Lithuanian Mathematical Journal, Vol. 51, No. 3, July, 2011, pp. 385–401
WEAK APPROXIMATION OF CIR EQUATION BY DISCRETE RANDOM VARIABLES Vigirdas Mackeviˇcius Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225 Vilnius, Lithuania (e-mail:
[email protected]) Received September 3, 2010
√ Abstract. For the CIR equation dXt = (θ − k Xt ) dt + σ Xt dBt , we propose positive weak first- and second-order approximations that use, at each step, generation of discrete (respectively two- and three-valued) random variables (Theorems 3 and 4). √ The equation is split into deterministic part dDt = (θ − kDt ) dt, which is solved exactly, and stochastic part dSt = σ St dBt , which is actually approximated in distribution. MSC: 65H35, 65C30 Keywords: CIR equation, simulation, weak approximations, split-step approximations
1
INTRODUCTION
We consider the solution of the CIR equation (Cox–Ingersoll–Ross [4]) dXt = (θ − kXt ) dt + σ Xt dBt , X0 = x0 0,
(1.1)
where B is a standard Brownian motion, and θ, σ 0, k ∈ R. The CIR equation, which was initially introduced to model the short interest rate [4], seems to be most widely used in finance and financial mathematics. Mathematically, its main qualitative features are the positivity of the solution and known analytic expressions for its moments. Note that the distribution (of increments) of CIR process is known explicitly as a noncentral χ2 distribution (see, e.g., [6, pp. 130–131]), and thus its exact simulation is possible; however, it is too slow in comparison with discretization schemes. Therefore, construction and analysis of positivity-preserving approximation methods for CIR and other square-root diffusion equations is still a topic of great interest; see, e.g., [5]. The main problem with developing numerical methods for square-root diffusions is the square-root itself, which has unbounded derivatives near zero. Therefore, discretization schemes that (explicitly or implicitly) involve the derivatives of the coefficients—even when they assure the positivity of approximation—usually lose their accuracy near zero, especially, for large σ ; the larger σ is, the more concentrated near zero the valuedistributions of CIR process are. One way to get around this difficulty is modifying the scheme considered c 2011 Springer Science+Business Media, Inc. 0363-1672/11/5103-0385
385
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V. Mackeviˇcius
by switching near zero to another scheme, which (1) is sufficiently “regular” and (2) sufficiently accurate near zero; we refer, for example, to [3, 10] and references therein. Typically, the approximation near zero is done by discrete random variables matching two or three moments with the true solution. In this paper, we extend this idea by constructing first- and second-order discrete-valued approximations for the distributions of the CIR process on the whole positive √ line. This is done by (1) first constructing approximations for the stochastic part of the equation dSt = σ St dBt in the form of two- or three-valued discrete random variables that match respectively two or four moments with the true solution and then (2) using splitting with the exact solution of the deterministic part dDt = (θ − kDt ) dt (for the idea of splitting the equations, we refer to, e.g., [1, 3, 9, 10, 11] and references therein). 2
PRELIMINARIES
Consider the general one-dimensional SDE Xtx
t =x+
b(Xsx ) ds
t +
0
σ Xsx dBs ,
t 0,
(2.1)
0
for x ∈ D ⊂ R. We assume that the equation has a unique weak solution X x such that P{Xtx ∈ D, t 0} = 1 for all x ∈ D. For example, for Eq. (1.1), one can take D = [0, ∞), provided that θ, σ > 0. On the fixed time interval [0, T ], we consider equidistant time discretizations Δh = {ih, i = 0, 1, . . . , [T /h]}, where [a] is the integer part of a. By a discretization scheme for Eq. (2.1) we mean a family of timeˆ h = {X ˆ h (x, t), x ∈ D, t ∈ Δh }, h > 0, with initial homogeneous (discrete-time) D-valued Markov chains X h ˆ ˆ h by ph (t, x, dz), t ∈ Δh , x ∈ D; for the values X (x, 0) = x. We denote the transition probability of X ˆ h in one step, we omit the superscript h and denote p(h, x, dz) = ph (h, x, dz). For transition probability of X convenience, we consider only the steps taking the values h = T /n with n ∈ N. With some abuse of notation, ˆ tx or X(x, ˆ ˆ h (x, t). t) instead of X sometimes we also write X ∞ (D) respectively the spaces of C ∞ functions f : D → R, of funcWe denote by C ∞ (D), C0∞ (D), and Cpol ∞ tions f ∈ C (D) with compact support in D, and of f ∈ C ∞ (D) having all the derivatives with polynomial ∞ (D) if growth. In other words, f ∈ Cpol (n) f (x) Cn 1 + |x|kn ,
x ∈ D, n ∈ N0 = {0, 1, 2, . . .},
for some sequence {Cn , kn } of positive numbers Cn and nonnegative integers kn . Following [2, 3], we say that {(Cn , kn ), n = 0, 1, 2, . . .} is a good sequence for f . ˆ h is a weak ν th-order approximation for the solution X x of Eq. (2.1) D EFINITION 1. A discretization scheme X if, for every f ∈ C0∞ (D), there exists K > 0 such that
x x x h ˆ = Ef X − Ef X ˆ (x, T ) Khν , Ef XT − Ef X T T
h > 0.
