Combined Mutiplicative-Heston Model for Stochastic Volatility
aa r X i v : . [ q -f i n . M F ] J u l Combined Mutiplicative-Heston Model for Stochastic Volatility
M. Dashti Moghaddam a , R. A. Serota a,1 a Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221-0011
Abstract
We consider a model of stochastic volatility which combines features of the multiplicative model for largevolatilities and of the Heston model for small volatilities. The steady-state distribution in this model is aBeta Prime and is characterized by the power-law behavior at both large and small volatilities. We discussthe reasoning behind using this model as well as consequences for our recent analyses of distributions ofstock returns and realized volatility.
Keywords:
Volatility, Heston, Multiplicative, Beta Prime, Distribution Tails
1. Introduction
Distributions of stock returns (SR) have long fascinated researchers – see [1] for a good summary of earlierworks, dating back to Mandelbrot in the early 60s. It is widely believed that, at least for daily returns,SR distributions have fat (power-law) tails. Accordingly, SR distributions were fitted with a number offat-tailed distributions, such as stable, Student’s t , and generalized t – see [1] and references therein anda more recent [2]. Studies of intra-day returns also argue in favor of the power-law-tail hypothesis [3, 4].Alternatively, the long multi-day returns seem to be described just as well with an exponentially decayingdistribution [5, 6].Student’s t distribution has an appeal of being underpinned by a simple multiplicative stochastic volatil-ity model [7], which leads to an Inverse Gamma (IGa) steady-state distribution for the variance of thevolatility[8, 9, 10, 6]. Its drawback, however, is that IGa decays exponentially quickly for small values ofvolatility. Another widely used stochastic volatility model is the Heston model [11, 5], which leads to aGamma (Ga) steady-state distribution for the variance [5, 6]. Ga scales as power law for small volatilitiesand serves as an underpinning for the exponentially decaying SR distribution that may also be suitable forfitting multi-day returns. In fact, the Kolmogorov-Smirnov (KS) test does not give a clear advantage toeither multiplicative or Heston model [6].In this paper we propose a stochastic volatility model that marries the properties of multiplicative andHeston models and results in a Beta Prime (BP) steady-state distribution that replicates the power-lawproperties of Ga for small volatilities and of IGa for large volatilities. We discuss what consequences thismodel has for our previous results on SR distributions [6] and realized variance [12]. We also consider thequestion of whether the stochastic equation for SR should be understood in Stratonovich or Ito context [13].This paper is organized as follows. In Section 2 we introduce the combined multiplicative-Heston modeland discuss its steady-state BP distribution. In Section 3 we discuss Startonovich versus Ito interpretationof the SR equation and its consequences for SR and leverage. We discuss SR distribution fitting and itsmoments in light of the new model. In Section 4 we discuss the theoretical value of the variance of realizedvariance (RV) [12] for this model and compare it with the numerical results from the market data. [email protected] Preprint submitted to arXiv July 31, 2018 . Models of Volatility
