Complex logarithms and the piecewise constant extension

of the Heston model

Shamim Afshani∗

May 25, 2010

Abstract

The Heston [1993] model features an analytical characteristic function. It is well known that discon-

tinuities can arise in the original formulation of the function, when the complex logarithm therein is

restricted to its principal branch. In recent years, however, an alternative formulation has emerged. For

this alternative, it has been established that discontinuities cannot arise, within the strip of regularity,

when restricting the corresponding complex logarithm in this manner. For a region of this strip, we

extend the analysis by allowing for piecewise constant parameters. Within this semi-analytical frame-

work, our results cater for both European and Forward Starting options.

JEL Classification: C63, G13

1 Introduction

The Heston model allows for semi-analytical valuation of European options. This result relies on the

fact that an analytical expression exists for the Fourier transform of the option value, featuring the con-

ditional characteristic function (CF) for the log underlying XT . The CF is defined as

Et0

(

ei[u−i[α+1]]XT

)

≡ E

(

ei[u−i[α+1]]XT |Xt0 , Vt0

)

(1)

where X and V are the affine state variables in the model. The strip of regularity is specified by the

complex pair (u,−α) where α is restricted to the range (αmin, αmax) such that the CF exists for u ∈ R.

It is well known that discontinuities can arise in Heston’s original formulation of the CF, when the

complex logarithm featured therein is restricted to its principal branch. The issue is noted in Schobel

and Zhu [1999] and a practical solution is provided in Kahl and Jackel [2005]. However, in Lord and

Kahl [2006], the authors observe that an alternative formulation for the CF has emerged in the literature

which appears to be free of any discontinuous behavior, when restricting the complex logarithm in this

∗Global Markets, Standard Bank, Johannesburg - Shamim.Afshani@standardbank.co.za

The results presented were derived as part of the author’s Masters dissertation. Thanks to Dr. Graeme West for supervising the

dissertation. Thanks to Roelof Sheppard, Roger Lord and Garith John Botha for providing useful comments and suggestions.

1

manner. In Gatheral [2006], this is stated as a conjecture. In Lord and Kahl [2006], a proof is presented

for α ∈ (αmin, αmax) subject to certain parameter restrictions. In Albrecher et al. [2007] a proof free of

any parameter restrictions is presented for α ∈ (0, αmax). In Fahrner [2007] the displaced Heston model

is considered and a proof is presented for α = −0.5. This is subject to, effectively, the same parameter

restrictions as those arising in Lord and Kahl [2006]. Finally, in Lord and Kahl [2008] a proof free of any

parameter restrictions is presented for α ∈ (αmin, αmax) making use of results in Lord and Kahl [2006]

and Albrecher et al. [2007].

For α ∈ [−1, 0], we extend the analysis by introducing piecewise constant parameters. By means of

an analytical expression for the conditional joint characteristic function (JCF), semi-analytical formulae

can be obtained for European options, which feature piecewise constant parameters. This result is well

known and is made use of in Andersen and Andreasen [2002], Mikhailov and Nogel [2005] and Elices

[2009]. Semi-analytical valuation of Forward Starting Options has also been considered in Lucic [2003],

Hong [2004], Kruse and Nogel [2005] and Elices [2009]. Having obtained such formulae for European

options within the piecewise constant framework, results can be obtained almost immediately for For-

ward Staring Options. For the range of α specified, our analysis also confirms that complex logarithms,

featured in these formulae, do not yield any discontinuities when restricted to their principal branches.

In Section 2, we present the pricing formula for a European Call option, the JCF and the piecewise

constant CF (PCCF). The cause of any discontinuous behavior in the original formulation is also dis-

cussed. In Section 3, we present our main results, confirm that semi-analytical formulae for Forward

Starting Options do not complicate the analysis and note the application to the displaced Heston model.

In Section 4, we present a numerical example (with piecewise constant parameters) where the original

formulation yields a discontinuous pricing integrand. In Section 5, we conclude.

2 Semi-analytical pricing formulae

For the period (t0, T ], the underlying St and Xt = ln(St), the dynamics of the Heston model are

dXt =

[

r − q − 1

2Vt

]

dt+√

VtdWX

t (2)

dVt = κ[θ − Vt]dt+ ν√

VtdWV

t (3)

dWX

t dWV

t = ρdt (4)

where r, q, κ, θ, ν > 0 and |ρ| < 1.

