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Journal of Computational Finance manuscript No.(will be inserted by the editor)
Positive stochastic volatility simulation
William Halley · Simon J.A. Malham
· Anke Wiese
Received: 29th February 2008
Abstract We present a positivity preserving numerical scheme for the pathwise solu-
tion of nonlinear stochastic differential equations driven by a multi-dimensional Wiener
process and governed by non-commutative linear and non-Lipschitz vector fields. This
strong order one scheme uses: (i) Strang exponential splitting, an approximation that
decomposes the stochastic flow separately into the drift flow, and the pure diffusion
flow governed by the diffusion vector fields; (ii) an implicit Euler method to approx-
imate the drift flow; and (iii) an implicit Milstein method to approximate the pure
diffusion flow. The separate approximations for the drift and pure diffusion flows pre-
serve positivity. Therefore the Strang exponential splitting approximation does also.
We demonstrate the efficacy of our method by applying it to the Heston model and
a variance curve model, and compare it against well-established positivity preserving
schemes.
Keywords stochastic volatility · positivity preservation · implicit Milstein method ·exponential splitting
Mathematics Subject Classification (2000) 60H10 · 60H35 · 93E20
William Halley (Current address)Scottish Widows Investment PartnershipEdinburgh One, 60 Morrison StreetEdinburgh EH3 8BE, UKTel.: +44-131-6558500Fax: +44-131-6620293E-mail: W [email protected]
William Halley · Simon J.A. Malham · Anke WieseMaxwell Institute for Mathematical Sciencesand School of Mathematical and Computer SciencesHeriot-Watt University, Edinburgh EH14 4AS, UKTel.: +44-131-4513200Fax: +44-131-4513249E-mail: [email protected]: [email protected]
2 Halley, Malham and Wiese
1 Introduction
We present a numerical scheme for the strong approximation of stochastic differential
equations that preserves positivity. Our method applies to linear and non-Lipschitz
governing vectors fields characteristic of stochastic volatility models in finance. Broadly,
such models have the following features. The asset price is modelled as a standard Ito
process with linear drift and a diffusion coefficient that depends on the asset price and
its volatility with the dependency linear or sublinear (non-Lipschitz) in the variables.
The volatility is modelled as a mean-reverting Ito process, pulled towards the long
term value θ with the speed of convergence scaled by the parameter α. The diffusion
coefficient is typically of square-root (p = 12 ) or sublinear ( 1
2 < p < 1) functional form
and scaled by the parameter β. Hence when thought of as a system, the drift vector
field is typically linear and the diffusion vector fields typically have fractional powers.
The non-Lipschitz form of the diffusion vector field means that standard existence
and uniqueness results for the exact model cannot be applied. However Yamada and
Watanabe [66] proved existence and uniqueness for scalar Ito diffusions with fractional
powers p ≥ 12 (see Karatzas and Shreve [41, p. 291], and also Andersen and Piter-
barg [5]). We assume throughout existence and uniqueness for the class of vector fields
we consider, in particular this is true for our applications, the Heston and variance
curve models. Another typical feature of stochastic volatility models is that from a
modelling perspective, the volatility should be non-negative, and the asset price should
be also. The exact solutions of the stochastic differential systems representing these
models should preferably exhibit these properties for each component. For example for
the Heston model, the volatility is strictly positive for αθ ≥ 12β
2 and non-negative for
0 < αθ < 12β
2. In the latter case, the zero boundary is attainable but instantaneously
reflecting. Further the asset price process is a pure exponential process and hence
globally strictly positive. These results follow from the Feller boundary classification
criteria, see Feller [25], Karlin and Taylor [43, Chapter 15], Andersen and Piterbarg [5]
and Revuz and Yor [60, p. 412]. Non-Lipschitz diffusion vector fields create another
difficulty when we attempt to simulate. A strong Euler–Maruyama approximation step
applied to the square-root (p = 12 ) mean-reverting Ito process, will generate a negative
next-step value with positive probability. Therefore not only does the numerical solu-
tion lose a fundamental property of the model, integration must stop as the square-root
diffusion coefficient becomes complex.
We propose a strong order one method based on Strang exponential splitting (see
Strang [62], Hairer, Lubich and Wanner [33, Chapter II]). Explicitly, across the inter-
val [tn, tn+1] with h = tn+1 − tn, in the case of a stochastic system governed by a
Stratonovich drift V0, and two diffusion vector fields V1 and V2, we approximate the
exact flow-map by
exp`1
2hV0´
◦ exp`
∆W 1(tn)V1 +∆W 2(tn)V2 + A12(tn)[V1, V2]´
◦ exp`1
2hV0´
.
Here A12(tn) represents a sufficiently accurate approximation of the Levy chordal area
between the two driving Wiener processes W 1 and W 2 across the interval [tn, tn+1],
and ∆W 1(tn) and ∆W 2(tn) are the corresponding Wiener increments. Strang splitting
provides a convenient framework to decompose the stochastic flow naturally into the
drift and pure diffusion flows. This splitting itself generates a strong order one error.
The central flow indicated represents an approximation to the pure diffusion flow of
strong order one. This approach, not including the Lie bracket term, was used to gen-
erate strong order one-half approximations, to the Cox, Ingersoll and Ross model [23]
Positive stochastic volatility simulation 3
by Misawa [55], and to the Heston model by Ninomiya and Victoir [58] and Alfonsi [3].
In both cases, the separate flows indicated in the Strang decomposition can be com-
puted exactly. This not only results in an extremely efficient approximation method,
but each separate flow is positivity preserving and therefore the method overall is
positivity preserving.
However our concern here is for the case when the separate flows in the Strang
splitting cannot be computed exactly. We approximate the central pure diffusion flow
in the Strang decomposition by an implicit Milstein approximation. Implicitness is
strategically chosen in the evaluation of the Lie bracket vector field [V1, V2] and in parts
of the product vector fields V 21 and V 2
2 that appear in the Milstein approximation. The
vector fields V1 and V2 and other parts of the product vector fields V 21 and V 2
2 are
evaluated explicitly; they combine to produce positive terms—this borrows ideas from
Kahl, Gunter and Roßberg [38]. We prove that the resulting semi-implicit Milstein
step for the pure diffusion flow is positivity preserving and has finite moments of all
order, as does the underlying system that is approximated. The non-Lipschitz property
of the diffusion vector fields is crucial to establishing these two properties. Further
since we introduce implicitness in the higher order terms involving the Levy area and
products of the Wiener increments, our method converges to the solution with strong
order one error without the need to modify the drift vector field (see Kloeden and
Platen [44], Burrage, Burrage and Tian [16] and Alfonsi [2]). For the Stratonovich drift
flow we use a semi-implicit Euler step: we evaluate the non-constant negative terms
in the Stratonovich drift vector field implicitly. We prove this guarantees positivity of
the drift flow under the Stratonovich drift positivity condition (defined below). The
implicit Milstein step for the pure diffusion flow and implicit Euler method for the drift
flow each generate strong order one contributions to the global mean square error when
used in the Strang decomposition. Hence the Strang exponential splitting method with
these flow approximations represents a positivity preserving strong order one method.
Many simulation methods have been proposed to preserve positivity in stochastic
volatility models. These methods roughly fall into the following categories:
– Euler projections: strong approximations based on the Euler–Maruyama method
that use projection in the drift and/or diffusion vector fields to ensure arguments
remain non-negative;
– Balanced implicit techniques: strong balanced, implicit and Milstein methods, which
include implicitness via additional control terms;
– Drift implicit methods: strong Milstein methods adapted with implicitness in the
drift vector field;
– Exponential splitting : strong methods constructed by composing flows of individual
vector fields (as above);
– Exact simulation: sampling from the known distribution of the volatility.
Of the Euler projections, Deelstra and Delbaen [24] provided one of the first. They
used a cut-off for the argument in the diffusion coefficient, making the argument zero
whenever it became negative. They proved the convergence of their adapted Euler
scheme to the model. Subsequently Bossy and Diop [10] suggested taking the absolute
value of the proposed Euler increment at each step (also see Berkaoui, Bossy and
Diop [9]). Further, for the mean-reverting square-root process, Higham and Mao [34]
proposed taking the absolute value of the argument in the diffusion coefficient. These
methods and their properties are summarized in Lord, Koekkoek and Van Dijk [48,
Table 1] who proposed the full truncation method. In this method a cut-off is used for
4 Halley, Malham and Wiese
the argument in both the diffusion and drift coefficients. Of these methods only that of
Bossy and Diop preserves non-negativity. However they all apply in the full parameter
range for the mean-reverting process, i.e. for all p ∈ [12 , 1) with no further restrictions.
