Positive stochastic volatility simulation arXiv:0802 ...simonm/psdesarxiv.pdf · volatility in the Heston model is a non-central chi-squared distribution (see Cox, In-gersoll and

arX

iv:0

802.

4411

v1 [

mat

h.PR

] 2

9 Fe

b 20

08

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]

http://arXiv.org/abs/0802.4411v1

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


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


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 ,


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.


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


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.


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

«

,


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 .


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


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.


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.


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

!

.


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 .


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.


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


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.


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.


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.


−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

)


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.


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.


−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

)


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

)


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.


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.



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 .


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.


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.

References

1. C. Albanese and A. Kuznetsov, Transformations of Markov processes and classificationscheme for solvable driftless diffusions, arXiv:0710.1596v1 8 Oct 2007.

2. A. Alfonsi, On the discretization schemes for the CIR (and Bessel squared) processes,Monte Carlo Methods and Applications, 11(4) (2005), pp. 355–384.

3. A. Alfonsi, A second-order discretization scheme for the CIR process: application to theHeston model, Preprint, 2007.

4. L. Andersen, Efficient simulation of the Heston stochastic volatility model, preprint, 2007.5. L.B.G. Andersen and V.V. Piterbarg, Moment explosions in stochastic volatility mod-

els, Finance and Stochastics, 11(1) (2007), pp. 29–50.6. R. Azencott, Formule de Taylor stochastique et developpement asymptotique d’integrales

de Feynman, Seminar on Probability XVI, Lecture Notes in Math., 921 (1982), Springer,pp. 237–285.

7. F. Baudoin, An introduction to the geometry of stochastic flows, Imperial College Press,2004.

8. G. Ben Arous, Flots et series de Taylor stochastiques, Probab. Theory Related Fields,81 (1989), pp. 29–77.

9. A. Berkaoui, M. Bossy and A. Diop, Euler scheme for SDEs with non-Lipschitz diffu-sion coefficient: strong convergence, INRIA Rapport de recherche n◦ 5637, 2005.

10. M. Bossy and A. Diop, An efficient discretization scheme for one dimensional SDEswith a diffusion coefficient function of the form |x|α, α ∈ [1/2, 1), INRIA Rapport derecherche n◦ 5396, 2007.

11. M. Broadie and O. Kaya, Exact simulation of stochastic volatility and other affine jumpdiffusion processes, Operations Research, 54(2) (2006), pp. 217–231.

12. E. Buckwar, R. Horvath–Bokor and R. Winkler Asymptotic mean-square stabilityof two-step methods for stochastic ordinary differential equations, BIT Numerical Math-ematics, 46(2) (2006), pp. 261–282.

http://arXiv.org/abs/0710.1596


13. H. Buhler, Volatility markets: Consistent modeling, hedging and practical implementa-tion, PhD Thesis, Berlin, 2006.

14. H. Buhler, Consistent variance curve models, Finance and Stoch., 10 (2006), pp. 178–203.15. P.M. Burrage and K. Burrage, A variable stepsize implementation for stochastic dif-

ferential equations, SIAM J. Sci. Comput., 24(3) (2002), pp. 848–864.16. K. Burrage, P.M. Burrage and T. Tian, Numerical methods for strong solutions

of stochastic differential equations: an overview, Proc. Roy. Soc. Lond. A, 460 (2004),pp. 373–402.

17. P.M. Burrage, R. Herdiana and K. Burrage, Adaptive stepsize based on control theoryfor stochastic differential equations, Journal of Computational and Applied Mathematics,170 (2004), pp. 317–336.

18. F. Castell, Asymptotic expansion of stochastic flows, Probab. Theory Related Fields, 96(1993), pp. 225–239.

19. F. Castell and J. Gaines, An efficient approximation method for stochastic differentialequations by means of the exponential Lie series, Math. Comput. Simulation, 38 (1995),pp. 13–19.

