Implementation Problems and Solutions in Stochastic Volatility Models of the Hesto

n Type

Jia-Hau Guo and Mao-Wei Hung

The Discontinuity Problem

• The complex logarithm contained in the formula of the Heston model is the primary problem.

• The logarithm of a complex variable is multi-value.

• If one restricts the logarithm to its principal branch (similar to most software packages, such as C++, Gauss, Mathematica, and others), it is necessarily discontinuous at the cut (see Figure 1).

)2)(arg(||lnln nzizz

,)arg( z

n

Heston’s stochastic volatility modelHeston’s stochastic volatility model

)(tdWSVdtrSdS Stttt

)(tdWVdtVdV VtVtt

(1)

(2)

variance the:V

pricestock the:S

t

t

rateinterest free-risk theis :r y volatilitofy volatilitthe:

reversionmean of speed theis:

ncemean variarun -long theis:

V

dtdWdW VS ,Cov

Lewis’ illustrationLewis’ illustration• The Heston model is represented in the transform-based solution

• The type of financial claim is entirely decoupled from the calculation of the Green function

• Different payoffs are then managed through elementary contour integration over functions and contours that depend on the payoff

• The issue is fundamentally related to the Green function component of the solution

02

1

2

12

22

2

2

22

rCV

CV

S

CrS

V

CV

VS

CSV

S

CVS

t

CVV (3)

• The Heston partial differential equation for a European-style claim

The Transform-based SolutionThe Transform-based Solution

• Reduce (3) from two variables to one and entirely separate every (volatility independent) payoff function from the calculation of the Green function.

tT

)()log( tTrSx )(),,(),,( tTreVxWtVSC

V

WV

V

WV

Vx

WV

x

W

x

WV

WVV

)(2

1

2

12

22

2

2

2

(4)

• Define the Fourier transform of ),,( VxW

dxVxWeVG xi ),,(),,( (5)

• The inversion formula is

]Im[

]Im[),,(

2

1),,(

i

i

xi dVGeVxW (6)

]Im[

]Im[),,(

2

1),,(

i

i

xi dVGeix

VxW

The Fundamental Transform The Fundamental Transform

• A solution to (7), which satisfies , is called a fundamental transform. ),,( VG 1)0,,( VG

• Given the fundamental transform, a solution for a particular payoff can be obtained by

Im

Im

~)( ),,()0,,(

2

1),,(

i

i

xitTr dVGVWeetVSC (8)

• Taking the derivative of both sides of (5), and then replacing inside the integral by the right-hand side of (4),

W

V

GViVGiV

V

GV

GVV

)(

2

1

2

1 22

22

(7)

• is the Fourier transform of the payoff function at maturity.)0,,(~

VW

• At maturity, the payoff of a vanilla call option with strike is

The Payoff Function of a Call OptionThe Payoff Function of a Call Option

K

0,KSMax T

• In terms of the logarithmic variables, we have

0,)0,,( KeMaxVxW x

• The Fourier transform of the payoff is

21

]log[

~

)0,,()0,,(

iKdxKeedxVxWeVW i

K

xxixi (9)

• (9) does not exist unless 1Im

Other Common Types of Other Common Types of Payoff Functions and Restrictions Payoff Functions and Restrictions

• Put Option: 21 iK i 0Im

• Digital Call:

Payoff Transform Payoff Restriction

iK i KSH T

0,TSKMax

0Im

• Cash-secured Put: KSMin T , iK i 21 1Im0

The Fundamental Solution The Fundamental Solution

• The fundamental solution of (7) is in the form

VBAeVG ),(),(),,( (10)

• After substituting (10) into (7), a pair of ordinary differential equations

for and is obtained),( A ),( B

BA

iiBBB VV

222

2

1

2

1

(11)

(12)

• The solutions can be expressed by

(13)

)(

)(

2 )(1

1)(),(

d

d

V

V

eg

ediB

1)(

1)(log2)(),(

)(

2

g

egdiA

d

VV

(14)

• The auxiliary functions

)(

)()(

di

dig

V

V

222)( VV iid (15)

The Discontinuity Problem in the Formula

• Figure 2 illustrates the discontinuity problem in the implementation of the fundamental solution.