Balanced implicit methods were introduced by Milstein, Platen and Schurz [54].
Kahl and Schurz [41] have also introduced the balanced Milstein method (also see Kahl
and Jackel [39,40]). These methods require the determination of control functions that
are model dependent (also see Schurz [61] for more details).
Kahl, Gunter and Roßberg [38] used the drift implicit Milstein method to preserve
positivity (also see Higham, Mao and Stuart [35] for similar implicit ideas involving
splitting also). By choosing negative non-constant terms in the drift vector field to
be implicit, their method naturally preserves positivity for the general mean-reverting
process provided p > 12 , or when p = 1
2 provided αθ ≥ 14β
2. Indeed these conditions
are required for positivity preservation for all the balanced, drift-implicit and splitting
methods. They result from ensuring that in the Stratonovich drift vector field, com-
ponentwise, constant terms or terms not involving that component variable explicitly,
are non-negative. They ensure the Stratonovich drift vector field is positivity preserv-
ing, hence we shall call them the Stratonovich drift positivity conditions. Anderson and
Piterbarg [5] and Lord, Koekkoek and Van Dijk [48] point out that for FX markets in
particular, fitted values for the parameters typically satisfy αθ < 14β
2. This parame-
ter set is thus not covered by the methods just described, as well as the method we
propose.
Exact simulation methods use that the distribution (transition density) of the
volatility in the Heston model is a non-central chi-squared distribution (see Cox, In-
gersoll and Ross [23] and Glassermann [28, Section 3.4]). The exact exponential form
of the asset price process is used to update its value using the volatility sample. It
has successfully been applied by Broadie and Kaya [11], and more recently by Ander-
son [4]. Moro and Schurz [56] have combined this technique with exponential splitting.
All of these exact simulation methods, like the Euler projection methods, avoid the
Stratonovich drift positivity condition.
In this paper the focus is on strong direct discretization methods. Taking a leaf
from Deelstra and Delbaen [24], with more general models and contexts in mind, some
applications require the sample paths themselves and for example, path-dependent
options require more than weak convergence (see Deelstra and Delbaen [24, p. 78]).
Indeed, as Higham and Mao [34] point out, direct discretization methods are: widely
used in practice; typically more efficient for path-dependent options; and more easily
adaptable from model to model (the analytic transition density may not be available
in some cases).
Strong order one numerical methods with non-commuting diffusion vector fields
require suitably accurate approximations of the Levy area. The overall computational
effort associated with these approximations is proportional to h−2, see Clark and
Cameron [22], Kloeden and Platen [44, p. 367], Gaines and Lyons [26] and Lord,
Malham and Wiese [47] for more details. Wiktorsson [65] has recently derived a new
approximation method with an improved scaling which has been implemented in prac-
tice in strong algorithms by Gilsing and Shardlow [27]. Lord, Malham and Wiese [47]
have shown that inspite of the additional computational effort associated with order
one methods, they are competitive compared to order one-half methods, delivering su-
perior accuracy for the same overall computational cost. We demonstrate this in our
applications, simulating the Heston model [31] and a variance curve model within the
class specified by Buhler (see Buhler [14, p. 197] and [13, Chapter 6], and also Overhaus
Positive stochastic volatility simulation 5
et al. [59]). Indeed, in summary, the advantages of the Strang exponential splitting
method with the implicit Milstein step, in practice, are:
– Strong order one method;
– Superior accuracy for given computational effort;
– Positivity preserving under the Stratonovich drift positivity condition;
– Universal, simple and easy to implement.
Our paper is organised as follows. In Section 2 we review the exponential Lie series.
We define positivity preserving flow-maps in Section 3 and present classical conditions
for individual vector fields to have positivity preserving flow-maps. We also briefly
review the conditions for scalar Ito processes to be strictly positive or non-negative. In
Section 4 we derive the Strang exponential splitting method. Subsequently, we prove in
Section 5, that our proposed implicit Milstein algorithm for the pure diffusion flow, and
implicit Euler algorithm for the drift flow, are both positivity preserving. We review
the Heston and variance curve models in Section 6. We summarize all the comparison
positivity preserving methods we implement in Section 7 and apply them to both
models in Section 8. Finally in Section 9 we present some concluding remarks.
2 Lie series
We are interested in nonlinear Stratonovich stochastic differential equations of the form
yt = y0 +dX
i=0
Z t
0Vi(yτ ) dW i
τ . (1)
Here W 1t , . . . ,W
dt are d independent scalar Wiener processes and W 0
t ≡ t and y ∈ RN .
We suppose that the vector fields Vi, i = 0, 1, . . . , d, are smooth and when expressed
in local coordinates are Vi =PN
j=1 Vji ∂yj .
The flow-map ϕt : RN → R
N of the stochastic differential equation (1) is defined
as the map taking the initial data y0 to the solution yt at time t, i.e. yt = ϕt ◦ y0. An
explicit expression for the flow-map ϕt associated with the solution of the stochastic
differential equation (1) is given by the stochastic Taylor expansion
ϕt =∞X
m=0
X
α∈Pm
Jα1···αm(t)Vα1 · · ·Vαm .
Here Pm is the set of all combinations of multi-indices α = (α1, . . . , αm) of length
m with αi ∈ {0, 1, . . . , d} and we have adopted the standard notation for multiple
Stratonovich integrals
Jα1···αm (t) =
Z t
0· · ·Z τm−1
0dWα1
τm· · · dWαm
τ1.
The logarithm of of the flow-map ϕt is called the exponential Lie series or Lie
series for short. Hence we can express the flow-map in the form
ϕt = expψt ,
6 Halley, Malham and Wiese
with the Lie series given by
ψt =
dX
i=0
Ji(t)Vi +
dX
j>i=1
Aij(t)[Vi, Vj ] + · · · ,
where Aij(t) ≡ 12
`
Jij(t)−Jji(t)´
are the Levy chordal areas. Here [· , ·] is the Lie bracket
on the Lie algebra of vector fields on RN . See Yamato [67], Kunita [45], Azencott [6],
Sussmann [64], Ben Arous [8], Castell [18] and Baudoin [7] for the derivation and
convergence of the stochastic Lie series.
Recall that when the driving process has dimension d ≥ 2, the Universal Limit
Theorem tells us that the Ito map (W 1, . . . ,W d) 7→ y is continuous in the p-variation
topology, in particular for 2 ≤ p < 3 (see Lyons [49], Lyons and Qian [50] and Malli-
avin [52]). A Wiener path with d ≥ 2 has finite p-variation for p > 2. This means
that from a pathwise perspective, approximations to y constructed using successively
refined approximations to (W 1, . . . ,W d) are only guaranteed to converge to the correct
solution y, if we include information about the Levy chordal areas of the driving path
process. Note however that the L2-norm of the 2-variation of a Wiener process is finite.
To guarantee pathwise convergence our approximation must thus include the Levy area
and thus also the Lie bracket of the diffusion vector fields [Vi, Vj ] for all i, j = 1, . . . , d.
With this in mind we consider strong order one numerical schemes.
3 Non-negativity and strict positivity
Given a d-dimensional driving Wiener process, for what class of governing drift and as-
sociated diffusion vector fields can we prove that the exact flow is positivity preserving?
By this we mean that the exact flow maps non-negative components to non-negative
components, and strictly positive components to strictly positive components, at any
later time for which the flow-map exists. We now state sufficient conditions for an
individual vector field V to be positivity preserving.
Lemma 1 (Sufficient conditions for vector fields) A sufficient condition for the
flow associated with the vector field V to be positivity preserving is that for each j ∈{1, . . . , N}, there exists a Lebesgue integrable function g on [0, T ] such that as yj → 0+
we have the asymptotic equivalence condition
V j ◦ y ∼ g(t) yαj
j , (2)
for some αj ∈ R such that either, αj < 1 and 11−αj
is an even integer, in which
case the corresponding components yj will be non-negative, or αj = 1, for which the
components yj will be strictly positive.