From the work of Carr and Madan [1999], Lewis [2001] and Lee [2005], we have the semi-analytical,

undiscounted value of a European Call option

1

π

∫ ∞

0

Re

([

e−i[u−iα]k

−[u− iα][u− i[α+ 1]]

]

Et0

(

ei[u−i[α+1]]XT

)

)

du

+ Et0(ST )

[

I[α≤0] −1

2I[α=0]

]

−K

[

I[α≤−1] −1

2I[α=−1]

]

(5)

where k = ln(K) and I is the indicator function. For the complex pair (u,−α), points of singularity exist

at (0, 0) and (0, 1).

2

An analytical expression for the CF allows us to evaluate (5) by means of a single integration. The shape

of the integrand is, however, sensitive to the value of α passed through it. For a given strike, maturity

pair, significant peaks or oscillations can arise over the valid range of α. Referring to the ’Lewis-Lipton’

formula of Lewis [2001] and Lipton [2002] and the investigation in Schoutens et al. [2005], α is specified

as −0.5 and 0.75, respectively. In Lee [2005] and Lord and Kahl [2007], however, the task of identifying

an optimal α is tackled. Specifically, in Lord and Kahl [2007], an optimal choice is advocated for each

strike, maturity pair and is defined as that value, in the range (αmin, αmax), which minimizes the absolute

value of the integrand at its maximum point. We must highlight that our results cannot accommodate

for this optimal approach, in its entirety.

2.1 The conditional joint characteristic function

To introduce piecewise constant parameters into the valuation formula, we must make use of the JCF

Et0

(

eizXT+izvVT

)

(6)

where

z = u− iζ (7)

zv ∈ C (8)

and, for the sake of convenience, we specify ζ = α+ 1.

Proposition 1. For the period (t0, T ], the JCF has the affine form

exp (izXt0 +D(τ, iz, izv)Vt0 + C(τ, iz, izv)) (9)

where

D(τ, iz, izv) =e−γ(iz)τ

[

b(iz)− γ(iz)− ν2izv]

[b(iz) + γ(iz)]−[

b(iz) + γ(iz)− ν2izv]

[b(iz)− γ(iz)]

e−γ(iz)τ [b(iz)− γ(iz)− ν2izv] ν2 − [b(iz) + γ(iz)− ν2izv] ν2

(10)

C(τ, iz, izv) = iz[r − q]τ +κθ

ν2[b(iz)− γ(iz)] τ − 2κθ

ν2log (ψ(τ, iz, izv)) (11)

ψ(τ, iz, izv) =

[

b(iz) + γ(iz)− ν2izv]

− e−γ(iz)τ[

b(iz)− γ(iz)− ν2izv]

2γ(iz)(12)

γ(iz) =√

b(iz)2 − ν2iz[iz − 1] (13)

b(iz) = κ− ρνiz (14)

with

limγ(1)=|b(1)|→0

D(τ, 1, izv) =izv

1− izv12ν

2τ(15)

limγ(1)=|b(1)|→0

ψ(τ, 1, izv) = 1− izv1

2ν2τ (16)

Furthermore, −Im (z) = ζ ∈ [0, 1] ∧ Im (zv) ≥ 0 are sufficient conditions for the existance of the JCF.

An outline of the proof is presented in Appendix A.

3

2.2 The principal branch of the complex logarithm

Regarding (11), the JCF features a complex logarithm in the form

exp

(

2κθ

ν2log (ψ(τ, iz, izv))

)

= exp

(

2κθ

ν2[ln (|ψ(τ, iz, izv)|) + iArg (ψ(τ, iz, izv))]

)

exp

(

i2πn2κθ

ν2

)

(17)

where Arg (ψ(τ, iz, izv)) ∈ (−π, π], n ∈ Z and a branch cut exists along (−∞, 0]. When evaluated within

a software package, such as MatLab, the multi-valued complex logarithm is restricted to its principal

branch, obtained by setting n = 0.