The two widely used mean-reverting models of stochastic volatility σ t , expressed in terms of stochasticvariance v t = σ t , are multiplicative (MM) [10]d v t = − γ ( v t − θ )d t + κ M v t d W (2) t (1)and Heston (HM) [5] d v t = − γ ( v t − θ )d t + κ H √ v t d W (2) t (2)where d W (2) t is the normally distributed Wiener process, d W (2) t ∼ N(0 , d t ). The steady-state distributionsfor v t and σ t are respectively IGa( v t ; αθ + 1 , α ) and 2 σ t · IGa( σ t ; αθ + 1 , α ) for MM and α Ga( αv t ; α, θ ) and2 σ t · α Ga( ασ t ; α, θ ) for HM, where α = 2 γθκ M,H (3)for both models, with α > κ M and κ H : κ M θ ≈ κ H √ θ or κ H /κ M ≈ θ .Here we introduce a new combination multiplicative-Heston model (MHM), given byd v t = − γ ( v t − θ )d t + q κ M v t + κ H v t d W (2) t (4)Its steady-state distribution is a BP, BP ( v t ; p, q, β ) = (1 + v t β ) − p − q ( v t β ) − p βB ( p, q ) (5)where B ( p, q ) is the beta function, p = 2 γθκ H (6)and q = 1 + 2 γκ M (7)are the shape parameters and β = κ H κ M (8)is the scale parameter and, according to the above, β ≈ θ = α ( θ/α ) is the product of the scale parametersof MM and HM. The limiting behaviors of BP is BP ( v t ; p, q, β ) ∝ ( v t β ) − q − , v t ≫ β (9)and BP ( v t ; p, q, β ) ∝ ( v t β ) p − , v t ≪ β (10)that is the same as in MM and HM respectively ( p > p ≫
1, BP canmimic IGa for v t ≪ β and, for q ≫
1, BP can mimic Ga for v t ≫ β . Stochastic volatility is, accordingly,distributed as 2 σ t BP ( σ t ; p, q, β ).We would like to point out that, obviously, BP has an extra shape parameter relative to IGa and Gaand that it is non-trivial to extract the latter two from BP as limits. We also point out that ordinarily itis assumed that the equations for stochastic variance should be understood in the Ito sense. However, sincethe term that couples to the Gaussian noise contains powers of v t it is appropriate to ponder a Stratonovichinterpretation as well. We observe, however, that for MHM transition from Stratonovich to Ito involves asimple renormalization of constants γ and θ – or just one of them in its MM and HM limits.2 . Stock Returns Distributions and Moments The standard equation for the stock price reads [14]d S t S t = µ d t + σ t d W (1) t (11)where d W (1) t ∼ N(0 , d t ). As is for volatility, this equation is almost always interpreted as Ito but aStratonovich interpretation also needs explored. In the latter case, the transformation to Ito yieldsd S t S t = µ d t + σ t t + σ t d W (1) t (12)Using Ito calculus, eqs. (11) and (12) can be rewritten asd log S t = µ d t − σ t t + σ t d W (1) t (13)and d log S t = µ d t + σ t d W (1) t (14)respectively.The first conclusion that can be drawn is that the equation used to estimate the implied volatility indexVIX [15, 16] d S t S t − d log S t = σ t t (15)is not affected by which interpretation – Ito or Stratonovich - is used. It is also obvious that the Black-Scholes equation is not affected either, since it assumes a constant – or at least a non-stochastic – volatility;see [13] for a detailed analysis.Denoting r t = ln( S t /S ) and x t = r t − µt , the equations (13) and (14) for log returns becomed x t = − σ t t + σ t d W (1) t (16)and d x t = σ t d W (1) t (17)respectively.It should be pointed out that in general d W (1) t and d W (2) t are correlated asd W (2) t = ρ d W (1) t + p − ρ d Z t (18)Where d Z t is independent of d W (1) t , and ρ ∈ [ − ,
1] is the correlation coefficient. The latter can be evaluatedfrom leverage correlations [17, 18]. We showed, howewver, that SR distributions are not effected by thesecorrelations and that one can set ρ = 0 [6].The SR distribution in (17) can be evaluated as the product distribution (PD) of volatility and normaldistribution [6]. We also showed that in (16) the first term in the r.h.s. does not yield significant correctionsto the SR distribution until very long periods of returns [6]. Nonetheless, we evaluated the distribution in(16) as a joint probability (JP) distribution and found that PD fit of the market data had lower KS valuesthan JP [6], which points to that (11) should be interpreted as Startonovich and reduced to Ito as accordingto (12). Furthermore, comparing (11) and (14), if the former is interpreted as Stratonovich, it is the latterthat should