Further if for any j ∈ {1, . . . , N}, we have that for yj sufficiently small and positive
there exists a Lebesgue integrable function g on [0, T ] such that
V j ◦ y ≥ g(t) yαj
j , (3)
for any αj ∈ R withR t0 g(τ ) dτ ≥ 0 for all t ∈ [0, T ], then the corresponding components
yj will be strictly positive.
Positive stochastic volatility simulation 7
Proof Consider the scalar ordinary differential equation u′ = g(t)uα with u(0) = u0
where α ∈ R and g ∈ L1`[0, T ]´
. The solution at any time t ∈ [0, T ] is given by
u(t) =
8
<
:
“
u1−α0 +
R t0 g(τ ) dτ
”1
1−α, α 6= 1 ,
exp“
R t0 g(τ ) dτ
”
u0 , α = 1 .
The positivity preserving conclusions for the flow-map when condition (2) or when
condition (3) holds, are now immediate. ⊓⊔
Remark 1 If αj > 1 and initial data for that component is positive, then the solution
will blow-up in finite time.
Corollary 1 For the system of Stratonovich stochastic differential equations (1), sup-
pose that the Stratonovich drift vector field V0 satisfies either condition (2) or condi-
tion (3) and the diffusion vector fields, together with all the Lie bracket pairs of the
diffusion vector fields, satisfy the condition (2). Then the flow-map associated with the
Stratonovich stochastic differential system (1) is non-negativity preserving.
Proof For a given driving path, we construct an approximation to the exact solution
on the global interval [0, T ] as follows. On successive subintervals [tn, tn+1] of fixed
length h, approximate the solution by the Strang splitting (4) using the exact drift and
Euler–Levy flows (see Section 4). Under the conditions stated, each flow is positivity
preserving and their composition on the global interval is as well. In the limit h → 0,
the exact solution is recovered as well as non-negativity preservation. ⊓⊔
We now state the standard result for positivity of a scalar Ito diffusion process
based on the Feller classification of boundary points—we refer the reader to Ikeda and
Watanabe [36], Karlin and Taylor [43, Chapter 15], Kahl [37, Chapter 1] and Albanese
and Kuznetzov [1, p. 3] for more details.
Lemma 2 (Sufficient conditions for scalar Ito diffusion processes) For the
scalar Ito diffusion process given by the stochastic differential equation
dyt = V0(yt) dt+ V1(yt) dWt ,
define the scale function s and speed density m by
s(y) = exp
„
−Z y
y0
2V0(ξ)
V 21 (ξ)
dξ
«
and m(y) =1
s(y)V 21 (y)
.
The boundary 0 is attracting if for any y > 0,R y0 s(ξ) dξ < ∞, and attainable if for
any y > 0,R y0
R yξ m(η) dη s(ξ) dξ < ∞. In particular the boundary 0 is non-attracting
and unattainable if these respective integrals are not finite.
4 Strang exponential splitting
Let ψtn,tn+1denote the Lie series corresponding to the flow-map generated across
the interval [tn, tn+1] where h = tn+1 − tn. A natural decomposition or split of the
8 Halley, Malham and Wiese
governing vector fields is between the drift and the diffusion vector fields. Let ψdifftn,tn+1
denote the modified Lie series defined through the flow decomposition
expψtn,tn+1= exp
`12hV0
´
◦ exp`
ψdifftn,tn+1
´
◦ exp`1
2hV0´
,
where V0 denotes the drift vector field. The precise form of ψdifftn,tn+1
can be computed
using the Baker–Campbell–Hausdorff formula and involves the drift vector field V0.
However a natural approximation is to take ψdifftn,tn+1
to be the Lie series generated
solely by the set of governing diffusion vector fields. By the Baker–Campbell–Hausdorff
formula this is, in general, an order one approximation (the order representing the
global order of any numerical scheme based on this approximation). The numerical
method we propose here is based on this natural splitting approximation, together
with an approximation of the Lie series ψdifftn,tn+1
generated by governing diffusion vector
fields that guarantees a strong order one scheme. The idea is that the separate flows in
the decomposition above are more easily approximated. They might even be computed
exactly and, if they individually preserve positivity, their composition does also.
Definition 1 (Euler–Levy vector field) We define the Euler and Levy vector fields
on the interval [tn, tn+1] to be
V euler ≡dX
i=1
Ji(tn)Vi and V levy ≡dX
j>i=1
Aij(tn)[Vi, Vj ] ,
where Ji(tn) are Wiener increments, and Aij(tn) ≡ 12
`
Jij(tn) − Jji(tn)´
are the Levy
areas or suitably accurate approximations of them. We consequently define the Euler–
Levy vector field to be
V EL ≡ V euler + V levy .
The corresponding Euler–Levy flow for any τ ≥ 0 is exp`
τV EL´.
Constructing flows such as the Euler–Levy flow in this manner originates from Ben
Arous [8] and Castell [18], and in practice, approximation of such flows was considered
by Castell and Gaines [19,20] and Ninomiya and Victoir [58].
In the flow decomposition above we approximate exp`
ψdifftn,tn+1
´
≈ exp`
V EL´. Our
numerical approximation across the interval [tn, tn+1] is thus given by
expψtn,tn+1≈ exp
`
12hV0
´
◦ exp`
V EL´ ◦ exp`
12hV0
´
. (4)
This is known as the Strang exponential splitting and is preferred for its symmetry
properties. It is also an example of the Stormer–Verlet or explicit one-step leap-frog
method—see Strang [62] and Hairer, Lubich and Wanner [33]. When the drift and
Euler–Levy flows are given exactly, then the Strang exponential splitting generates a
strong L2 global error of order h and represents a method of order one. This follows by
estimating the L2-norm of the difference between the Strang generated flow and the
stochastic Taylor expansion—see Lord, Malham and Wiese [47] for more details. Sup-
pose we approximate the drift and Euler–Levy flows across the interval [tn, tn+1]. The
error in the Strang exponential splitting due to these approximations across [tn, tn+1],
at leading order, is given by the sum of the three individual errors associated with each
separate flow approximation. Next order corrections generate higher order contribu-
tions to the L2 global error.
Positive stochastic volatility simulation 9
5 Positive implicit Milstein approximation
The main idea in this paper is to construct a strong order one positivity preserv-
ing stochastic integrator by using the Strang exponential splitting (4). For a given
stochastic flow which we know to be exactly positivity preserving, we assume that
the associated exact drift and Euler–Levy flows are individually positivity preserving.
Composing the exact flows in the Strang exponential splitting will therefore generate a
positivity preserving scheme. In some cases these flows can be computed analytically,
however in general the exact flows may not be available.
For the central pure diffusion flow we therefore use an implicit Milstein approxi-
mation given by:
Un+1 = Un + V euler ◦ Un + V levy ◦ (Un, Un+1) + 12
`
V euler´2 ◦ (Un, Un+1) , (5)
where by V ◦ (Un, Un+1) we indicate that the vector field will be evaluated at both
present tn and forward tn+1 points in the interval [tn, tn+1].
We restrict ourselves here to only two diffusion vector fields Vi for i = 1, 2 which
have the form, for j = 1, . . . , N :
V ji = cij X
j , (6)
i.e. each diffusion vector field has the same functional form Xj in each component,
but with a different multiplicative factor cij . This is typical of correlation structures
in stochastic volatility models. Without loss of generality, we assume for each j that
Xj ◦y ≥ 0 for y ≥ 0 (by which we mean each component of y is non-negative). Further
we will consider here a two component system U = (u, v)T, as our argument generalizes
to systems with more components. Consequently we see that
V euler =
„`
c11J1(tn) + c21J2(tn)´
X1`
c12J1(tn) + c22J2(tn)´
X2
«
and
[V1, V2] =`
c11X1∂u + c12X
2∂v´
„
c21X1
c22X2
«
−`
c21X1∂u + c22X
2∂v´
„
c11X1
c12X2
«
= (c12c21 − c11c22)
„
X2∂vX1
−X1∂uX2
«
.
These forms suggest we set
J1(tn) = c11J1(tn) + c21J2(tn) ,
J2(tn) = c12J1(tn) + c22J2(tn) ,
A1(tn) = (c12c21 − c11c22)A12(tn) ,
A2(tn) = −(c12c21 − c11c22)A12(tn) .