D EFINITION 2. Let Lf = bf + 12 σ 2 f be the generator of the solution Xtx of Eq. (2.1). Suppose that Lf ∈ ∞ (D) for all f ∈ C ∞ (D), i.e., b, σ 2 ∈ C ∞ (D). The ν th-order remainder of a discretization scheme X ˆx Cpol t pol pol ∞ (D) → C(D) defined by for Xtx is the operator Rνh : Cpol
x
ˆ − f (x) + Rνh f (x) := Ef X h
ν Lk f (x) k=1
k!
hk ,
x ∈ D, h > 0.
(2.2)
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Weak approximation of CIR equation
ˆ x is a local ν th-order weak approximation of Eq. (2.1) if A discretization scheme X t
Rνh f (x) = O hν+1 ,
h → 0,
(2.3)
∞ (D) and x ∈ D. for all f ∈ Cpol
Remark 1. The motivation of the above definition is as follows. Iterating the Dynkin formula Ef (Xhx ) =
h f (x) + 0 ELf (Xsx ) ds, we have Ef
Xhx
= f (x) +
ν Lk f (x)
k!
k=1
k
h s1
sν ···
h +
0
0
0
ν+1
ELν+1 f Xsxν+1 dsν+1 . . . ds2 ds1 .
If Lν+1 f behaves “well” (for example, b, σ, f ∈ C0∞ (D), and thus ELν+1 f (·) is bounded), then, for a local ˆ x , we have ν th-order weak approximation discretization scheme X t x x ˆ = O hν+1 , Ef X − Ef X h
h
h → 0.
(2.4)
Therefore, one may expect that in “good” cases, similarly to the deterministic case, a local ν th-order weak approximation discretization scheme is indeed a ν th-order (“global”) weak approximation. For rigorous results, one needs certain uniformity in f of property (2.4) and regularity of the generator L. Here, for brevity, we below state sufficient conditions only in the case of CIR equation, without formulation of known general results. ˆ tx is a potential ν th-order weak approximation of Eq. (2.1) if D EFINITION 3. A discretization scheme X
h Rν f (x) Chν+1 1 + |x|k ,
∞ f ∈ Cpol (D), x ∈ D, h h0 ,
for some C > 0, k ∈ N0 , and h0 ∈ (0, T ] depending only on a good sequence for f . ˆ h , h > 0, has uniformly bounded moments if there exists h0 > 0 D EFINITION 4. A discretization scheme X such that h ˆ (x, t)n < +∞, n ∈ N, x ∈ D. sup sup EX 0 0, C3 > 0, and k ∈ N and for some C4 > 0 and l ∈ N depending only on a good ∞ ( R) : sequence for f ∈ Cpol E Sˆhx − x = 0, 2 E Sˆhx − x = xah, x E Sˆ − x 3 C3 h2 1 + xk , h 4 E max f (4) (s) Sˆhx − x C 4 h2 1 + x l , 0sSˆhx
x ∈ D, 0 < h < h0 ,
(3.2)
x ∈ D, 0 < h < h0 ,
(3.3)
x ∈ D, 0 < h < h0 ,
(3.4)
x ∈ D, 0 < h < h0 .
(3.5)
Let us first look for Sˆhx satisfying Eqs. (3.2) and (3.3). In fact, we follow Alfonsi [2, 3, Sect. 2.2], who used positive two-valued discrete random variables matching the first two moments for second-order approximation of the CIR equation in a neighborhood of zero. We shall see that his scheme, when applied to the stochastic part of the equation only, appears to be rather simple, satisfies conditions (3.4)–(3.5) as well, and, together with composition (2.7), gives a potential first-order weak approximation for CIR equation. Thus, we define Sˆhx by x1 = x1 (x, h) := x + ah −
(x + ah)ah > 0,
x 0,
x2 = x2 (x, h) := x + ah + (x + ah)ah > 0, x 0, x P Sˆhx = x1,2 = p1,2 := , x > 0, P Sˆh0 = 0 = 1. 2x1,2 Lith. Math. J., 51(3):385–401, 2011.
(3.6) (3.7) (3.8)
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V. Mackeviˇcius
Then, for x > 0, we have x x x x1 + x2 x 2(x + ah) + = = = 1, 2x1 2x2 2 x1 x2 2 (x + ah)2 − (x + ah)ah x x + x2 − x = 0, E Sˆhx − x = x1 p1 + x2 p2 − x = x1 2x1 2x2 2 x x x E Sˆhx − x = x21 p1 + x22 p2 − x2 = x21 + x22 = (x1 + x2 ) − x2 2x1 2x2 2 2 = x(x + ah) − x = xah, x 3 x x + (x2 − x)3 = 2xa2 h2 , E Sˆh − x = (x1 − x)3 2x1 2x2 4 x x E Sˆhx − x = (x1 − x)4 + (x2 − x)4 = x(x + 4ah)a2 h2 . 2x1 2x2 p1 + p2 =
From this we immediately get (3.2)–(3.4). Note that if, for example, f (4) is bounded, we also get (3.5). In fact, ∞ (D). To check this, we shall use, for higher central moments of S ˆx , the (3.5) is also true for every f ∈ Cpol h expressions similar to those for Shx in Proposition 1: Proposition 2. For p ∈ N, 2p 2p+1 ˆ p (x, ah), E Sˆhx − x = x(ah)p Pˆp (x, ah), E Sˆhx − x = x(ah)p+1 Q ˆp = Q ˆ p (x, y) are (p − 1)th-order homogeneous two-variable polynomials with where Pˆp = Pˆp (x, y) and Q ˆ p at the power xp−1 are 1 and 2p, respecpositive integer coefficients; moreover, the coefficients of Pˆp and Q tively.