20. F. Castell and J. Gaines, The ordinary differential equation approach to asymptoticallyefficient schemes for solution of stochastic differential equations, Ann. Inst. H. PoincareProbab. Statist., 32(2) (1996), pp. 231–250.

21. V. Chickarmane, A. Ray, H. M. Sauro and A. Nadim, A model for p53 dynamicstriggered by DNA damage, SIAM J. Applied Dynamical Systems, 6(1) (2007), pp. 61–78.

22. J. M. C. Clark and R. J. Cameron, The maximum rate of convergence of discreteapproximations for stochastic differential equations, in Lecture Notes in Control and In-formation Sciences, Vol. 25, 1980, pp. 162–171.

23. J.C. Cox, J.E. Ingersoll and S.A. Ross, A theory of the term structure of interestrates, Econometrica, 53(2) (1985), pp. 385–407.

24. G. Deelstra and F. Delbaen, Convergence of discretized stochastic (interest rate) pro-cesses with stochastic drift term, Appl. Stochastic Models Data Anal., 14 (1998), pp. 77–84.

25. W. Feller, Two singular diffusion problems, Annals of Mathematics, 54 (1951), pp. 173–182.

26. J. G. Gaines and T. J. Lyons, Variable step size control in the numerical solution ofstochastic differential equations, SIAM J. Appl. Math., 57(5) (1997), pp. 1455–1484.

27. H. Gilsing and T. Shardlow, SDELab: stochastic differential equations with MATLAB,MIMS EPrint: 2006.1, ISSN 1749-9097.

28. P. Glasserman, Monte Carlo methods in financial engineering, Applications of Mathe-matics, Stochastic Modelling and Applied Probability, 53, Springer, 2004.

29. I. Gyongy and N. Krylov, On splitting-up method and stochastic differential equations,Ann. Probab., 31 (2003), pp. 564–591.

30. W. Halley, Positivity preserving numerical schemes for stochastic differential equations,Masters Thesis, Heriot–Watt University, 2007.

31. S.L. Heston, A closed-form solution for options with stochastic volatility with applicationsto bond and currency options, Review of Financial Studies, 6(2) (1993), pp. 327–343.

32. D. Hobson, Stochastic volatility models, correlation, and the q-optimal measure, Mathe-matical Finance, 14 (2004), pp. 537-556.

33. E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Springer Seriesin Computational Mathematics, 2002.

34. D.J. Higham and X. Mao, Convergence of Monte–Carlo simulations involving the mean-reverting square root process, Journal of Computational Finance, 8 (2005), pp. 35–61.

35. D.J. Higham, X. Mao and A.M. Stuart, Strong Convergence of Numerical Methods forNonlinear Stochastic Differential Equations, SIAM J. Num. Anal, 40 (2002), pp. 1041–1063.

36. N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes,North-Holland and Kodansha, 1981.

37. C. Kahl, Positive numerical integration of stochastic differential equations, Diploma The-sis, Wuppertal, 2004.

38. C. Kahl, M. Gunther and T. Roßberg, Structure preserving stochastic integrationschemes in interest rate derivative modeling, Preprint BUW-AMNA 04/10, 2004.

39. C. Kahl and P. Jackel, Fast strong Monte–Carlo schemes for stochastic volatility mod-els, preprint, 2006.

40. C. Kahl and P. Jackel, Not-so-complex logarithms in the Heston Model preprint, 2006.41. C. Kahl and H. Schurz, Balanced Milstein methods for ordinary SDEs, preprint, 2006.


42. I. Karatzas and S.E. Shreve, Brownian motion and stochastic calculus, Graduate textsin mathematics, 2nd Edition, Springer, 1988.

43. S. Karlin and H.M. Taylor, A second course in stochastic processes, Academic Press,1981.

44. P. E. Kloeden and E. Platen, Numerical solution of stochastic differential equations,Springer, 1999.

45. H. Kunita, On the representation of solutions of stochastic differential equations, LectureNotes in Math. 784, Springer–Verlag, 1980, pp. 282–304.