As effectively stated in Lee [2005] and Albrecher et al. [2007], the JCF is an even function of the complex

square root γ(iz) so replacing γ(iz) in Proposition 1 with −γ(iz) would yield an algebraically equiva-

lent formulation. However, as effectively pointed out in Lord and Kahl [2008], ψ(τ, iz, izv) exp (γ(iz)τ)

would replace ψ(τ, iz, izv) in the complex logarithm. This is the defining difference between Heston’s

formulation and the alternative.

The possibility of complex discontinuities arises because of the scaling factor 2κθν2 . If n 2κθ

ν2 ∈ Z for all

n then exp(

i2πn 2κθν2

)

= 1 and (17) is a single-valued expression. Otherwise, as we evaluate (17) along

the integration path, we must keep track of the function ψ(τ, iz, izv) in case it crosses the branch cut to

determine the value of n, at each point along the path, that ensures (17) is a continuous function. It has

been proved, most notably in Lord and Kahl [2008], that ψ(τ, iz, 0) 6∈ (−∞, 0]. Therefore, discontinuities

will not arise in the CF when restricting the complex logarithm to its principal branch. In section 3, we

address this issue for the PCCF, focussing on the function ψ(τ, iz, izv).

2.3 The conditional piecewise constant characteristic function

Proposition 2. For the period (t0, tn], divided into n partitions, the PCCF has the affine form

Et0

(

Et1

(

...Etn−1

(

eizXtn

)

...))

= exp

(

izXt0 +D(τ1, iz,Dτ2)Vt0 +

n∑

m=1

C(τm, iz,Dτm+1)

)

(18)

where τm = tm − tm−1 for partition m and

Dτm := D(τm, iz,Dτm+1) (19)

Dτn+1:= 0 (20)

Proof. Expression (9) yields the result by means of induction where zv = −iDτm+1for partition m.

For T = tn, we replace the CF in (5) with the PCCF in (18). The constant model parameters in (2)-(4)

are replaced by the time-dependent parameters rt, qt, κt, θt, νt and ρt. For any parameter x with xm the

constant parameter, seen at t0, that applies over the period (tm−1, tm]

xt =

n∑

m=1

xmI[tm−1<t≤tm] (21)

The expression in (18) can be determined entirely from the analytical expression for the JCF in Proposi-

tion 1, as we need only to evaluate the recursive expressions C(τm, iz,Dτm+1) and D(τm, iz,Dτm+1

).

4

3 The principal branch in the extended model

3.1 The principal branch

The PCCF features a series of complex logarithms

−n∑

m=1

2κmθmν2m

log(

ψ(τm, iz,Dτm+1))

(22)

To confirm that each complex logarithm in (22) can be restricted to its principal branch, without intro-

ducing discontinuities into the PCCF, we make use of Propositions 3 and 4 to prove Theorem 1. The

final result follows in Corollary 1.

Proposition 3. For ζ ∈ [0, 1]

Re (γ(iz)) ≥ |Re (b(iz)) | (23)

Equality holds when u = 0 ∧ ζ ∈ {0, 1}.

The proof is presented in Appendix B.

Proposition 4. For ζ ∈ [0, 1] ∧ Im (zv) ≥ 0

Re (D (τ, iz, izv)) ≤ 0 (24)

Equality holds when u = 0 ∧ ζ ∈ {0, 1} ∧ zv = 0.

The proof is presented in Appendix C.

Theorem 1. For ζ ∈ [0, 1] ∧ Im (zv) ≥ 0

ψ(τ, iz, izv) 6∈ (−∞, 0] (25)

Proof. Regarding the origin for ζ ∈ [0, 1], consider that from (49), (46) and Proposition 4

Re (C(τ, iz, izv)) = ζ[r − q]τ + κθ

∫ τ

0

Re (D(s, iz, izv)) ds ≤ [r − q]τ (26)

From (11), Re (C(τ, iz, izv)) features − 2κθν2 ln

(∣

∣ψ(τ, iz, izv)∣

∣

)

so if ψ(τ, iz, izv) reaches the origin,

Re (C(τ, iz, izv)) will explode to +∞ (the remaining terms in Re (C(τ, iz, izv)) all exist at this point). This

contradicts (26) and so confirms that the origin cannot be attained.