Proposition 3. For p ∈ N,
p ˆ p (x, ah), E Sˆhx = xR
ˆp = R ˆ p (x, y) is a (p − 1)th-order homogeneous two-variable polynomial with positive integer coeffiwhere R ˆ p at the power xp−1 is 1. cients; moreover, the coefficients of R
The proofs of Propositions 2 and 3 are given in the Appendix. ∞ (D) with constants C > 0 and l ∈ N depending only Corollary 1. Inequality (3.5) is satisfied for all f ∈ Cpol 4 on a good sequence of f . ∞ (D), there exist M > 0 and q ∈ N (depending only on a good sequence of f ) such Proof. Since f ∈ Cpol 1 that (f (4) (s))2 M1 (1 + s2q ), s ∈ D. Also note that, for a pth-order homogeneous two-variable polynomial P (x, y), we have the simple estimate |P (x, y)| C(|x|p + |y|p ) for all x, y ∈ R with some finite constant C . Using Propositions 2 and 3, by the Cauchy–Schwarz inequality, we have that, for any fixed a > 0 and h0 > 0, the left-hand side of (3.5) does not exceed 2 8 E1/2 max f (4) (s) E1/2 Sˆhx − x ¯x 0 s X h 1/2
2q 1/2 1/2 1 + E Sˆhx x(ah)4 Pˆ4 (x, ah) ˆ 2q (x, ah) 1/2 x x3 + (ah)3 1/2 (ah)2 M2 1 + x R 1/2 M3 1 + x x2q−1 + (ah)2q−1 1 + x2 h2 C4 1 + xq+2 h2 , M1
0 < h < h0 ,
with the constants Mi , i = 1, 2, 3, and C4 depending only on a good sequence for f .
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From Proposition 3 we also easily obtain that, for all a > 0, h0 > 0, and p ∈ N, there exists a constant C such that p E Sˆhx xp (1 + Ch) + Ch, x ∈ D, 0 < h h0 . By Proposition 1.4 of [2] this is exactly what we need to state the following: Corollary 2. The discretization scheme Sˆtx has uniformly bounded moments. Summarizing the above, by Theorem 2 we have the following: ˆ x be defined by composition (2.6), where the two-valued random Theorem 3. Let the discretization scheme X t ˆ tx is a first-order variables Sˆhx take the values x1,2 with probabilities p1,2 defined by Eqs. (3.6)–(3.8). Then X ∞ discretization scheme for the CIR equation (1.1), i.e., for every f ∈ C0 (D), there exist K > 0 and h0 ∈ (0, T ] such that x x ˆ Kh, 0 < h < h0 . Ef XT − Ef X T
4
A POTENTIAL SECOND-ORDER APPROXIMATION OF THE STOCHASTIC PART
Continuing Taylor’s expansion (3.1) for any discretization scheme Sˆtx and any f ∈ C 6 (R), we have f (x) x 2 f (x) x 3 Ef Sˆhx = f (x) + f (x)E Sˆhx − x + E Sˆh − x + E Sˆh − x 2 6 x Xˆ h (5) (x) 5 1 f 1 4 5 + Ef (4) (x) Sˆhx − s + E Sˆhx − x + E f (6) (s) Sˆhx − s ds. 4! 5! 5! x
Since a2 x a2 x2 (4) f (x) + f (x), 2 4 by Eq. (2.2) we have that the second-order remainder is L20 f (x) =
R2h f (x)
h2 x 2 ˆ = Ef Sh − f (x) + L0 f (x)h + L0 (x) 2 f (x) x 3 3 1 2 = f (x)E Sˆhx − x + f (x) E Sˆhx − x − axh + E Sˆh − x − a2 xh2 2 6 2 4 f (5) (x) x 5 f (4) (x) ˆx + E Sh − s − 3a2 x2 h2 + E Sˆh − x + r2 (x, h), x ∈ D, h > 0, 4! 5!
where Sˆhx 1 5 r2 (x, h) = E f (6) (s) Sˆx − s ds 1 E max f (6) (s) Sˆx − x 6 . h h 6! 5! 0sSˆhx x
Therefore, Sˆ is a potential second-order approximation of the stochastic part (2.5) if the following conditions are satisfied for some h0 > 0, C5 > 0, and k ∈ N and for some C6 > 0 and l ∈ N depending only on a good Lith. Math. J., 51(3):385–401, 2011.
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V. Mackeviˇcius
∞ ( R) : sequence of f ∈ Cpol
E Sˆhx − x = 0, 2 E Sˆhx − x = xah, 3 3 E Sˆhx − x = a2 xh2 , 2 4 E Sˆhx − x = 3a2 x2 h2 , x E Sˆ − x 5 C5 h3 1 + xk , h 6 E max f (6) (s) Sˆhx − x C 6 h3 1 + x l , 0sSˆhx
x ∈ D, 0 < h < h0 ,
(4.1)
x ∈ D, 0 < h < h0 ,
(4.2)
x ∈ D, 0 < h < h0 ,
(4.3)
x ∈ D, 0 < h < h0 ,
(4.4)
x ∈ D, 0 < h < h0 ,
(4.5)
x ∈ D, 0 < h < h0 .
