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-Vasicek
bond prices, and fitting with first-order perturbation corrected Vasicek bond
prices. We presented a detailed derivation that gives us the expression for
the correction term P1 . Since the Fong-Vasicek prices are exact in accounting
for stochastic interest rate volatility, it serves as an analytic benchmark for
the perturbation scheme. The fitting test is conducted on an S-shaped and a
monotonically increasing yield curves. We found that in the former case, while
the perturbation scheme slightly underperforms relative to the analytic scheme
in 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 the
CIR model. Unlike the perturbation corrected Vasicek model under which the
correction 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 process
coefficients from the group parameters, as well as extending the technique to
non-affine interest rate models.
22
25. Appendix A: Integrating Factor for D1 Under CIR
Model
The full nonhomogeneous ODE is
µ ¶
∂ ¡ 2 ¢
+ σ B + a D1 = V3 B 3 (4.1)
∂τ
The general solution u should solve
Z Z
du ¡ 2 ¢
=− σ B (τ ) + a dτ + c (4.2)
u
Evaluating the LHS and substituting Equation (3.42) into the RHS, we have
Z ¡ ¢
2 eθτ − 1
log u = −σ 2 dτ − aτ (4.3)
(θ + a) (eθτ − 1) + 2θ
With the change of variable
¡ ¢
dχ ≡ d eθτ − 1 = θeθτ dτ (4.4)
Z
2χ dχ
=⇒ log u = −σ 2 − aτ
(θ + a) χ + 2θ θ (χ + 1)
it can be shown that the general solution to Equation (4.1), which is also the
integrating factor for solving the original nonhomogeneous ODE, is
" ¡ ¡ ¢ ¢ #
2 (θ + a) θτ − 2θ log (θ + a) eθτ − 1 + 2θ
u (τ ) = exp 2σ ¡ ¢ − aτ (4.5)
θ θ2 − a2
which can be readily verified by taking derivative with respect to τ .
23
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