Regarding the negative real line and making use of Proposition 4, in particular, we consider two cases:

Case 1: u = 0 ∧ ζ ∈ {0, 1} ∧ zv = 0

In this case iz = ζ. Making use of Proposition 3 it can be shown that

Re (ψ(τ, ζ, 0)) = exp (b(ζ)τ) for b(ζ) < 0 (27)

= 1 for b(ζ) ≥ 0 (28)

where the result for b(1) = 0 follows from (16).

5

Case 2: u 6= 0 ∨ ζ 6∈ {0, 1} ∨ zv 6= 0

From Proposition 4 we have Re (D(τ, iz, izv)) < 0 and from Proposition 1, we can write

D(τ, iz, izv) =izv

ψ(τ, iz, izv)+

[b(iz) + γ(iz)]

ν2

[

1− 1

ψ(τ, iz, izv)

]

(29)

Suppose that for some ϕ ∈ (0,∞), ψ (τ, iz, izv) = −ϕ. This yields

Re (D(τ, iz, izv)) =Im (zv)

ϕ+

[Re (b(iz)) + Re (γ(iz))]

ν2

[

1 +1

ϕ

]

(30)

Making use of Proposition 3 and the restriction Im (zv) ≥ 0, ϕ > 0 implies that (30) is non-negative,

yielding a contradiction.

Together, cases 1 and 2 confirm that the negative real line cannot be attained, completing the proof.

Corollary 1. For ζ ∈ [0, 1] and m = 1, ..., n

ψ(τm, iz,Dτm+1) 6∈ (−∞, 0] (31)

Proof. Referring to Proposition 2, we set zv = −iDτm+1for the mth partition and so

Im (zv) = −Re(

Dτm+1

)

(32)

For the nth partition, Dτn+1:= 0 so Proposition 4 and (32) yield, by means of induction,

Re(

D(

τm, iz,Dτm+1

))

≤ 0 (33)

where ζ ∈ [0, 1]. Hence, the conditions of Theorem 1 are satisfied for each partition.

3.2 Forward Starting Options

For t0 < ts < tn, we consider Forward Starting Options with what we refer to as the % payoff(

Stn

Sts

−K)+

and the $ payoff Sts

(

Stn

Sts

−K)+

. Within the same framework as that resulting in (5), semi-analytical

pricing formulae can be obtained for Forward Starting Options which feature the forward characteristic

function (FCF). Essentially, this would seem to have been considered first in Hong [2004].

Proposition 5. The semi-analytical, undiscounted value of a % or $ Forward Starting Options is expressed as

1

π

∫ ∞

0

Re

([

e−i[u−iα]k

−[u− iα][u− i[α+ 1]]

]

Et0

(

ei[u−i[α+1]][Xtn−Xts

]+I$Xts

)

)

du

+ Et0

(

eXtn−Xts

+I$Xts

)

[

I[α≤0] −1

2I[α=0]

]

−KEt0

(

eI$Xts

)

[

I[α≤−1] −1

2I[α=−1]

]

(34)

where I$ is the indicator function equal to 1 when valuing a $ option.

Proof. The result follows from use of the tower property and the fact that, at ts, both options are Euro-

pean and so can be valued by means of (5).

6

The result features the FCF Et0

(

eiz[Xtn−Xts

]+I$Xts

)

. Again, an analytical expression for the JCF allows

us to obtain an analytical expression for the FCF. Furthermore, the piecewise constant FCF (PCFCF) can

be obtained in exactly the same manner as that described for the PCCF.

Proposition 6. For the period (t0, tn], divided into n partitions and 0 < s < n, the PCFCF has the affine form

Et0

(

...Ets−1

(

e−izXts+I$XtsE

ts

(

...Etn−1

(

eizXtn

)

...))

...)

= exp

(

I$Xt0 +D(τ1, I$, Dτ2)Vt0 +n∑

m=1

C(τm, iz, Dτm+1)

)

(35)

where Dτm := D(τm, iz, Dτm+1), Dτn+1

:= 0 and for each partition m = 1, ..., n

z = z if m > s (36)

= −iI$ if m ≤ s (37)

Proof. The expression in (9) yields the result by means of induction.