(4.6)
Now we proceed similarly to the first-order case. We find discrete random variables Sˆhx matching the first four moments of the solution to the stochastic part, that is, satisfying Eqs. (4.1)–(4.4), and then check that, happily, they satisfy conditions (4.5)–(4.6). Alfonsi [3, Lemma 3.5] gives a construction of a two-valued random variable matching three moments of any nonconstant random variable. We present here an analogous lemma giving a construction of a three-valued nonnegative random variable, with one zero value of the three, matching the first four moments of a given nonnegative random variable: Lemma 1. Let X be a nonnegative random variable nonconcentrated in {0, 1} with mi := EX i < ∞, i = 1, 2, 3, 4. (i) Let s :=
m1 m4 − m2 m3 m1 m3 − m22
and
q :=
m2 m4 − m23 . m1 m3 − m22
Then s > 0, q > 0, Δ := s2 − 4q > 0, and thus the polynomial P (x) := x2 − s x + q has two positive roots √ √ s− Δ s+ Δ x1 = and x2 = . 2 2 (ii) p1 :=
m 1 x 2 − m2 > 0, x1 (x2 − x1 )
p2 :=
m2 − m1 x1 > 0, x2 (x2 − x1 )
and
p0 := 1 − p1 − p2 > 0.
ˆ be a random variable taking values 0, x1 , and x2 with probabilities p0 , p1 , and p1 , respectively. (iii) Let X Then ˆ i = mi , i = 1, 2, 3, 4. EX
The proof of Lemma 1 is given in the Appendix. We now apply the lemma to X = Shx . In this case, mi = E(Shx )i , i = 1, 2, 3, 4, that is, m1 = x,
m2 = x2 + xah,
3 m3 = x3 + 3x2 ah + x(ah)2 , 2
m4 = x4 + 6x3 ah + 9x2 (ah)2 + 3x(ah)3 .
Thus, in view of Lemma 1, as approximations of Shx , we consider the three-valued random variables Sˆhx taking the values √ √ s− Δ s+ Δ x1 = x1 (x, h) = (4.7) , x2 = x2 (x, h) = , and 0 2 2
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Weak approximation of CIR equation
with probabilities p1 =
m1 x2 − m2 , x1 (x2 − x1 )
p2 =
m2 − m1 x1 , x2 (x2 − x1 )
and p0 = 1 − p1 − p2 ,
(4.8)
respectively, where 4x2 + 9xah + 3(ah)2 and 2x + ah ah(16x3 + 33x2 ah + 18xa2 h2 + 3a3 h3 ) Δ= . (2x + ah)2 s=
(4.9) (4.10)
Since the random variables Sˆhx match the first four moments with Shx , they also match the first four central moments that coincide with the right-hand sides of Eqs. (4.1)–(4.4): E Sˆhx − x = E Shx − x = 0, 3 3 3 E Sˆhx − x = E Shx − x = a2 xh2 , 2
2 2 E Sˆhx − x = E Shx − x = xah, 4 4 E Sˆhx − x = E Shx − x = 3a2 x2 h2 .
Unfortunately, the estimates in Eqs. (4.5)–(4.6) are not satisfied; we have only O(h2 ) instead of the required rate O(h3 ): 2 3 2 2 3 x E Sˆ − x 5 = − x(ah) (16x − 90x ah − 102x(ah) − 27(ah) ) h 4(2x + ah) C5 xh2 1 + x2 , x ∈ D, 0 < h < h0 , 6 x(ah)2 (32x5 − 36x4 ah + 588x3 (ah)2 + 864x2 (ah)3 + 405x(ah)4 + 63(ah)5 ) E Sˆhx − x = 4(2x + ah)2 2 3 C6 xh 1 + x , x ∈ D, 0 < h < h0 , and, in general,
p E Sˆhx − x Cp xh2 1 + xp−3 ,
x ∈ D, 0 < h < h0 .
(4.11)
(The expressions here and below were all obtained with the help of the MAPLE package.) However, below, we shall see that this approximation will be very useful when approximating the CIR equation near zero. Moreover, as one can expect, it gives rather good results for tests functions that are (or can be well approximated by) fourth-order polynomials. To get an approximation satisfying conditions (4.5)–(4.6), first note that we can use the formulas of Lemma 1 for random variables that are not nonnegative, provided that s and q are well defined (i.e., m1 m3 − m22 = 0) and that Δ, p0 , p1 , and p2 are all positive. Clearly, in such a case, x1 and x2 are not necessarily positive. We apply these formulas for centered random variables Y = Shx − x, thus taking the centered moments κi of Shx instead of mi , i = 1, 2, 3, 4. The corresponding approximating random variable Yˆ matching the first four moments with Y takes the values y1 < 0, y2 > 0, and 0 with probabilities p1 , p2 , and p0 , respectively, defined as follows1 : √ √ 3ah 21 s c − Δc s c + Δc 2 sc = y1 = , Δc = (ah) + 12xah, , y2 = , 2 4 2 2 1
The subscript “c” stands for centered.
Lith. Math. J., 51(3):385–401, 2011.
