A Note on the Discontinuity Problem in Heston's Stochastic Volatility

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Applied Mathematical Finance

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A Note on the Discontinuity Problem in Heston's Stochastic Volatility Model

a School of Business, Soochow University, Taipei, Taiwan b College of Management, National Taiwan

University, Taipei, Taiwan

Online Publication Date: 01 September 2007

To cite this Article Guo, Jia-Hau and Hung, Mao-Wei(2007)'A Note on the Discontinuity Problem in Heston's Stochastic Volatility Model',Applied Mathematical Finance,14:4,339 — 345

To link to this Article: DOI: 10.1080/13504860601170534 URL: http://dx.doi.org/10.1080/13504860601170534

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A Note on the Discontinuity Problem in

Heston’s Stochastic Volatility Model

JIA-HAU GUO* & MAO-WEI HUNG**

*School of Business, Soochow University, Taipei, Taiwan, **College of Management, National Taiwan University, Taipei, Taiwan

(Received 20 June 2006; in revised form 23 November 2006)

ABSTRACT Although quasi-analytic formulas can be derived for European-style financial claims in Heston’s stochastic volatility model, the inverse Fourier integration involved makes the calculation somewhat complicated. This challenge has puzzled practitioners for many years because most implementations of Heston’s formula are not robust, even for customarily-used Heston parameters, as time to maturity is increased. In this article, a simplified approach is proposed to solve the numerical instability problem inherent to the fundamental solution of the Heston model. Specifically, the solution does not require any additional function or a particular mechanism for most software packages or programming library routines to correctly evaluate Heston’s analytics.

KEYWORDS: Stochastic volatility model, Heston, discountinuity, options

Introduction

The objective of this paper is to contribute to the methodology debate on the discontinuity problem arising from the evaluation of Heston’s formula. The literature on asset pricing using Heston’s stochastic volatility framework has expanded dramatically over the last decade to successfully describe the empirical leptokurtic distributions of asset price returns. However, implementations of these formulas are not as straightforward as they may appear and most numerical procedures are not reported in detail (Scho¨bel and Zhu, 1999; Lee, 2005). The complex logarithm contained in the solution of the Heston model is one of the primary problems. This paper offers a simple and efficient solution to the discontinuity problem of the Heston approach in the Lewis representation in which the type of financial claim is entirely decoupled from the calculation of the fundamental Green’s function. Once the discontinuity problem in the Green’s function component of the solution has been solved, the robustness of formulae for all European-style financial claims in Heston’s model can be assured.

Correspondence Address: Professor Mao-Wei Hung, College of Management, National Taiwan University, No. 1, Section 4, Roosevelt Road, Taipei, Taiwan. Email: [email protected]

Vol. 14, No. 4, 339–345, September 2007

1350-486X Print/1466-4313 Online/07/040339–7 # 2007 Taylor & Francis DOI: 10.1080/13504860601170534

If the complex, multi-valued logarithm is restricted to the principal branch only (similarly to most software packages such as C++, Gauss, Mathematica and others), discontinuities are necessarily incurred at the cut of the complex logarithm along the integration path, resulting in an incorrect value of Heston’s formula (Mikhailov and No¨gel, 2003). In the literature, various authors propose the idea to carefully keep track of the branch by monitoring the complex logarithm function for each step along a discretized integral path to remedy phase jumps. As described in Kruse and No¨gel (2005), if the imaginary value of the complex logarithm for one step differs from the previous one by more than 2p, the jump of 2p is added or subtracted to recover the continuity of phase. However, using this approach, the already complex integrals of Heston’s formula may become too complicated in practice (Broadie and Kaya, 2004). To make matters worse, discontinuities arise quite naturally for customarily-used Heston parameters simply as time to maturity is increased, thereby illustrating the importance of the correct treatment of phase jumps of Heston’s formula. Our simple solution was inspired by Kahl and Ja¨ckel (2005); however, they remedy discontinuities using the rotation-corrected angle of the phase of a complex variable, which is not necessary in our approach. From a computational and convenience point of view, our simplified approach can be implemented easily and is thereby very suitable for practical application.