Then the Euler and Levy vector fields are given by
V euler =
„
J1(tn)X1
J2(tn)X2
«
and V levy =
„
A1(tn)X2∂vX1
A2(tn)X1∂uX2
«
,
10 Halley, Malham and Wiese
and further, we have
`
V euler´2 =`
J1(tn)X1∂u + J2(tn)X2∂v´
„
J1(tn)X1
J2(tn)X2
«
=
„
J21 (tn)X1∂uX
1 + J1(tn)J2(tn)X2∂vX1
J1(tn)J2(tn)X1∂uX2 + J2
2 (tn)X2∂vX2
«
.
Our implicit strategy is to evaluate the vector fields in the Milstein step (5) as
follows:
V levy ◦ (Un, Un+1) =
„
A1(tn)`
X2∂vX1´(un+1, vn)
A2(tn)`
X1∂uX2´(un, vn+1)
«
,
and
`
V euler´2 ◦ (Un, Un+1)
=
„
J21 (tn)
`
X1∂uX1´(un, vn) + J1(tn)J2(tn)
`
X2∂vX1´(un+1, vn)
J1(tn)J2(tn)`
X1∂uX2´(un, vn+1) + J2
2 (tn)`
X2∂vX2´(un, vn)
«
.
In other words, as already indicated, we evaluate the Euler vector field explicitly, while
in the Levy vector field we evaluate each component implicitly in that component
variable. We are interested in the case when X1∂uX1 ◦ y ≥ 0 and X2∂vX
2 ◦ y ≥ 0 for
y ≥ 0; this will be true for the class of vector fields we consider below. Hence in the
Euler vector field product, we evaluate those terms explicitly, and for the remaining
terms, apply the same strategy as we did for the Levy vector field. This strategy takes
a partial cue from the drift implicit method of Kahl, Gunter and Roßberg [38]: the
hope is that combined in the Euler step (5), the resulting quadratic forms for J1(tn)
and J2(tn) imply that the explicit terms are positive.
To show this, first we need the following important result.
Lemma 3 For any z ≥ 0 and γ ∈ R define the nonlinear function
P(ω) ≡ ω − γωp − z .
We assume 0 < p < 1. If z > 0 then P has a unique real, strictly positive root. If z = 0
then P has a unique real, strictly positive root except when γ ≤ 0, in which case its
largest non-negative root is ω = 0.
Proof Assume z > 0. All statements assume ω ≥ 0. Note that P(0) = −z < 0 and
P(ω) ∼ ω as ω → +∞ and so P has at least one strictly positive root. If γ ≤ 0 then
P ′(ω) = 1 + p|γ|/ω1−p > 0, i.e. P grows monotonically and therefore has a unique
strictly positive root. If γ > 0 then P ′(ω) = 1−pγ/ω1−p and so P ′ has a single strictly
positive root at ωmin = (pγ)1/(1−p) which corresponds to a local minimum. Hence Pitself again has a unique strictly positive root.
If z = 0, then P(0) = 0. When γ ≤ 0 then P grows monotonically and its largest
non-negative root is ω = 0. If γ > 0 then P still has a single strictly positive local
minimum at ωmin and therefore P itself has a unique strictly positive root. ⊓⊔
Remark 2 For any z > 0 Newton’s method will converge to the strictly positive root of
P : for γ ≤ 0 for any initial guess ω0 > 0; while for γ > 0 we should choose ω0 > ωmin.
For z = 0: when γ ≤ 0 the only non-negative solution is ω = 0; while for γ > 0, we
should again choose ω0 > ωmin to guarantee Newton’s method converges to the strictly
positive root of P .
Positive stochastic volatility simulation 11
Second we restrict the class of diffusion vector fields further. For this class of vector
fields there is a unique positive solution to the nonlinear equation representing the
implicit Milstein step constructed using the strategy we have outlined.
Theorem 1 Assume that the diffusion vector fields (6) have the form
Xj ◦ U = uαj1vαj2 , (7)
where for each j, k = 1, 2 we assume αjk ∈ R with αjj ∈ [12 , 1] and αjk ∈ (0, 1] for
j 6= k. We also assume that α1k + α2k ≤ 1 for each k, and exclude the special cases
α11 = α11 + α21 = 1 and α22 = α12 + α22 = 1. Then the resulting implicit Milstein
scheme we propose is given by:
un+1 = un + J1(tn)uα11n vα12
n + 12 J
21 (tn)α11u
2α11−1n v2α12
n
+`
A1(tn) + 12 J1(tn)J2(tn)
´
α12uα11+α21
n+1 vα12+α22−1n , (8a)
vn+1 = vn + J2(tn)uα21n vα22
n + 12 J
22 (tn)α22u
2α21n v2α22−1
n
+`
A2(tn) + 12 J1(tn)J2(tn)
´
α21uα11+α21−1n vα12+α22
n+1 . (8b)
This scheme is preserves strict positivity in the first component if α11 >12 and non-
negativity if α11 = 12 , and strict positivity in the second component if α22 > 1
2 and
non-negativity if α22 = 12 . Both un+1 and vn+1 are m-integrable for all m ≥ 1. Further,
used in the Strang decomposition, this approximation contributes an error of order h
to the L2 global error.
Proof First we prove positivity. For positive coefficients c0, c1 and c2 we have:
g(ξ; c0, c1, c2) ≡ c0 + c1ξ + 12c2ξ
2 ≥ g(−c1/c2; c0, c1, c2) = c0 − c21/2c2 .
Hence we see that g`
J1;un, uα11n vα12
n , α11u2α11−1n v2α12
n´
> 0 if α11 >12 , with equality
possible if α11 = 12 . Similarly g
`
J2; vn, uα21n vα22
n , α22u2α21n v2α22−1
n´
> 0 if α22 > 12 ,
with equality possible if α22 = 12 . This shows that the explicit terms in each component
of our algorithm are strictly positive if αjj >12 and non-negative if αjj = 1
2 . Let us
now consider the nonlinear equation for un+1, the argument is the same for vn+1.
Assume α11 >12 . Solving the nonlinear equation for un+1 is equivalent to finding the
real, strictly positive roots of P . We see this if we set: α11 + α21 = p; ω = un+1;
z = g`
J1(tn);un, uα11n vα12
n , α11u2α11−1n v2α12
n´
; and finally γ to be the coefficient of
(un+1)α11+α21 . Then by Lemma 3 we know that only one real, strictly positive root
of P exists. Hence Newton’s method with initial guess max˘
un, ωmin
¯
—see Remark 2
above—generates a robust scheme to achieve the implicit forward step whilst preserving
strict positivity independent of the stepsize h. Note that if α11 = 12 so that z is possibly
zero then we set un+1 to be: zero if γ ≤ 0; and the strictly positive root found by
Newton’s method with initial guess max˘
un, ωmin
¯
if γ > 0.
Second we prove the stated integrability property. To see this consider the nonlinear
equation un+1 = z+γupn+1 (the argument is the same for vn+1), where 0 < p < 1, and
z and γ depend on (un, vn), on the Wiener increments and on the Levy area across
the time interval [tn, tn+1]. Note that if (un, vn) are m-integrable for all m ≥ 1, so are
12 Halley, Malham and Wiese
z and γ. For all m ≥ 1 we have
Eˆ
|un+1|m˜
≤ Cm
“
Eˆ
|z|m˜
+ Eˆ
|γupn+1|
m˜”
≤ Cm
„
Eˆ
|z|m˜
+“
Eˆ
|γ|m
1−p˜
”1−p“
Eˆ
|un+1|m˜
”p«
≤ Cm max
1,“
Eˆ
|z|m˜
+`
Eˆ
|γ|m
1−p˜´1−p
”1
1−p
ff
,
where we have used the Holder inequality, and Cm is a generic constant. Thus un+1 ∈Lm(P ) for all m ≥ 1 provided (un, vn) ∈ Lm(P ).