be used for evaluations of leverage. In [19], we show that indeed this approach gives a betterstatistical fit of the market data. 3sing now 2 σ t BP ( σ t ; p, q, β ) as the distribution of stochastic volatility σ t and taking a PD with thenormal distribution in (17) in a manner explained in [6], we obtain the following SR distribution: ψ MH ( z ) = Γ (cid:0) q + (cid:1) U (cid:16) q + , − p, z βτ (cid:17) √ πβτ B ( p, q ) (19)where U is the confluent hypergeometric function, d x t was replaced with z and and d t was replaced with τ – the number of days over which the returns are calculated.We use the same fitting procedures as in [6] to find the parameters p , q and β . Figs. 1 - 5 are plotted asa function of τ , the number of days over which the returns are calculated. Fig. 1 shows q of MHM vis-a-vis αθ + 1 of MM, which reflects long tails of the stochastic variance per (9). Fig. 2 shows p of MHM vis-a-vis α of HM, which reflects small stochastic variance behavior per (10). Fig. 3 shows θ , the mean value of thestochastic variance, for all three models; for BP it is determined using (21) below and for IGa and Ga fromdirect fitting [6]. We also show θ calculated directly from the variance of SR, z = θτ – see (20) below – andparameter β to confirm (see above) that β ≈ θ . Fig. 4 gives KS values for SR fits, the only new elementrelative to [6] being the fit using (19).Fig. 5 contains reduced moments for n = 1 and n = 2 z n E ( z n ) ! n (20)where z n is numerically calculated average from the market data and E ( z n ) is its analytical value calculatedfrom all three models. E M ( z n ) and E H ( z n ) are given in [6] so here we only list the MHM values: E MH ( z ) = pβτq − θτ (21) E MH ( z ) = 3 p ( p + 1) β τ ( q − q −
2) = 3(2 γθ + κ H θ ) τ γ − κ M (22)We point out that one must have 2 γ > κ M . We recall that in MM γκ M = αθ defines the exponent of thepower-law tail. This parameter is greater than one [6] – see also (9) and Fig. 1.JP results in Fig. 4 is shown to illustrate Stratonovich versus Ito discussion in this Section. While KStest does not give an advantage to either of the three models – MM, HM, and MHM – the latter describesthe moments in Fig. 5 clearly better than the other two. Finally, using (7) and (8), we can find κ M and κ H once we know γ , which can be found by fitting the market data correlation function – see Sec. 4; κ M and κ H are plotted in Fig. 6.
4. Application to Realized Variance
The correlation function of stochastic variance [19] E [ v t v t + τ ] = E [ v t ] + var [ v t ] e − γτ (23)can be used (along with leverage [17, 18], with minor differences in the result [19]), to determine γ . Here E [ v t ] = θ (24)for the mean-reverting models (for BP it can be obtained by integration with (5), and var [ v t ] = E [ v t ] − ( E [ v t ]) (25)4
50 100 150 2001.522.533.5 pa r a m s D J I A pa r a m s SP Figure 1: Parameter q in the BP distribution (5) versus parameter αθ + 1 in the IGa distribution pa r a m s D J I A p BP Ga pa r a m s SP p BP Ga Figure 2: Parameter p in the BP distribution (5) versus parameter α in the Ga distribution
50 100 150 2000.50.60.70.80.911.1 D J I A -4 BPGa
IGadata SP -4 BPGa
IGadata D J I A -4 BPGa
IGadata SP -4 BPGa
IGadata
Figure 3: Mean value of stochastic variance θ for MM, HM, MHM and data; β ≈ θ also included. Bottom row is the same astop row on smaller scale. KS s t a t i s t i c D J I A BP PDGa PDGa JPIGa PDIGa JPN KS s t a t i s t i c SP BP PDGa PDGa JPIGa PDIGa JPN
Figure 4: KS test results. PD statistic is better than JP pointing to (12) as a preferred interpretation.
50 100 150 2000.9511.05 n D J I A = BPGaIGa n SP = BPGaIGa n D J I A = BPGaIGa n SP = BPGaIGa
Figure 5: Reduced moments of stock returns, per (20). D J I A SP Figure 6: κ M (BP) and κ H (BP) found from (7) and (8) with γ DJIA = 0 .
042 and γ S&P = 0 .