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V. Mackeviˇcius
p1 =
−κ2 2xah √ , =√ y1 (y2 − y1 ) Δc ( Δc − s c )
(4.12)
p2 =
κ2 2xah √ , =√ y2 (y2 − y1 ) Δc ( Δc + s c )
(4.13)
p0 = 1 − p1 − p2 =
8 3x
+ ah . 4x + ah
(4.14)
√ √ 2 = 21 ah > s , and the positivity (ah) The positivity of p1 easily follows from the inequality Δc > 21 c 4 2 of p0 and p2 is obvious. Thus, alternatively, as approximations of Shx , we now consider the three-valued random variables Sˆhx taking the values x 1 = x + y1 ,
x 2 = x + y2 ,
and x
with probabilities p1 , p2 , and p0 defined by Eqs. (4.12)–(4.14). By the construction, Sˆhx match the first four moments with Shx , and thus conditions (4.1)–(4.4) are satisfied. Moreover, 5 45 E Sˆhx − x = x(ah)3 9x + ah , 8 x 6 99 171 3 2 2 ˆ E Sh − x = x(ah) 9x + xah + (ah) , 4 16 and, in general, 2p E Sˆhx − x = x(ah)p Pˆp (x, ah), 2p+1 ˆ p (x, ah), = x(ah)p+1 Q E Sˆhx − x p ˆ p (x, ah), E Sˆhx = xR
p 3, p 2, p 1,
ˆp = Q ˆp = R ˆ p (x, y), and R ˆ p (x, y) are (p − 1)th-order homogeneous two-variable where Pˆp = Pˆp (x, y), Q ˆ p at the power xp−1 is 1. So, conditions polynomials with positive coefficients, and the coefficient of R (4.5)–(4.6) are satisfied (cf. Corollaries 1 and 2 in Section 3). Now, the only obstacle to use the new “centered” approximation Sˆhx is that it can take negative values, since x1 may be negative. This is the case if 3 x + ah < 4
that is,
21 (ah)2 + 12xah, 4
√ x 3 + 21 < = 1.896 . . . . ah 4
Thus, we may use Sˆhx only for x 2ah.2 However, for “small” values x < 2ah, we may recall the “noncentered” nonnegative approximation Sˆhx taking values x1 , x2 , and 0 with probabilities p1 , p2 , and p0 defined in 2
For simplicity, we round the constant 1.896. . . up to the nearest integer 2.
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Weak approximation of CIR equation
Eqs. (4.7)–(4.10). By estimate (4.11), in the region x ∈ [0, 2ah], for this approximation, we have x E Sˆ − x p 2Cp xh3 1 + xp−3 , h
p 5,
and, thus, in this region, conditions (4.5)–(4.6) are also satisfied. Summarizing the above, by Theorem 2 we now have the following: ˆ tx be defined by composition (2.7), where the three-valued random Theorem 4. Let the discretization scheme X x variables Sˆh take the values x1 , x2 , and x0 with probabilities p1 , p2 , and p0 = 1 − p1 − p2 defined as follows: • If x 2ah, then √ s− Δ x1 = x + , 2 2xah , p1 = √ √ Δ( Δ − s)
√ s+ Δ x2 = x + , 2 2xah p2 = √ √ , Δ( Δ + s)
x0 = x,
where s=
3ah , 2
Δ=
21 (ah)2 + 12xah. 4
• If 0 < x < 2ah, then √ √ s− Δ s+ Δ x1 = , x2 = , x0 = 0, 2 2 √ √ x(2ah − s − Δ) x(2ah − s + Δ) , p2 = √ √ , p1 = √ √ Δ( Δ − s) Δ( Δ + s)
(4.15) (4.16)
where s=
4x2 + 9xah + 3(ah)2 , 2x + ah
Δ=
ah(16x3 + 33x2 ah + 18xa2 h2 + 3a3 h3 ) . (2x + ah)2
(4.17)
ˆ x is a second-order discretization scheme for the CIR equation (1.1), i.e., for every f ∈ C ∞ (D), there Then X t 0 exist K > 0 and h0 ∈ (0, T ] such that
x x ˆ T Kh2 , Ef XT − Ef X 5
0 < h < h0 .
SIMULATION EXAMPLES
We illustrate the constructed approximations for the test function f (x) = e−x and three sets of parameters θ, k, σ, x0 , with σ = 1 (“small volatility”), 2.5 (“high volatility”), and 4 (“very high volatility”); moreover, x in the third one, the initial state x0 = 0.3 is chosen so that the Laplace transform F (λ, t) = EeλXt is not monotone in t. We prefer to show and compare the behavior of approximations “dynamically,” that is, by plotting the true and approximate expectations Ef (Xt ) as functions of time t for some moderately small h. The approximate expectations are obtained by averaging over 100,000 samples. For comparison, we also include the (full-truncated) Euler scheme (see [7]). The simulation results are shown in Figs. 1–3, where the following legend is used for graph lines: A1 denotes the first-order approximation (Theorem 3), A2c denotes the (“centered”) second-order approximation (Theorem 4), and A2 denotes the “uncentered” approximation defined by using Eqs. (4.15)–(4.17) for all x > 0, instead of “small” x ∈ (0, 2ah). Lith. Math. J., 51(3):385–401, 2011.
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V. Mackeviˇcius
0.5
↑Ef (Xt ) .
. . . . . . . . . . . . . . . . .
. . . . . ........ ........... ........... ......... ...... ....... ........... .......... ......... ........... .......... ........... ........... .......... ......... ........... ........... ......... ............ .......... .......... ......................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... . . . . ............... . . . . ..... . ...... ...... . .... ...
............................................................ ............................. ...... . . .. 0.4 ... .. .................... exact . . . . . . .......... ........................ ........................................
0.3
1
2
Euler A1 A2 A2c
3
4
→ t
↑Ef (Xt )
0.5
.
.
. . . . . . . . . . . . . . . .