Third we estimate the error in the implicit Milstein scheme. There are contributions
from two sources, we: (i) truncated the Taylor expansion in the Milstein step (5), and
in so doing have dropped terms proportional to terms of the form Ji(tn)Jj(tn)Jk(tn)
and Ji(tn)Ajk(tn); and (ii) replaced explicit terms in the Euler–Levy vector field and
product of the Euler vector field which are proportional to Ji(tn)Jj(tn) or Aij(tn) by
implicit terms, and using the stochastic Taylor expansion to make these replacements
thus generates errors proportional to Ji(tn)Jj(tn)Jk(tn) or Aij(tn)Jk(tn). The expec-
tation of each of these terms is zero while their standard deviation is proportional to
h3/2. Hence the contribution of these terms to the strong L2 global error will be pro-
portional to h; see Milstein [53] or Lord, Malham and Wiese [47]. ⊓⊔
We now consider how to approximate the drift flow exp`
12hV0
´
. We assume that
the drift vector field components can be written as a linear combination of terms of
the form (7). We do however, allow that one term in the linear combination can be an
additive positive constant. We apply an Euler step using the following semi-implicit
strategy. For a given component with several additive terms: if the fixed multiplicative
coefficient is negative then we make that term implicit in that component variable,
and explicit in the other multiplicative variables in that term. The resulting implicit
Euler scheme is positivity preserving; the argument is a special case of that for the
implicit Milstein scheme above if we include any additive positive constant terms in
our identification of z. The local error at leading order is proportional to h2 (from
the second order term we dropped and making the implicit replacements). Hence the
global error across [0, T ] generated by the accumulation of this local error will be of
order h.
Remark 3 Some further qualifications are required.
1. The non-Lipschitz property of the diffusion vector fields and in particular the frac-
tional powers of the components are crucial, not only to proving that all the mo-
ments of the approximation at the next step are finite, but also to guarantee posi-
tivity.
2. On the discrete time set of the numerical scheme, the probability that z = 0 when
either α11 = 12 in the case of un, or α22 = 1
2 in the case of vn, is zero.
3. In the case of linear diffusion vector fields, implicit Euler schemes are not directly
applicable as the next step approximations do not have finite moments (see Kloe-
den and Platen [44], Burrage, Burrage and Tian [16] and Burrage, Herdiana and
Burrage [17]).
4. In Theorem 1 we excluded the special linear cases when α11 = α11 + α21 = 1 or
α22 = α12 + α22 = 1. In either of these cases we would apply an exponential algo-
rithm which also preserves positivity. Care must be taken to generate the correct
exponent as some terms are automatically included by the exponential ansatz.
Positive stochastic volatility simulation 13
5. We introduce implicitness in the higher order terms involving the Levy area and
products of the Wiener increments. Hence our method converges to the solution
with strong order one error.
6. For each implicit step, if the implicit nonlinear equation for the next-step value
cannot be solved explicitly, then a Newton rootfinding algorithm may be required
to solve the implicit equation. The additional computational cost is not significant
compared to the cost of evaluating the Levy areas.
7. We could combine the implicit Euler step for the drift vector field additively with
the implicit Milstein step for the pure diffusion, replacing the Strang decomposition
completely (indeed, Alfonsi [2] established such a method of order one-half for the
Heston model, based on an Euler–Maruyama step with the diffusion coefficient in
the volatility evaluated implicitly). In particular if the drift vector field were linear,
all the conclusions above for the method we have proposed would apply to this
more direct method (we would still use the function P to establish positivity and
so forth). If the drift vector field also involved a fractional power term then we
could, in principle still prove analogous results (we would now need the properties
of the function Q(ω) ≡ ω− γωp − δωq − z to establish positivity; see Appendix A).
8. However, separating the drift and diffusion flows is quite natural. We can take ad-
vantage of analytical solutions available for either of the separated flows. In partic-
ular this approach points towards breaking down the strong order one flow further
if the vector fields are more complicated or we have more than two components. For
example suppose we have a three component stochastic differential system driven
by a three-dimensional Wiener process. In this case we could approximate the pure
diffusion central flow in the Strang decomposition by
exp`
J1V1 + A12[V1, V2]´
◦ exp`
J2V2 + A23[V2, V3]´
◦ exp`
J3V3 + A13[V1, V3]´
,
which still represents a strong order one splitting. We prove in Appendix A, using
the properties of the function Q, that our strategy applied to the individual flows
in the above decomposition generates a positivity preserving scheme. A four com-
ponent system with four diffusion vector fields requires further astute splitting—for
example composing the separate flows associated with each diffusion and each Lie
bracket vector field.
Algorithm
In summary, for diffusion vector fields of the form (6),(7), we thus propose the following
algorithm. First, analytically compute the form of the Euler, Levy, and Euler product
vector fields required for the implicit Milstein step. Second, decompose the global
interval of integration into successive subintervals. Then for each multi-dimensional
Wiener path and each subinterval:
(i) Approximate the Levy areas using any suitable pathwise approximation—for
example the method of Gaines and Lyons [26];
(ii) Step forward across the subinterval with the Strang exponential splitting, using
the implicit Milstein method to approximate the Euler–Levy flow and the implicit Euler
method to approximate the drift flow.
(iii) For each implicit step, a Newton rootfinding algorithm may be required to
solve the implicit equation for the next-step value.
14 Halley, Malham and Wiese
6 Stochastic volatility models
We present two stochastic volatility models: the Heston model (Heston [31]) and a
double mean-reverting variance curve model in the class suggested by Buhler [13].
6.1 Heston model
The Heston model is a two-factor model, in which the first component u describes
the evolution of a stock price, and the second component v, its stochastic volatility.
Expressed as an Ito integral, the Heston model is given by
dut = µut dt+√vt ut dW 1
t ,
dvt = α(θ − vt) dt+ βρ√vt dW 1
t + βp
1 − ρ2√vt dW 2
t ,
where and W 1t and W 2
t are independent scalar Wiener processes. The parameters
µ, α, θ and β are all positive and ρ ∈ (−1, 1). In the context of option pricing, an
equivalent martingale measure must be specified. We will consider the model in its
original form above, this corresponds to the choice of the minimal martingale measure
(see Hobson [32] for other measures). By the Yamada condition this model has a unique
strong solution—see Karatzas and Shreve [42, p. 291]. In particular, the volatility v is
non-negative, and since the stock price u is a pure exponential process, it is strictly
positive. Further, if αθ ≥ 12β
2 then the volatility is in fact strictly positive. This follows
from the Feller classification of boundary points in Lemma 2; also see for example
Andersen and Piterbarg [5]. When αθ < 12β
2 the zero boundary is attainable but
strongly reflecting—if the process reaches zero it leaves it immediately—see Revuz and
Yor [60, p. 412].
When we reformulate the Heston model in Stratonovich form the system becomes
dut =`
µ− 12vt − 1
4βρ´
ut dt+√vt ut dW 1
t ,
dvt =`
α(θ − vt) − 14β
2´ dt+ βρ√vt dW 1
t + βp
1 − ρ2√vt dW 2
t .
We identify the Stratonovich drift, V0, and diffusion vector fields, V1 and V2, as
V0 ◦ y ≡„
(µ− 12v − 1
4β1)u
α(θ − v) − 14β
2
«
, V1 ◦ y ≡„
u√v
β1√v
«
and V2 ◦ y ≡„
0
β2√v
«
,
where y = (u, v)T, β1 = βρ and β2 = βp
1 − ρ2. Analogously to Ninomiya and
Victoir [58], also see Lander [46] and Halley [30], the flows associated with the two
diffusion vector fields can be computed exactly. The Euler–Levy vector field for the
Heston model is
V EL ◦ y =
„
J1(tn)u√v − 1
2β2A12(tn)u
J2(tn)√v
«
,
where J1(tn) = J1(tn) and J2(tn) = β1J1(tn) + β2J2(tn). The Euler–Levy flow is
exp`
τV EL´ ◦ y =
exp`
τ J1(tn)√v + 1
4τ2J1(tn)J2(tn) − 1
2τβ2A12(tn)´
u`√v + 1
2τ J2(tn)´2
!
.
Positive stochastic volatility simulation 15
The Stratonovich drift flow is given by
exp(tV0) ◦ y =
u exp“
`
µ− 14β1 − 1
2H´
t+ 12α (v −H)(e−αt − 1)
”
ve−αt +H(1 − e−αt)
!
,
where H =`
αθ− 14β
2´/α. Note that for the Strang exponential splitting (4), the drift
flow and Euler–Levy flow can be computed using either these exact formulae for the
flows, or the implicit Euler and Milstein methods, respectively.