From Proposition 6, the PCFCF has exactly the same form as the PCCF, for partitions m > s while for

partitions m ≤ s, z is a special case of z = u− iζ where u = 0 ∧ ζ ∈ {0, 1}. Hence, the results of Section

3.1 can be applied to the complex logarithms in the PCFCF.

3.3 The displaced Heston model

In Fahrner [2007], the issue at hand is considered for the dynamics

dSt = σ [βSt + [1− β]L]√

VtdWX

t (38)

dVt = κ[θ − Vt]dt+ ν√

VtdWV

t (39)

dWX

t dWV

t = ρdt (40)

where 0 < β ≤ 1, σ > 0 and L > 0.

For ζ = 0.5 and ρ < 2κνσβ

, it is confirmed that discontinuities will not arise when restricting the complex

logarithm, in the corresponding CF, to its principal branch. However, for Xt = ln (βSt + [1− β]L) and

Vt = σ2β2Vt, Ito’s formula yields

dXt = −1

2Vtdt+

√

VtdWX

t (41)

dVt = κ[θ − Vt]dt+ ν√

VtdWV

t (42)

where θ = σ2β2θ > 0 and ν = σβν > 0. Hence, the results of Section 3.1 can be applied directly to the

displaced Heston model, for ζ ∈ [0, 1], without introducing any parameter restrictions.

When allowing for piecewise constant parameters, it is worth noting that we restrict β to a constant since

(ST −K)+

=1

β

(

eXT − ek)+

(43)

where k = ln (βK + [1− β]L) so β will not feature exclusively in the PCCF.

7

4 A numerical example

For the initial values St0 = 1, Vt0 = 0.1 and the time points t0 = 0, t1 = 1, t2 = 3 and t3 = 5, we specify

rt = 0, qt = 0, θt = 0.1, νt = 1, ρt = −0.9 and κt =∑3

m=1 κmI[tm−1<t≤tm] where κ1 = 4, κ2 = 2 and

κ3 = 1. Setting νt = 0.2 and ρt = −0.3 would yield the parameter set made use of in Mikhailov and

Nogel [2005] Table 11. We can confirm the analytical results presented therein but do not consider the

specification any further as discontinuities do not arise when making use of the original formulation.

We value European Call options, maturing at t3, with ζ = 0.5. The pricing formula is evaluated by

means of MatLab’s adaptive quadrature routine quadl with a tolerance level of 10−12. We transform the

domain of integration from u ∈ [0,∞) to x ∈ [0, π2 ] where u(x) = 12 tan(x) and compare our results to a

finite-difference (FD) implementation.2 In Table 1, we present semi-analytical (SA) results for both the

original and the alternative formulation, as well as results from the FD implementation. We also present

the absolute error between the respective SA values and the FD value as a percentage of the FD value.

For out the money cases, in particular, use of the original formulation results in significant differences.

Strike SA value SA value FD value Absolute error Absolute error

Original PCCF Alternative PCCF Original PCCF Alternative PCCF

0.50 0.530659 0.548724 0.549383 3.41% 0.12%

0.75 0.364044 0.370421 0.371007 1.88% 0.16%

1.00 0.256576 0.230355 0.230393 11.36% 0.02%

1.25 0.182311 0.129324 0.128985 41.34% 0.26%

1.50 0.122790 0.063974 0.063377 93.75% 0.94%

Table 1: European Call option values

In Figure 1 we present the pricing integrand for both the original (a) and the alternative (b) formulation,

given a strike of 1.5 and restricting all complex logarithms to their principal branches.

0 1.57−2

1.5

x

inte

gra

nd

(a)

0 1.57−2

1.5(b)

x

inte

gra

nd

Figure 1: Integrand in the semi-analytical formula

1We confirmed the partitioning of the 5 year maturity with the second author as this is not explicitly stated in the article.2Thanks to Garith John Botha for providing the finite-difference implementation in MatLab.

8

5 Conclusion

We have shown that complex logarithms, featured in the alternative formulation of the PCCF and the

PCFCF, can be restricted to their principal branches without introducing any discontinuities, for α ∈[−1, 0] or equivalently ζ ∈ [0, 1]. This specifies a region of the strip of regularity that is useful as it is

always contained within the strip and includes the ’Lewis-Lipton’ specification.