............................................................ ............................. ...... . . .. 0.4 ... .. .................... . ................. ................ ............... ................ ................. .......... ................ ............... ............... .............. ............... ................ . .............. ....... ............ ........................... . . . . ............... ... . .......... .. ....... . ..... . . . ... ... exact . . . . . Euler .......... A 1 ........................ A 2 ........................................ A
2c
0.3
1
2
3
4
→ t
√ Figure 1. Approximation of CIR equation dXt = (θ − kXt ) dt + σ Xt dBt with x0 = 1, θ = k = 1, σ = 1 (small volatility); h = 0.5 (left) and h = 0.25 (right).
↑Ef (Xt ) .
...................................................................................... ...... 0.6 .... .. ... .. 0.5 .. .................... .. .... .. .. . 0.4 . ... . ........... ........... .......... ........... ......... ........... .......... ......... ........... ........... ......... .......... ........... ......... ...... ............ ....... ........... ......... ........... ....... . ................ ........... . ..... ..... . . . . .... . ... . .... ..................... ......... . .... ... . ..... .. . . . ... .. . . .... .. exact . ..... .. . . . . . . ... .. Euler . .... .. .......... A . . . 1 . .. . ........................ A . ....... 2 ....... ........................................ A . . 2c . .... . . ..... ...
1
2
3
4
→ t
↑Ef (Xt ) .
...................................................................................... ...... .... .. ... .. 0.5 .. .................... .. .... .. .. . 0.4 . ... 0.6
.............. ............... .............. ............... .............. ............... ........ ............... .............. ............... ............... .............. .............. ..... ............ . ........................ ........ . ..... . .... . . . . . . . . . . . . . . . . . . . . . . . . . .. ..... . ... ... . . .... . . . . . ... . . ... .. . ..... .. . .... .. . ..... .. . exact . ..... . . . . . . . ...... Euler ...... .......... A . . . 1 .... . ........................ A ...... 2 . ....... ........................................ A . . 2c . .... . .... ...
1
2
3
4
→ t
√ Figure 2. Approximation of CIR equation dXt = (θ − kXt ) dt + σ Xt dBt with x0 = 1, θ = k = 1, σ = 2.5 (high volatility); h = 0.5 (left) and h = 0.25 (right).
. (Xt ) ↑Ef .
. . ..... ........ ......... ........... ........... . ......................... ............ ......... .......... .......... .......... ........... .. ........... ........... .......... .......... ......... . ........... .......... ............ ........... ......... ..... ........ ........... ....... . ..... . . . . . . . . . . . . ...... . . . . ... .. ....... . . ........ ...... .. exact .... . . . .
........................................................... 0.8 ........ ........................... ... .. ................ .
. (Xt ) ↑Ef .
0.8
Euler
. ... .... A
1 2 .............................. A 2c
0.7 0.0
0.4
0.6
......... .......... ........................................................................ ...... . ................ .. Euler
. ... .... A
1 2 2c
0.7
.................. A
0.2
. . ..... ........ ......... ........... ........... . ......................... ............ ......... .......... .......... .......... ........... .. ........... ........... .......... .......... ......... . ........... .......... ............ .......... .......... ..... ........ ........... ....... . ..... . . . . . . . . . . . . ...... . . . . ... .. ....... . . ........ ...... .. exact .... . . . .
0.8
→ t
0.0
.................. A .............................. A
0.2
0.4
0.6
0.8
→ t
√ Figure 3. Approximation of CIR equation dXt = (θ − kXt ) dt + σ Xt dBt with x0 = 0.3, θ = k = 1, σ = 4 (very high volatility); h = 0.1 (left) and h = 0.05 (right).
Acknowledgment. The author would like to thank the anonymous referee for carefully reading the manuscript, pointing out a significant number of misprints, and giving several useful suggestions.
397
Weak approximation of CIR equation
APPENDIX Proof of Proposition 1. We use induction in p. The statement clearly holds for p = 1. Suppose that it holds for some p ∈ N. Then, for even powers of Stx − x, we have 1 κ2p+2 (t, x) = (2p + 2)(2p + 1)a 2
t = (p + 1)(2p + 1)a
t
κ2p+1 (s, x) + xκ2p (s, x) ds
0
x(as)p+1 Rp−1 (x, as) + x2 (as)p Qp−1 (x, as) ds
0
= (p + 1)(2p + 1)xa
t
p+1
asp+1 Rp−1 (x, as) + xsp Qp−1 (x, as) ds
0
= (p + 1)(2p + 1)xa
t
p+1
as 0
p−1
t
= (p + 1)(2p + 1)xap+1
= (p + 1)(2p + 1)x(at)
p−1
i
ri x (as)
p−1−i
p−1
p−1
i
qi x (as)
p−1−i
i=0
ri xi ap−i s2p−i +
p−1
rˆi xi (at)p−i +
i=0
p−1
ds
qi xi+1 ap−1−i s2p−1−i ds
i=0
rˆi xi ap−i t2p+1−i +
i=0 p−1 p+1
+ xs
p
i=0
i=0
0
= (p + 1)(2p + 1)xap+1
p+1
qˆi xi+1 ap−1−i t2p−i
i=0 p−1
qˆi xi+1 (at)p−1−i
i=0
= x(at)p+1 Qp (x, at).