The drift vector field satisfies the conditions of Lemma 1 which imply that the sec-
ond component of the drift flow-map is strictly positive for αθ ≥ 14β
2. Consequently the
first component is also strictly positive. Both these properties can be seen directly from
the exact flow-map. The Euler–Levy vector field satisfies the conditions of Lemma 1
which imply that the second component is non-negative while the first component is
strictly positive, which is also clear from the Euler–Levy flow-map. Hence if we use
the exact drift and Euler–Levy flows in the Strang decomposition, then for αθ > 14β
2
the volatility will be strictly positive, while for αθ = 14β
2 it will be non-negative.
As is well known, there are thus three parameter regimes of interest for αθ namely,
(0, 14β
2), [14β2, 1
2β2) and [12β
2,∞); see for example Lord, Koekkoek and Van Dijk [48]
or Andersen [4].
6.2 Variance curve model
We consider a double mean-reverting variance curve model suggested by Buhler [14,
p. 197]. The process u is the stochastic volatility and the process v resembles the
volatility of the volatility. More details can also be found in Buhler [13, Chapter 6] and
Overhaus et al. [59]. Expressed as an Ito integral, this model is given by
dut = κ(vt − ut) dt+ ν√utvt dW 1
t ,
dvt = c(wt − vt) dt+ βρ√vtwt dW 1
t + βp
1 − ρ2√vtwt dW 2
t ,
dwt = η1wt dW 1t + η2wt dW 2
t + η3wt dW 3t ,
where W 1t , W 2
t and W 3t are independent scalar Wiener processes and the parameters
κ, c, ν and β are all positive and ρ ∈ (−1, 1). Also we have set η1 = η1, η2 = η2,
and η3 = ηq
1 − 21 − 22 with η positive and 1, 2 ∈ (−1, 1). Again using the Yamada
condition, this model has a unique strong solution. Note that w is a pure exponential
process and therefore strictly positive. Further v is non-negative, and so is the volatility
u. Again using Lemma 2, if c ≥ 12β
2 then v is strictly positive and if κ ≥ 12ν
2 then
u is strictly positive also. If c < 12β
2 then the zero boundary for v is attainable but
strongly reflecting, and similarly for u if κ < 12ν
2.
When we reformulate this variance curve model in Stratonovich form, then the
Stratonovich drift vector field is
V0 ◦ y ≡
0
@
κ(v − u) − 14ν
2v − 14νβ1
√uw
c(w − v) − 14β
2w − 14 (β1η1 + β2η2)
√vw
− 12η
2w
1
A .
16 Halley, Malham and Wiese
where y = (u, v, w)T, β1 = βρ and β2 = βp
1 − ρ2, and the diffusion vector fields are
V1 ◦ y ≡
0
@
ν√uv
β1√vw
η1w
1
A , V2 ◦ y ≡
0
@
0
β2√vw
η2w
1
A and V3 ◦ y ≡
0
@
0
0
η3w
1
A .
The flows associated with each diffusion vector field can be computed exactly. The
Euler–Levy vector field is
V EL ◦ y =
0
@
J1(tn)√vu+ 1
2 A1(tn)√wu
J2(tn)√wv + 1
2 A2(tn)√wv
J3(tn)w
1
A ,
where J1(tn) = νJ1(tn), J2(tn) = β1J1(tn) + β2J2(tn) and
J3(tn) = η1J1(tn) + η2J2(tn) + η3J3(tn) ,
A1(tn) = − β2νA12(tn) ,
A2(tn) = (η1β2 − η2β1)A12(tn) − β2η3A23(tn) − β1η3A13(tn) .
The Euler–Levy flow can be computed exactly and is given by
exp`
τV EL´ ◦ y =
0
B
B
B
@
“√u+ 1
2 J1(tn)R τ0
p
v(ξ) dξ + 14 A1(tn)
R τ0
p
w(ξ) dξ”2
“√v + 1
2
`
J2(tn) + 12 A2(tn)
´ R τ0
p
w(ξ) dξ”2
w exp(τ J3(tn))
1
C
C
C
A
,
whereZ τ
0
p
w(ξ) dξ =√w`
exp( 12τ J3(tn)) − 1
´
/` 1
2 J3(tn)´
,
Z τ
0
p
v(ξ) dξ =√v τ + 1
2
`
J2(tn) + 12 A2(tn)
´
Z τ
0
Z ξ1
0
p
w(ξ2) dξ2 dξ1 ,
and
Z τ
0
Z ξ1
0
p
w(ξ2) dξ2 dξ1 =√w“
`
exp( 12τ J3(tn)) − 1
´
/`
12 J3(tn)
´
− τ”
/`
12 J3(tn)
´
.
The flow associated with the Stratonovich drift vector cannot be computed exactly, and
we employ the implicit Euler method to approximate the flow and preserve positivity.
For the variance curve model, the Stratonovich drift positivity conditions separate the
parameter interval regimes as follows: for c we distinguish (0, 14β
2), [14β2, 1
2β2) and
[12β2,∞); while for κ we distinguish (0, 1
4ν2), [14ν
2, 12ν
2) and [12ν2,∞). Note that u, v
and w are globally m-integrable—see Appendix B for a proof. We have not included
the underlying asset price here which may suffer moment explosion in particular pa-
rameter regimes (as shown for the stock price in the Heston model by Andersen and
Piterbarg [5]).
7 Algorithms
We outline the algorithms we use in simulations and their properties.
Positive stochastic volatility simulation 17
7.1 Heston model
Partial truncation, reflection and full truncation
The partial truncation method of Deelstra and Delbaen uses an Euler–Maruyama step
for the volatility with the argument vn of the diffusion coefficient replaced by v+n ≡max{vn, 0}:
un+1 = un exp
„
h(µ− 12v
+n ) + J1(tn)
q
v+n
«
,
vn+1 = vn + hα(θ − vn) + (β1J1(tn) + β2J2(tn))
q
v+n .
In the reflection method of Bossy and Diop, the absolute value of the whole of the
standard Euler–Maruyama increment for the volatility is taken at each step. The full
truncation method of Lord, Koekkoek and Van Dijk [48] looks similar to the partial
truncation algorithm but the argument in the linear form of the drift coefficient of the
volatility is also replaced by v+n . Note that we naturally model the asset value u by the
exponential approximation shown and we use this approximation for this component
for all the Euler projection methods (see Lord, Koekkoek and Van Dijk).
Drift implicit
If we consider a stochastic Taylor expansion that includes the first set of repeated
integral terms, we generate the strong order one scheme:
un+1 = un exp“
h`
µ− 14β1 − 1
2vn´
+ J1(tn)√vn + 1
4β1
`
J1(tn)´2
+ 12β2J21(tn)
”
,
vn+1 = vn − hαvn+1 + h(αθ − 14β
2) + J2(tn)√vn + 1
4 J22 (tn) .
Note that we have used the natural exponential approximation for the asset price. For
the volatility, as suggested in Kahl, Gunther and Roßberg [38, pp. 7–8], we take the
negative linear drift term to be implicit. Since the quadratic form vn+√vnξ+
14ξ
2 ≥ 0,
this guarantees positivity provided αθ ≥ 14β
2.
Strang implicit Milstein
In the Strang decomposition, on each step [tn, tn+1], we must compose three successive
flows. For the Heston model the flows associated with drift and Euler–Levy vector fields
can be computed exactly and we simply successively compose the flows. However,
suppose we use the implicit Euler and Milstein methods to compute the Stratonovich
drift and Euler–Levy flows, respectively. For the drift flow exp`
12hV0
´
, using the implicit
Euler strategy, we thus step forward across [tn, tn+1] using:
un+1 =`
1 + 12hµ
´
un/`
1 + 12h(
14β1 + 1
2vn)´
,
vn+1 =`
vn + 12h(αθ − 1
4β2)´
/`
1 + 12hα
´
.