The rest of this article proceeds as follows: Section 2 provides a brief description of the numerically-induced discontinuity problem in the fundamental solution of the Heston model. Section 3 introduces the simplified approach and gives some numerical examples to illustrate its usefulness. Section 4 concludes the paper.

The Discontinuity Problem in the Fundamental Solution of the Heston Model Heston’s stochastic volatility model is based on the system of stochastic differential equations, which represent the dynamics of the stock price and the variance processes under the risk-neutral measure

dSt~rStdtzSt ffiffiffiffiffi Vt p dWS(t) ð1Þ dVt~k h{Vð tÞdtzsV ffiffiffiffiffi Vt p dWV(t) ð2Þ

Stand Vtdenote the stock price and its variance at time, t, respectively; r is the

risk-free interest rate. The variance evolves according to a square-root process: h is the long-run mean variance, k is the speed of mean reversion, and sV is the

parameter which controls the volatility of the variance process. WSand WVare two

standard processes of Brownian motion having the correlation, r. The Heston partial differential equation for a European-style claim C(St, Vt, t) with expiration,

T, is LC Lt z 1 2VS 2L 2_C LS2zsVrSV L2C LSLVz 1 2s 2 VV L2C LV2zrS LC LSzk h{Vð Þ LC LV{rC~0 ð3Þ 340 J.-H. Guo and M.-W. Hung

with the terminal condition, C(St, Vt, t5T), at expiration. The fundamental

transform method proposed by Lewis (2000) is used to obtain the solution. The fundamental Green’s function, G(w, V, t), adhering to the Heston specification must satisfy LG Lt~ 1 2s 2 VV L2G LV2{ 1 2V (v 2_{{iv)Gz k h{Vð Þ{ivs} VrV ð ÞLG LV ð4Þ

subject to the condition, G(w, V, 0)51, where t5T2t. Given Green’s function, CtðSt,Vt,tÞ~ 1 2pe {r(T{t)ðiIm w ½
z? iIm w½
{? e{ivx W (w,V ,0)G(w,V ,T {t)dw,~ ð5Þ where x5log(S)+r(T2t) and W w,V ,0~ð Þ is the Fourier transform of the payoff function. For a standard call with strike, K, W w,V ,0~ð Þ~Kð1ziwÞ._iw{w2

and Im[w].1. The fundamental solution of (4) is in the form

G(w,V ,t)~eA(t,w)zB(t,w)V ð6Þ After substituting (6) into (4), a pair of ordinary differential equations for A(t, w) and B(t, w) is obtained A. ~hkB ð7Þ B.~1 2B 2_s2 V{B kziwsð VrÞ{ 1 2 w 2 {iw   ð8Þ The solutions can be expressed by

B t,wð Þ~ðkzirsVwzd wð ÞÞ s2 V 1{ed wð Þt   1{g wð Þed wð Þt ð Þ ð9Þ A(t,w)~kh s2 V kzirsVwzd(w) ð Þt{2 log 1{g(w)e d(w)t 1{g(w)    ð10Þ using the auxiliary functions

g wð Þ~kzirsVvzd wð Þ kzirsVvzd wð Þ , d(w)~ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w2{iw   s2 Vz kzirsð VwÞ2 r ð11Þ The discontinuity problem of Heston’s model arises because most software packages restrict the logarithm to its principal branch. Figure 1 illustrates the discontinuity problem in the implementation of the fundamental solution. In this example, depicted in Figure 1, S05100, r50.0319, V050.010201, r520.70, k56.21, h50.019,

sV50.61, and Im[w]52.

Reasonable parameters in practice may incur the numerically-induced disconti-nuity such that the correct treatment of the phase jump is very crucial. In fact, in examples having long maturity periods, discontinuities are destined to arise from the formula presented in (10) for A(t, w) if the complex logarithm uses the principal branch only and khs2

V is not an integer (see Figure 2).