Similarly, for odd powers of Stx − x, using the already checked relation for κ2p+2 , we have 1 κ2p+3 (t, x) = (2p + 3)(2p + 2)a 2
t = (p + 1)(2p + 3)a
t
κ2p+2 (s, x) + xκ2p+1 (s, x) ds
0
x(as)p+1 Qp (x, as) + x2 (as)p+1 Rp−1 (x, as) ds
0
= (p + 1)(2p + 3)xa
p+2
t
sp+1 Qp (x, as) + xsp+1 Rp−1 (x, as) ds
0
= (p + 1)(2p + 3)xap+2
t
sp+1
0
= (p + 1)(2p + 3)xa
p+2
t p 0
Lith. Math. J., 51(3):385–401, 2011.
i=0
p
qi xi (as)p−i + xsp+1
i=0
ri xi (as)p−1−i ds
i=0
i p−i 2p+1−i
qi x a
p−1
s
+
p−1 i=0
ri x
i+1 p−1−i 2p−i
a
s
ds
398
V. Mackeviˇcius
= (p + 1)(2p + 3)xap+2
p
qˆi xi ap−i t2p+2−i +
i=0
= (p + 1)(2p + 3)x(at)p+2
p
qˆi xi (at)p−i +
i=0
= x(at)p+2 Rp (x, at).
p−1
rˆi xi+1 ap−1−i t2p+1−i
i=0 p−1
rˆi xi+1 (at)p−1−i
i=0
Proof of Propositions 2 and 3. Denoting for short y = ah and using the equalities x1 x2 = x(x + y)
and
(x1 − x)(x2 − x) = −xy,
for even-order moments of Sˆhx − x, we have 2p E Sˆhx − x
x x = (x1 − x)2p p1 + (x2 − x)2p p2 = (x1 − x)2p + (x2 − x)2p 2x1 2x2 x = (x1 − x)2p x2 + (x2 − x)2p x1 2x1 x2 1 = (x1 − x)2p (x2 − x) + (x2 − x)2p (x1 − x) + x (x1 − x)2p + (x2 − x)2p 2(x + y) −xy x = (x1 − x)2p−1 + (x2 − x)2p−1 + (x1 − x)2p + (x2 − x)2p 2(x + y) 2(x + y) 2p−1 −xy 2p−1 = y − (x + y)y + y + (x + y)y 2(x + y) 2p 2p x + y − (x + y)y + y + (x + y)y 2(x + y) p−1 i −xy 2p−1 2p − 1 2p−1−2i = y + y (x + y)y 2i x+y i=1 p x 2p i + y 2p + y 2p−2i (x + y)y 2i x+y i=1 p−1 2p 2p − 1 =x − y 2p−i (x + y)i−1 + (x + y)p−1 y p 2i 2i i=1 p−1 2p 2p − 1 p p−i i−1 p−1 = xy − y (x + y) + (x + y) 2i 2i i=1
p
= xy P¯p (x, y). 2p−1 Since 2p > 0 for all i = 1, . . . , p, we have that all the coefficients of P¯p are positive integers. 2i − 2i Moreover, the coefficient of P¯p at xp−1 equals 1. Similarly, for odd-order moments of Sˆhx − x, we have
2p+1 E Sˆhx − x = (x1 − x)2p+1 p1 + (x2 − x)2p+1 p2 = (x1 − x)2p+1
x x + (x2 − x)2p+1 2x1 2x2
399
Weak approximation of CIR equation
x (x1 − x)2p+1 x2 + (x2 − x)2p+1 x1 2x1 x2 1 = (x1 − x)2p+1 (x2 − x) + (x2 − x)2p+1 (x1 − x) + x (x1 − x)2p+1 + (x2 − x)2p+1 2(x + y) −xy x = (x1 − x)2p + (x2 − x)2p + (x1 − x)2p+1 + (x2 − x)2p+1 2(x + y) 2(x + y) 2p 2p −xy = y − (x + y)y + y + (x + y)y 2(x + y) 2p+1 2p+1 x + y − (x + y)y + y + (x + y)y 2(x + y) p p i −xy 2p 2p 2p−2i x 2p + 1 i = y + y (x + y)y + y 2p+1 + y 2p+1−2i (x + y)y 2i 2i x+y x+y i=1 i=1 p p 2p + 1 2p 2p + 1 2p 2p+1−i i−1 p+1 =x − y (x + y) = xy − y p−i (x + y)i−1 2i 2i 2i 2i =
i=1 p+1
= xy
i=1
¯ p (x, y), Q
¯ p at xp−1 is 2p+1 − 2p = 2p + 1 − 1 = 2p. where the coefficient of Q 2p 2p As for noncentral moments (Proposition 3), things are even simpler: p x p−1 x x E Sˆhx = xp1 p1 + xp2 p2 = xp1 + xp2 = x1 + xp−1 2 2x1 2x2 2 p−1 p−1 x = x + y − (x + y)y + x + y + (x + y)y 2 [(p−1)/2] [(p−1)/2] p−1 p − 1 i =x (x + y)p−1−2i (x + y)y =x (x + y)p−1−i y i 2i 2i i=0 i=0 p(p − 1) p−2 ¯ p−1 (x, y). = x xp−1 + x y + · · · = xR 2 Proof of Lemma 1. We look for a random variable taking three values 0 < x1 < x2 with probabilities p0 , p1 , and p2 such that xi1 p1 + xi2 p2 = mi , i = 1, 2, 3, 4. Solving the first two equations (i = 1, 2) with respect to p1 and p2 , we get p1 =
m1 x2 − m2 , x1 (x2 − x1 )
p2 =
m2 − m1 x1 , x2 (x2 − x1 )
p2 =
m4 − m3 x1 . x32 (x2 − x1 )
while solving the other two (i = 3, 4), we get p1 =
m3 x2 − m4 , x31 (x2 − x1 )
Equating the expressions obtained for p1 and p2 , we get the system
Lith. Math. J., 51(3):385–401, 2011.