A Proposition 1

Proof. From the Feynman-Kac Theorem it follows that the JCF Et

(

eizXT +izvVT

)

=: φ(Xt, Vt) must satisfy

the partial differential equation

∂φ

∂t+

[

r − q − 1

2Vt

]

∂φ

∂Xt

+1

2Vt∂2φ

∂X2t

+ κ[θ − Vt]∂φ

∂Vt+

1

2ν2Vt

∂2φ

∂V 2t

+ ρνVt∂2φ

∂Xt∂Vt= 0 (44)

We assume the solution has the form

exp (izXt +D(T − t, iz, izv)Vt + C(T − t, iz, izv)) (45)

and so must determine analytical expressions for C(T − t, iz, izv) and D(T − t, iz, izv) subject to

C(0, iz, izv) = 0 (46)

D(0, iz, izv) = izv (47)

By substituting (45) into (44) and grouping terms, we obtain

∂

∂sD(s, iz, izv) =

1

2ν2D(s, iz, izv)

2 − [κ− ρνiz]D(s, iz, izv) +1

2iz[iz − 1] (48)

∂

∂sC(s, iz, izv) = iz[r − q] + κθD(s, iz, izv) (49)

where s = T − t. From this, we can obtain analytical expressions.

Having obtained these expressions, we see that γ(1) = |b(1)| = |κ − ρν| and so an indeterminant form

arises in (10) and (12) when κ = ρν. L’Hopital’s rule yields the limits in (15) and (16).

From Jensen’s inequality and assuming Im (zv) ≥ 0

∣

∣Et0

(

eizXT+izvVT

)∣

∣ ≤ Et0

(∣

∣eizXT+izvVT

∣

∣

)

= Et0

(

eζXT−Im(zv)VT

)

(50)

≤ Et0

(

eζXT

)

(51)

where we note −Im (z) = ζ. The moment generating function (MGF) for XT in (51) is a strictly convex

function of ζ 3 where ζ = 1 yields the forward price of ST . Hence, the MGF exists at least for ζ ∈[0, 1].

3see Lee [2005] Theorem 4.1 and Lukacs [1960] Theorem 7.1.4.

9

B Proposition 3

Proof. For the complex square root γ(iz), we can write

Re (γ(iz)) =1√2

√

√

Re (Γ(iz))2+ Im (Γ(iz))

2+ Re (Γ(iz)) (52)

where

Re (Γ(iz)) = Re (b(iz))2 − ν2[ζ − 1]ζ + u2ν2[1 − ρ2] (53)

For ζ ∈ [0, 1], we have Re (Γ(iz)) ≥ 0 so

Re (γ(iz)) ≥√

Re (Γ(iz)) ≥ |Re (b(iz)) | (54)

From (13) and (14), γ(ζ) = |b(ζ)| for ζ ∈ {0, 1}.

C Proposition 4

Proof. The inequality in (50) must hold for any positive Vt0 . Making use of (9), we obtain

Re (D(τ, iz, izv)) ≤ D(τ, ζ,−Im (zv)) (55)

where equality holds when u = 0 ∧ Re (zv) = 0.

From (14), b(ζ) ∈ R and from (13), γ(ζ) ∈ R when ζ ∈ [0, 1]. From Proposition 1, we can write

D(τ, ζ,−Im (zv)) =Im (zv)

[

b(ζ)− γ(ζ)− [γ(ζ) + b(ζ)] e−γ(ζ)τ]

−[

γ(ζ)2 − b(ζ)2] [

1− e−γ(ζ)τ]

1ν2

Im (zv) ν2[

1− e−γ(ζ)τ]

+ [γ(ζ) + b(ζ)] + e−γ(ζ)τ [γ(ζ)− b(ζ)]

(56)

From (56), Proposition 3 and (15), we see that for ζ ∈ [0, 1] ∧ Im (zv) ≥ 0

D(τ, ζ,−Im (zv)) ≤ 0 (57)

where equality holds when ζ ∈ {0, 1} ∧ Im (zv) = 0. Together with (55), this yields the final result.

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