The Simplified Approach and Numerical Examples

Here, we introduce the simple solution to the discontinuity problem in the implementation of Heston’s formula. Recall that, given w, d(w) is a complex number presented by its radius/phase representation d(w)5l(w)exp[iv(w)] where l (w)>0 and

Figure 1. The discontinuity occurs in the fundamental solution of the Heston model if the logarithm with complex arguments is restricted to the principal branch. U n d e r l y i n g : dSt~rStdtz ffiffiffiffiffiVt p StdWSð Þ with St 051 00 an d r 50 . 0 3 1 9. V a r i a n c e : dVt~k h{Vð tÞdtzsV ffiffiffiffiffiVt p dWVð Þ with Vt 050.010201, k56.21, h50.019, sV50.61, and

r520.70. Time to maturity: T52.00. The red line was obtained by evaluating A(t, w) with the unfixed form given in (10). The green dashed line was obtained by evaluating A(t, w) with the adjusted formula given in (14), and is the correct curve. The logarithmic function of both cases is

restricted to using only the principal branch.

Figure 2. Discontinuities arise quite naturally for customarily-used Heston parameters which typically occur in practice, as time to maturity is increased. Other parameters are the same as

those specified in Figure 1.

342 J.-H. Guo and M.-W. Hung

v wð Þ[ {p,p½ Þ. If the logarithm function is restricted to the principal branch, it has a cut along the negative real axis in the complex plane which may lead to jumps for the logarithm of the power a of d(w) as w varies. As long as m(w) is not zero, it is trivial that

alog l w½ ð Þ
ziav wð Þ=a log l(w)½
ziv(w) ð12Þ where v wð Þ:av wð Þ{2pm wð Þ and v wð Þ[ {p,p½ Þ. Although the definition of the logarithm of the power a of d(w) consists of the expression on the left of the inequality sign in (12), most software packages using the principal branch only yield the answer of the expression on the right of the inequality sign in (12). For example, we specify d(w) with the same parameters as those in Figure 1 to calculate the imaginary part of log[d(w)a] for w523+2iR3+2i to obtain Figure 3. After adding the rotation-corrected term, 2p m(w), onto the imaginary part of the expression on the right of (12), the difference between the left expression and the right expression can be eliminated. The rotation count number, m(w), is given by

m(w)~int v(w)azp 2p

 

ð13Þ where int[N] denotes a Gauss integer bracket. However, an easier way to obtain the correct value of log[d(w)a] using the logarithm function provided by most software packages is to calculate its adjusted formula, a log[d(w)], which yields the expression on the left of (12). This example also illustrates that the branch switching of the complex logarithm is not the primary problem that causes phase jumps. In fact, the problem results from the complex power function in the complex logarithm.

Another insight into the discontinuity problem is that the subtraction of the number from a complex variable, , results simply in a shift parallel to the real axis. Because an imaginary component must be added to move a complex number across

Figure 3. The recovery of continuity of log[d(w)a] for w523+2iR3+2i where d(w) is given in (11). Parameters: a5100, k56.21, r520.70, and sV50.61. The red curve is implemented using

the principal branch only and discontinuities arise in the imaginary part of log[d(w)a_{]. One}

way to recover the continuity of log[d(w)a_{] is to calculate and add its rotation-corrected term}

to the imaginary part. An alternative approach is to directly calculate its adjusted formula a log[d(w)](see the green curve).

the negative real axis, the phases of c21 and c exist on the same phase interval. Therefore, the logarithm of c21 or c has the same rotation count number. These findings illuminate the simple solution to assure that the phase of A(t, w) is continuous without the rotation-corrected term. All that remains is to move exp[2d(w)t] into the logarithm of A(t, w) by simply adjusting A(t, w) as follows:

A(t,w)~kh s2 V kzirsVw{d(w) ð Þt{2 log 1{g(w)e d(w)t 1{g(w) e {d(w)t    ð14Þ The logarithm presented in (14) is the only term possibly giving rise to discontinuity. Trivially, log 1{g(w)e d(w)t 1{g(w) e {d(w)t  

~ log g(w)e d(w)t{1{ log g(w){1ð Þed(w)t _ð15Þ

Note that the subtraction of 1 does not affect the rotation count of the phase of a complex variable. The expression of log[g(w)ed(w)t21] has the same rotation count number as log[g(w)ed(w)t]. Nevertheless, log[g(w)ed(w)t]2log[(g(w)21)ed(w)t] needs no rotation-corrected terms for all levels of Heston parameters. Hence, the formula in (14), for A(t, w), provides a simple solution to the discontinuity problem for Heston’s stochastic volatility model.1Compared to the work of Kahl and Ja¨ckel (2005), the simple solution does not require any rotation-corrected terms in the already complex integral of Heston’s formula in order to recover its continuity.