(m1 x2 − m2 )x21 = m3 x2 − m4 ,
(A.1)
(m2 −
(A.2)
m1 x1 )x22
= m 4 − m3 x 1 .
400
V. Mackeviˇcius
Adding these equations, we have m1 x1 x2 (x1 − x2 ) + m2 x22 − x21 = m3 (x2 − x1 ), and thus m2 (x1 + x2 ) − m1 x1 x2 = m3 .
(∗)
On the other hand, multiplying the equations respectively by x2 and x1 and then adding, we get m2 x1 x2 (x2 − x1 ) = m3 x22 − x21 − m4 (x2 − x1 ), whence m3 (x1 + x2 ) − m2 x1 x2 = m4 .
(∗∗)
Solving the system (∗)–(∗∗) with respect to x1 + x2 and x1 x2 , we get x1 + x2 = s =
m1 m4 − m2 m3 , m1 m3 − m22
x1 x 2 = q =
m2 m4 − m23 . m1 m3 − m22
This means that x1 and x2 are the roots of the quadratic polynomial P (x) = x2 − sx + q . Thus, formally, we have obtained all the expressions stated for x1 , x2 , p1 , and p2 . To finish the proof, we need to check the inequalities s > 0, q > 0, Δ > 0, p0 > 0, p1 > 0, and p1 + p2 < 1. First, the positivity of s and q follows from the elementary moment inequalities for a nonnegative r.v. X m22 m1 m3 ,
m23 m2 m4 ,
m2 m3 m1 m4
(A.3)
that are all strict, unless X is concentrated in {0, 1}. We further have
m2 P m1
=
m2 m1
2
m2 −s m1
+q =
m22 − sm1 m2 + qm21 m22 − m1 m3 = < 0. m21 m21
Therefore, the polynomial P does have two real roots x1 < x2 . Since x1 + x2 = s > 0 and x1 x2 = q > 0, the roots are both positive; in particular, Δ = s2 − 4q > 0. Moreover, we have that x1 < m2 /m1 < x2 , and thus p1 =
m1 (x2 − m2 /m1 ) > 0, x1 (x2 − x1 )
p2 =
m1 (m2 /m1 − x1 ) > 0. x2 (x2 − x1 )
Now consider p0 = 1 − (p1 + p2 ): m1 x2 − m2 m 2 − m1 x 1 m1 (x22 − x21 ) − m2 (x2 − x1 ) − =1− x1 (x2 − x1 ) x2 (x2 − x1 ) x1 x2 (x2 − x1 ) m1 s − m2 m1 (x1 + x2 ) − m2 =1− =1− x1 x2 q 2 3 m m4 − 2m1 m2 m3 + m2 2m1 m2 m3 + m2 m4 − m21 m4 − m23 − m32 =1− 1 = . m2 m4 − m23 m2 m4 − m23
p0 = 1 −
Weak approximation of CIR equation
401
To show that the numerator of the last fraction is positive, consider three independent identically distributed random variables X1 , X2 , and X3 , each with the distribution of X . Then we directly check that 0 < E(X1 − X2 )2 (X2 − X3 )2 (X3 − X2 )2 = E 2 X1 X22 X33 + X1 X32 X23 + four similar terms + X14 X22 + X24 X14 + four similar terms − 2 X1 X2 X34 + X1 X3 X24 + X2 X3 X24 − 2 X13 X23 + X23 X33 + X33 X13 − 6X12 X22 X32 = 6 2m1 m2 m3 + m2 m4 − m21 m4 − m23 − m32 ,
and thus, indeed, p0 > 0. REFERENCES 1. A. Alfonsi, On the discretization schemes for the CIR (and Bessel squared) processes, Monte Carlo Methods Appl., 11(4):355–384, 2005. 2. A. Alfonsi, A second-order discretization scheme for the CIR process: Application to the Heston model, 2008, available from: http://hal.archives-ouvertes.fr/docs/00/25/83/50/PDF/2nd_ order_CIRHeston.pdf. 3. A. Alfonsi, High order discretization schemes for the CIR process: Application to affine term structure and Heston models, Math. Comput., 79(269):209–237, 2010. 4. J.C. Cox, J.E. Ingersoll, and S.A. Ross, A theory of the term structure of interest rates, Econometrica, 53:385–407, 1985. 5. W. Halley, S.J.A. Malham, and A. Wiese, Positive and implicit stochastic volatility simulation, 2009, available from: http://arxiv.org/PS_cache/arxiv/pdf/0802/0802.4411v2.pdf. 6. D. Lamberton and B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance, Chapman & Hall, London, 1996. 7. R. Lord, R. Koekkoek, and D. van Dijk, A comparison of biased simulation schemes for stochastic volatility models, Discussion paper 06-046/4, Tinbergen Institute, 2008. 8. V. Mackeviˇcius, On positive approximations of positive diffusions, Liet. Mat. Rink., 47(spec. issue):58–62, 2007. 9. V. Mackeviˇcius, On weak approximations of (a, b)-invariant diffusions, Math. Comput. Simul., 74(1):20–28, 2007. 10. V. Mackeviˇcius, On approximation of CIR equation with high volatility, Math. Comput. Simul., 80(5):959–970, 2010. 11. S. Ninomiya and N. Victoir, Weak approximation of stochastic differential equations and application to derivative pricing, Appl. Math. Finance, 15(2):107–121, 2008.
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