46. A. Lander, Positivity preserving numerical procedures for stochastic volatility models,MSc Thesis, Heriot–Watt University, 2006.

47. G. Lord, S. J. A. Malham and A. Wiese, Efficient strong integrators for linear stochasticsystems, submitted to SINUM, 2006.

48. R. Lord, R. Koekkoek and D. van Dijk, A comparison of biased simulation schemesfor stochastic volatility models, Tinbergen Institute Discussion Paper TI2006-046/4, 2006.

49. T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14(2)(1998), pp. 215–310.

50. T. Lyons and Z. Qian, System control and rough paths, Oxford University Press, 2002.51. W. Magnus, On the exponential solution of differential equations for a linear operator,

Comm. Pure Appl. Math., 7 (1954), pp. 649–673.52. P. Malliavin, Stochastic analysis, Grundlehren der mathematischen Wissenschaften 313,

Springer, 1997.53. G. N. Milstein, Numerical integration of stochastic differential equations, Mathematics

and its applications, Kluwer Academic Publishers, 1994.54. G. N. Milstein, E. Platen and H. Schurz, Balanced implicit methods for stiff stochastic

systems, SIAM J. Numer. Anal., 35(3) (1998), pp. 1010-1019.55. T. Misawa, A Lie algebraic approach to numerical integration of stochastic differential

equations, SIAM J. Sci. Comput., 23(3) (2001), pp. 866–890.56. E. Moro and H. Schurz, Boundary preserving semianalytical numerical algorithms for

stochastic differential equations, SIAM J. Sci. Comput., 29(4) (2007), pp. 1525–1549.57. M. Ninomiya and S. Ninomiya, A new weak approximation scheme of stochastic differ-

ential equations by using the Runge–Kutta method, arXiv:0709.2434v2, 5 Jan 2008.58. S. Ninomiya and N. Victoir, Weak approximation of stochastic differential equations

and application to derivative pricing, arXiv:math/0605361v3, 1 Oct 2006.59. M. Overhaus, A. Bermudez, H. Buhler, A. Ferraris, C. Jordinson, A. Lamnouar,

A. Puthu Recent developments in mathematical finance: a practitioner’s point of viewDMV Jahresbericht, 2006

60. D. Revuz and M. Yor, Continuous Martingales and Brownian motion, Springer-Verlag,1991.

61. H. Schurz, A brief introduction to numerical analysis of (ordinary) stochastic differentialequations without tears, in Handbook of Stochastic Analysis and Applications, V. Laksh-mikantham and D. Kannan, eds., Marcel Dekker, 2002, pp. 237–359.

62. G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer.Anal., 5(3) (1968), pp. 506–517.

63. R. S. Strichartz, The Campbell–Baker–Hausdorff–Dynkin formula and solutions of dif-ferential equations, Journal of Functional Analysis, 72 (1987), pp. 320–345.

64. H. J. Sussmann, Product expansions of exponential Lie series and the discretization ofstochastic differential equations, in Stochastic Differential Systems, Stochastic ControlTheory, and Applications, W. Fleming and J. Lions, eds., Springer IMA Series, Vol. 10,1988, pp. 563–582.

65. M. Wiktorsson, Joint characteristic function and simultaneous simulation of iterated Itointegrals for multiple independent Brownian motions, Ann. Appl. Probab., 11(2) (2001),pp. 470–487.

66. T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differentialequations, J. Math. Kyoto Univ., 11 (1971), pp. 155–167.

67. Y. Yamato, Stochastic differential equations and nilpotent Lie algebras, Z. Wahrsch. Verw.Gebiete, 47(2) (1979), pp,. 213–229.

http://arXiv.org/abs/0709.2434

http://arXiv.org/abs/math/0605361

Documents

Positive stochastic volatility simulation arXiv:0802 ...simonm/psdesarxiv.pdf · volatility in the Heston model is a non-central chi-squared distribution (see Cox, In-gersoll and