Table 1 is an illustration of the usefulness of this approach for evaluating European call options in Heston’s model on stochastic volatility with the complex logarithm using the principal branch only. The algorithm was verified using Monte Carlo simulation with the exact method proposed by Broadie and Kaya (2004) for the stochastic volatility process. This procedure is, of course, computationally more burdensome than our simple solution, but has the advantage that its convergence rate is much faster than the conventional Euler discretization method.

Table 1. Impacts of the discontinuity problem on the evaluation of European call options in Heston’s model on stochastic volatility.

T (year)

Monte Carlo simulation with the exact method

(10000 trials)

Fundamental solution of the Heston model Adjusted formula Using formula (14) Unfixed formula Using formula (10) 0.50 4.2658 4.2545 4.2555 1.00 6.7261 6.8061 6.4483 1.50 8.9510 8.9557 8.3286 2.00 10.9633 10.8830 9.7079 2.50 12.6100 12.6635 10.5542 3.00 14.2591 14.3366 10.9778

S05100.00, K5100.00, r50.0319, V050.010201, r520.70, k56.21, h50.019, and sV50.61. Note that the

evaluation of the fundamental solution of the Heston model using formula (14) still yields values consistent with those of the Monte Carlo simulation for all time-to-maturity cases although the complex logarithm is restricted to the principal branch.

344 J.-H. Guo and M.-W. Hung

Conclusions

This paper addressed numerical instabilities in the implementation of the semi-analytic formula for Heston’s stochastic volatility model due to the multi-valued nature of the complex logarithm and power functions. We found a simplified approach without rotation-corrected terms to solve this problem. The key insight is to transfer the fundamental solution of the Heston model into its adjusted formula for the calculation. This simple solution assures the robustness of the evaluation of formulae for all European-style financial claims for all levels of Heston parameters, even for long maturity periods, with the complex logarithm using the principal branch only.

Acknowledgement

The authors thank William Shaw, Co-Editor-in-Chief, for useful comments. Note

1

Shaw (2006) pointed out that another useful way to avoid the branch cut difficulties arising from the choice of the branch of the complex logarithm was to perform direct numerical integration of A(t, w) according to (7).

References

Broadie, M. and Kaya, O¨ . (2004) Exact simulation of stochastic volatility and other affine jump diffusion processes, Working paper, Columbia University, New York.

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

Kahl, C. and Ja¨ckel, P. (2005) Not-so-complex logarithm in the Heston model, Working paper, Wuppertal University, London.

Kruse, S. and No¨gel, U. (2005) On the pricing of forward starting options in Heston’s model on stochastic volatility, Finance and Stochastics, 9, pp. 223–250.

Lee, R. (2005) Option pricing by transform methods: extensions, unification, and error control, Journal of Computational Finance, 7(3), pp. 51–86.

Lewis, A. (2000) Option Valuation under Stochastic Volatility, Finance Press.

Mikhailov, S. and No¨gel, U. (2003) Heston’s stochastic volatility model- implementation, calibration and some extensions, Wilmot Magazine.

Shaw, W. (2006) Stochastic volatility: models of Heston type. Lecture Notes, http://www.mth.kcl.ac.uk/ ,shaww/web_page/papers/StoVolLecture.pdf http://www.mth.kcl.ac.uk/,shaww/web_page/papers/ StoVolLecture.nb.

Scho¨bel, R. and Zhu, J. (1999) Stochastic volatility with an Ornstein Uhlenbeck process: an extension, European Finance Review, 3, pp. 23–46.