We could use an exponential approximation for the first component, but we do not
implement this. The implicit Milstein method applied to the Euler–Levy flow is the
exact Euler–Levy flow for the Heston model. This is a consequence of the natural
18 Halley, Malham and Wiese
Table 1 For the Heston model, we indicate for each numerical method, if the volatility ispreserved strictly positively (+), non-negatively (0) or not at all (−) in the three numericallydistinguished parameter regimes.
method/parameter regime αθ > 14β2 αθ = 1
4β2 αθ < 1
4β2
partial truncation − − −reflection 0 0 0
full truncation − − −Strang: exact flows + 0 −
drift implicit + 0 −Strang: drift Euler + 0 −
nilpotency in Lie algebra generated by the two diffusion vector fields. Consequently we
implement the exact Euler–Levy flow across the interval [tn, tn+1].
In Table 1 we have summarized the positivity preserving properties of the methods
we have implemented. We see that the drift implicit and Strang decomposition methods
replicate the behaviour of the exact flow for αθ ≥ 12β
2. However, for αθ ∈ ( 14β
2, 12β
2)
they never attain zero, which the exact flow does with probability one. Of those that
we have not implemented, the balanced implicit and Milstein methods (see Milstein,
Platen and Schurz [54] and Kahl and Schurz [41]) have the same properties as the drift
implicit method.
Remark 4 Note that if we take the power in the diffusion term in the volatility not to
be p = 12 but rather p > 1
2 , then by Lemma 2 the volatility v in the Heston model is
always strictly positive (see for example Andersen and Piterbarg [5]). In this scenario
we expect the Strang exponential splitting (4) method to perform particularly better
than the Euler projection methods.
7.2 Variance curve model
Truncation and reflection
The truncation method we have implemented for the variance curve model of Buhler
is based on the full truncation method of Lord, Koekkoek and Van Dijk. We use the
algorithm
un+1 = un + hκ(v+n − u+n ) + ν
q
u+n v
+n J1(tn) ,
vn+1 = vn + h c (wn − v+n ) +
q
wnv+n
`
β1J1(tn) + β2J2(tn)´
,
wn+1 = wn exp`
η1J1(tn) + η2J2(tn) + η3J3(tn) − 12hη
2´ .
Adapting the reflection method of Bossy and Diop, we take the absolute value of the
whole of the Euler–Maruyama increment at each step for both u and v. We use the
same exponential approximation for w. Both methods apply in the entire parameter
regime. They do not distinguish whether the origin is attainable or not for either u
or v. In the reflection method the numerical approximations for u and v are always
non-negative; for the truncation method they can become negative.
Positive stochastic volatility simulation 19
Strang implicit Milstein
For the variance curve model the drift flow exp`1
2hV0´
cannot be computed exactly
and we approximate it using the implicit Euler approach. We thus step forward across
[tn, tn+1] using:
un+1 = un + 12h`
κvn − κun+1 − 14νβ1
√wn
√un+1
´
,
vn+1 = vn + 12h`
cwn − cvn+1 − β√wn
√vn+1
´
,
wn+1 = wn exp`
− 14hη
2´ ,
where κ ≡ κ− 14ν
2, c ≡ c− 14β
2 and β ≡ 14 (β1η1 + β2η2). Note that we can explicitly
solve the quadratic equations for√un+1 and
√vn+1 shown.
We approximate the Euler–Levy flow using the implicit Milstein method. Across
the interval [tn, tn+1] we implement:
un+1 = un + J1(tn)√unvn + 1
4 J21 (tn)vn +
`
12 A1(tn) + 1
4 J1(tn)J2(tn)´√wn
√un+1 ,
vn+1 = vn + J2(tn)√vnwn + 1
4 J22 (tn)wn +
`
12 A2(tn) + 1
4 J2(tn)J3(tn)´√wn
√vn+1 ,
wn+1 = wn exp(J3(tn)) ,
where the definitions for the Ji(tn) and Ai(tn) can be found at the end of Section 6.
Again, we can explicitly solve the quadratic equations for√un+1 and
√vn+1. Note
that we used the same strategy as that used to produce the algorithm in Theorem 1
to generate this scheme.
We use exponential integration to step forward w which is therefore guaranteed to
be strictly positive. The implicit Euler approximation for the drift flow can be seen
to generate strictly positive approximations for u and v from non-negative data. The
exact Euler–Levy flow and its implicit Milstein approximation are both non-negativity
preserving for u and v. Hence using either the exact or approximate Euler–Levy flow,
the Strang decomposition will preserve strict positivity for u and v provided c ≥ 14β
2
and κ ≥ 12ν
2.
8 Simulations
8.1 Heston model
We show in Figures 1 and 2 these methods applied to the Heston model. We use
the same parameter values and initial conditions as in Ninomiya and Victoir [58] and
Ninomiya and Ninomiya [57], namely: α = 2.0, θ = 0.09, β = 0.1, µ = 0.05 and
(u0, v0) = (1.0, 0.09). We also set ρ = 0.5 for the more general Heston model with
correlation between stock price and volatility. With these parameter values we have
αθ > 12β
2, and from Andersen and Peterbarg [5, p. 34], we know that the first two
moments of the stock price are globally finite. We see in the lower plot in Figure 1,
that when high accuracy is required, it is computationally less effort to simulate the
order one methods than the Euler projection methods. In Figure 2 we show for a
large stepsize h = 1 how all the methods perform. The Euler projection methods are
synchronous until the next Euler step generates a negative approximation. The order
one methods, which take into account intervening path information through the Levy
area, follow the true solution closely, while the Euler projection methods veer much
more wildly from it.
20 Halley, Malham and Wiese
−3.4 −3.2 −3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4−4.5
−4
−3.5
−3
−2.5
−2
log10
(stepsize)
log 10
(glo
bal e
rror
)
Number of sampled paths=100
Partial truncationReflectionFull truncationNinomiya−VictoirStrang: exact flowsDrift implicitStrang: Euler drift
−1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4−4.5
−4
−3.5
−3
−2.5
−2
log10
(CPU time)
log 10
(glo
bal e
rror
)
Number of sampled paths=100
Partial truncationReflectionFull truncationNinomiya−VictoirStrang: exact flowsDrift implicitStrang: Euler drift
Fig. 1 Global error vs stepsize (upper panel) and vs CPU clocktime (lower panel) for theHeston model at time t = 1.
Positive stochastic volatility simulation 21
0 5 10 15 20 25 30 35 40 45 50−0.05
0
0.05
0.1
0.15
0.2
time
vola
tility
Exact solutionEuler MaruyamaFull truncationReflectionStrang: exact flowsDrift implicitStrang: Euler drift
0 5 10 15 20 25 30 35 40 45 500
2
4
6
8
10
12
time
sto
ck p
rice
Exact solutionEuler MaruyamaFull truncationReflectionStrang: exact flowsDrift implicitStrang: Euler drift
Fig. 2 Volatility (upper panel) and stock price (lower panel) approximations for the Hestonmodel computed with a stepsize h = 1. Once the volatility for the Euler–Maruyama schemebecame negative, we froze the values for the volatility and stock price.
22 Halley, Malham and Wiese
−4 −3.5 −3 −2.5 −2 −1.5−4.5
−4
−3.5
−3
−2.5
−2
−1.5
log10
(stepsize)
log 10
(glo
bal e
rror
)
Number of sampled paths=100
ReflectionTruncationStrang: exact Levy flowStrang: semi−implicit Euler
−1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−4.5
−4
−3.5
−3
−2.5
−2
−1.5
log10
(CPU time)
log 10
(glo
bal e
rror
)
Number of sampled paths=100
ReflectionTruncationStrang: exact Levy flowStrang: semi−implicit Euler
Fig. 3 Global error vs stepsize (upper panel) and vs CPU clocktime (lower panel) for thevariance curve model at time t = 1.
Positive stochastic volatility simulation 23
0 1 2 3 4 5 6 7 8 9 10−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
time
u
Exact solutionEuler MaruyamaFull truncationReflectionStrang: exact flowsStrang: semi−implicit Euler
0 1 2 3 4 5 6 7 8 9 100
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
time
v
Exact solutionEuler MaruyamaFull truncationReflectionStrang: exact flowsStrang: semi−implicit Euler
Fig. 4 Approximate solution for the variance curve model computed with a stepsize h = 1/8.For the Euler–Maruyama scheme, once either u (upper panel) or v (lower panel) becamenegative, we froze their values.