• In fact, in examples with long maturity periods, discontinuities are certain to arise from the formula if the complex logarithm uses the principal branch only and is not an integer (see Figure 3). 22 V

• One may shift the problem from the complex logarithm to the evaluation of

VBdid

Veg

egVG ),()(

2)(

1)(

1)(),,(

where 2V

• However, discontinuities do not diminish in this way. Note that taking a complex variable to the power givesz

(16)

ierz (17)

• After restricting , the complex plane is cut along the negative real axis. Whenever crosses the negative real axis, the sign of its phase changes from to .

,)arg( zz

ii ee

ii

ii

ee

ee

if

if

(18)

Solutions to the Discontinuity Problem of HestoSolutions to the Discontinuity Problem of Heston’s Formula n’s Formula

• Kruse and Nögel [2005] tracked the complex logarithm function for each step along a discretised integral path to remedy phase jumps.

• Kahl and Jäckel [2005] remedied these discontinuities using the rotation-corrected angle of the phase of a complex variable.

• Shaw [2006] dealt with this problem by replacing the call to the complex logarithm by direct integration of the differential equation.

• Guo and Hung [2007] proposed a simple adjusted formula to solve this discontinuity problem.

• Broadie and Kaya [2004] considered the simulation as a practical alternative for finding option prices.

• Rotation-Corrected Angle

In order to guarantee the continuity of , an rotation-corrected term must be additionally calculated in advance. First, we introduce the notation

),( A

)()()( gig erg

)()()( dd ibad

(19)

(20)

The next step is to have a closer look at

)2)((*)( *

)(1)(1)( mig

ig

gg ererg

where

2

)(int gm

)1)(arg()(* gg

|1)(|)(* grg

(22)

(23)

(24)

(21)

The same calculation is done with

1)( )( deg

1)( )()()( dgd

biag eer

nigd

gder 2)(**

)(

where

2

)()(int dg b

n

1)(arg)( )(* dgd eg

|1)(|)( )(* dgd egr

(25)

(26)

(27)

(28)

Hence, we can compute the logarithm of quite simply as 1)(1)( )( geg d

mnirrg

egggdggd

d

2)()()()(log1)(

1)(log ****

)(

(29)

where is the rotation-corrected angle. mn 2

• Direct Integration

Another way to avoid the branch cut difficulties arising from the choice of the branch of the complex logarithm is to perform direct numerical integration of w.r.t. according to (11). Given , can be obtained by

),( A ),( A),( B

0

),(),( dssBA (30)

After replacing the call to the complex logarithm by direct integration of the differential equation, the complex logarithm can not be a problem any more and the continuity of is guaranteed. ),( A

• Simple Adjusted Formula

The solution is to move into the logarithm of by simply adjusting as follows:

)(exp d ),( A),( A

)(

)(

2 1)(

1)(log2)(),( d

d

VV

eg

egdiA (31)

Because an imaginary component must be added to move a complex number across the negative real axis, the phases of and exist on the same phase interval.

1c c

Therefore, the logarithms of and have the same rotation count number.

1c c

Obviously,

)()()()(

1)(log1)(log1)(

1)(log ddd

d

egegeg

eg

Because the subtraction of 1 does not affect the rotation count of the phase of a complex variable, has the same rotation count number as

1)(log )( deg )()(log deg

Nevertheless, needs no rotation-corrected terms for all levels of Heston parameters because

)()( 1)(log)(log dd egeg

1)(log)(log1)(

)(log

1)(

)(log

)(

)(

ggg

g

eg

egd

d

And, again, and has the same rotation count number. )(log g 1)(log g

Hence, the formula in (31), for , provides a simple solution to the discontinuity problem for Heston’s stochastic volatility model.

),( A

(32)

(33)

• Compared to the rotation-corrected angle method, the simple adjusted-formula method needs no rotation-corrected terms in the already complex integral of Heston’s formula to recover its continuity for all levels of Heston parameters.

Conclusions Conclusions

• Although the direct integration method neither needs the rotation-corrected terms to guarantee the continuity of Heston’s formula, it inevitably introduces the discretization bias into the evaluation of the Green function component of the solution.

• Many steps may be necessary to reduce the bias to an acceptable level and, hence, more computational effort is needed to guarantee that the bias is small enough.

• As a consequence, the direct integration method requires much more computing time than the simple adjusted-formula method.