24 Halley, Malham and Wiese
8.2 Variance curve model
We show in Figure 3 and 4 the four numerical methods we applied to the variance curve
model. We used the same parameter values and initial conditions as in Buhler [13,
pp. 121,122], namely: κ = 3.027, c = 0.013, ν = 2.442, β = 0.1563, η = 0.355, ρ = 0.0,
1 = −0.68, 2 = −0.60 and (u0, v0, w0) = (0.015, 0.030, 0.580). We have recalibrated
the values ν → ν/√v0 and β → β/
√w0 from those quoted in Buhler to account for
our different model system. Note that for the original and recalibrated values we have
c > 12β
2 and κ > 12ν
2. In the lower plot in Figure 3, we see that order one methods
are competitive with the Euler projection methods, delivering superior accuracy for a
given computational effort. When we consider an individual solution for a given driving
path, there is no significant difference in the approximations for w as all four schemes
are ultimately using similar exponential approximations for this component. However
for a relatively small stepsize h = 1/8, the volatility u and volatility of the volatility
v exhibit markedly different behaviour between the Euler projection and order one
schemes. As for the Heston model, the Euler projection methods separate when the
next Euler step approximation becomes negative in either component. Unsurprisingly
the order one methods preserve positivity and adhere to the true solution.
9 Concluding remarks
There are several aspects and extensions of the method we have proposed that remain
open, notably we would like to consider: (i) its numerical stability properties which
are typically better for implicit methods (see for example Buckwar, Horvath–Bokor
and Winkler [12]); (ii) the case of multiple components with a view to extending our
methods to stochastic partial differential equations (to which splitting methods have
been applied by Gyongy and Krylov [29]); (iii) implementing a variable step version (see
Gaines and Lyons [26] and Burrage and Burrage [15]); and (iv) applying it to more gen-
eral stochastic differential equations whose solution components are non-negative, such
as in molecular DNA damage dynamics (see Chickarmane, Ray, Sauro and Nadim [21]).
A Positivity for a three component system
We prove positivity for the approximate flow indicated in Remark 3 (item 8). Suppose thediffusion vector fields have the form (6) with
Xj ◦ U = uαj1vαj2 wαj3 .
Without loss of generality let us focus on approximating the flow generated by the vector fieldJ1(tn)V1 + A12(tn)[V1, V2]. The jth component of the Lie bracket is given by
[V1, V2]j =
X
ℓ 6=j
(c1ℓc2j − c2ℓc1j)Xℓ∂Uℓ
Xj ,
while the product vector field
(V1 ◦ V1)j =X
ℓ
c1ℓc1jXℓ∂UℓXj .
Positive stochastic volatility simulation 25
We define Jij(tn) = cijJi(tn) and Aijkl(tn) = (ciℓckj − ckℓcij)Aik(tn). Further let us focuson the first component j = 1. Then using our strategy, the implicit step we propose for thiscomponent is
un+1 = un + J11(tn)X1(un, vn, wn) + 12
`
J11(tn)´2
X1∂uX1(un, vn, wn)
+`
A1122 + 12J11(tn)J12(tn)
´
X2∂vX1(un+1, vn, wn)
+`
A1123 + 12J11(tn)J13(tn)
´
X3∂wX1(un+1, vn, wn) .
Further we have
X2∂vX1 ◦ U = α12 uα21+α11vα22+α12−1wα23+α13
X3∂wX1 ◦ U = α13 uα31+α11vα32+α12wα33+α13−1 ,
Hence the explicit terms will thus generate a non-negative quadratic form as before. Positivitycan now be established via the following result.
Lemma 4 For any z ≥ 0 and γ, δ ∈ R define the nonlinear function
Q(ω) ≡ ω − γωp − δωq − z .
We assume 0 < q < p < 1, γ 6= 0 and δ 6= 0 (otherwise Q ≡ P). If z > 0 then Q has a uniquereal, strictly positive root, except when γ > 0, δ < 0. In this exceptional case Q has at leastone and at most three strictly positive roots.
If z = 0, then Q has a unique real, strictly positive root, except when δ < 0, in which caseif γ < 0 then the largest non-negative root is ω = 0, while if γ > 0 one root is ω = 0 and theremay be two further strictly positive roots.
Proof Assume z > 0, the conclusions for z = 0 follow suit. All statements assume ω ≥ 0.Note that Q(0) = −z < 0 and Q(ω) → +∞ as ω → +∞ and so Q has at least one strictlypositive root. Note that Q′(ω) = 1 − pγωp−1 − qδωq−1. Define R(ω) ≡ pγωp−1 + qδωq−1.Hence Q′(ω) = 0 if and only if R(ω) = 1.
If γ < 0 and δ < 0, then Q′(ω) > 1. Therefore Q is monotonically increasing and has onlyone strictly positive root.
If γ > 0 and δ > 0, then R′(ω) < 0. Hence R is monotonically decreasing and Q has onlyone stationary point which is a local minimum. Hence Q has only one strictly positive root.
If γ < 0 and δ > 0, then R → 0− as ω → ∞ and R → +∞ as ω → 0+. Further R hasonly one stationary point at
ω∗ =“
q(q−1)|δ|p(p−1)|γ|
”
1p−q
,
which is a local minimum with R(ω∗) < 0. Hence there is a unique stationary point of Q whichis a local minimum and Q has only one strictly positive root.
If γ > 0 and δ < 0, then R → 0+ as ω → ∞ and R → −∞ as ω → 0+. Further R hasonly one stationary point at ω∗ (as defined above). However either: R(ω∗) ≤ 1 in which caseQ as none or one stationary point and thus a unique strictly positive root; or R(ω∗) > 1 inwhich case Q has two stationary points and therefore at most three strictly positive roots. ⊓⊔
Hence to establish positivity for the implicit numerical step above we set: α21 + α11 = p,α31 + α11 = q if p > q, or the other way round if not; ω = un+1; z equal to the explicit terms;
γ equal to the coefficient of uα21+α11
n+1 and δ equal to the coefficient of uα31+α11
n+1 with thesetwo swapped if p < q.
If α11 > 12
then z > 0 and there is only one strictly positive solution to the implicitnumerical step except for when γ > 0, δ < 0. In this latter case since γ, which is the coefficient
of uα21+α11
n+1 , is positive, we can in fact evaluate this term explicitly. We include it in thedefinition of z. The nonlinear function for ω = un+1 is then equivalent to P which has aunique strictly positive root.
If α11 = 12
then possibly z = 0 and there is only one strictly positive solution to theimplicit numerical step except for when δ < 0, in which case the largest non-negative solutionis ω = 0 in the case γ < 0, and in the case γ > 0 we use the same strategy as we did for thecase z > 0. Note that z = 0 with probability zero.
26 Halley, Malham and Wiese
B Integrability of the variance curve model
In the variance curve model, we apply Ito’s lemma to the function defined by f ◦ y =(|u|2m, |v|2m, |w|2m)T for any m ∈ N, and take the expectation. For the third componentwhich is a geometric Brownian motion, m-integrability is known:
Eˆ
|wt|2m
˜
= Eˆ
|w0|2m
˜
exp`
m(2m − 1)η2t´
.
For the second component we have
Eˆ
|vt|2m
˜
= Eˆ
|v0|2m
˜
+ m`
2c + β2(2m − 1)´
Z t
0E
ˆ
wτ v2m−1τ
˜
dτ − 2cm
Z t
0E
ˆ
|vτ |2m
˜
dτ .
Using the Holder and Young inequalities we see that
Eˆ
wtv2m−1t
˜
≤ 12m
Eˆ
|wt|2m
˜
+ 2m−12m
Eˆ
|vt|2m
˜
.
Using this inequality in our estimate for Eˆ
|vt|2m˜
above we have
Eˆ
|vt|2m
˜
≤ Eˆ
|v0|2m
˜
+ Cm
Z t
0E
ˆ
|vτ |2m
˜
+ Eˆ
|wτ |2m
˜
dτ ,
where Cm is a constant that depends on m, c, β and β. Using the Gronwall lemma we see that
Eˆ
|vt|2m
˜
≤
„
Eˆ
|v0|2m
˜
+CmEˆ
|w0|2m
˜
“
exp`
m(2m−1)η2t´
−1”
/`
m(2m−1)η2´
«
exp(Cmt) .
Hence all moments of v are globally finite, and using this, an almost identical argument estab-lishes that all moments of u are also globally finite.
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