041 found by fitting market data correlation function – see Sec. 4. κ M (IGa) and κ H (Ga) are values found using MM and HM respectively.7o find E [ v t v t + τ ] we must use d x t = v t d t , which follows from (17). We observe that E [d x t d x t + τ ] = ( E [ v t v t + τ ] + 2 E [ v t ])d t (26)and in particular, E [d x t ] = 3 E [ v t ]d t (27)The factor of 3 is purely combinatorial and is model-independent. It can be verified for any of the mentionedmodels. For instance, integrating v t with BP in (5), we find E [ v t ] = 2 γθ + κ H θ γ − κ M (28)in agreement with (22) and (27). When higher moments exist, we can similarly obtain E [d x nt ] = (2 n − E [ v nt ]d t n – see for instance those for HM in [6].Using (23), we obtain the following expression for the theoretical variance of RV: E [( 1 T Z T v t d t − θ ) ] = var [ v t ] f ( γT ) (29)where f ( γT ) describes the time dependence of the variance of RV: f ( γT ) = 2( − e − γT + γT )( γT ) ≈ ( γT ≪ γT ) − , γT ≫ var [ v t ] from the market data and plot their ratio, together with f ( γT ),in Fig. 7. We should mention that theoretical plots with γ found from correlations (see values in Fig. 6)and leverage, γ DJIA = 0 . , γ S&P = 0 .
5. Conclusions
Multiplicative and Heston model are simple stochastic volatility models, which successfully explain manyfeatures of stock returns, particularly over multiple days of accumulation. However they suffer from someshortcomings: multiplicative model seems to underestimate the effects of volatility for small volatilities andHeston the effects for large volatilities. The combined multiplicative-Heston model studied here breeches thetwo models and reproduces the power-law tails of the multiplicative model for large volatilities and Hestonmodel behavior at small volatilities.We also examined the even moments of the stock returns vis-a-vis the theoretical predictions of thismodel and found a good agreement. We discussed the fact that the theoretical moments can be derivedalternatively from the stock returns distribution function and stochastic variance distribution function.Towards this end, the distribution function of stock returns is best described by the product distribution ofstochastic volatility distribution function and normal distribution, indicating that the stock returns equationsshould be interpreted in the Stratonovich sense.Finally, we examined the correlation function of stochastic variance and used it to determine the relax-ation parameter and to calculate the time dependence of the variance of realized variance. We will addressthe distribution of realized variance, as well as various measures of comparing it to implied variance, in afuture publication [20]. 8 (T)-1.4-1.2-1-0.8-0.6-0.4-0.200.2 Log (r edu c ed v a r o f R V ) DataTheory (T)-1-0.8-0.6-0.4-0.200.20.4 Log (r edu c ed v a r o f R V ) DataTheory r edu c ed v a r o f R V DataTheory r edu c ed v a r o f R V DataTheory
Figure 7: Function f ( γτ ) (30) vis-a-vis market data for E [( T R T v t d t − θ ) ] /var [ v t ]. Straight line fits, corresponding to limitsof (30), are shown on log-log scale: slopes are, respectively, -0.0109 and -0.991 for DJIA and -0.0111 and -0.988 for S&P. eferences [1] D. F. Harris, C. C. Kucukozmen, The empirical distribution of uk and us stock returns, Journal of Business Finance &Accounting 28 (5-6) (2001) 715–740.[2] P. N. Rathie, M. Coutinho, T. R. Sousa, G. S. Rodrigues, T. B. Carrijo, Stable and generalized-t distributions andapplications, Communications in Nonlinear Science and Numerical Simulation 17 (12) (2012) 5088–5096.[3] A. Gerig, J. Vicente, M. A. Fuentes, Model for non-gaussian intraday stock returns, Physical Review E 80 (2009)065102R.[4] S. K. Behfar, Long memory behavior of returns after intraday financial jumps, Physica A: Statistical Mechanics and itsApplications 461 (2016) 716–725.[5] A. A. Dragulescu, V. M. 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Serota, Distributions of historic market data – implied and realized volatility,arXiv 1804.05279.[13] J. Perello, J. M. Porraa, M. Monteroa, J. Masoliver, Black–scholes option pricing within itˆo and stratonovich conventions,Physica A: Statistical Mechanics and its Applications 278 (1-2) (2000) 260–274.[14] M. F. Osborne, in The Random Character of Stock Market Prices, MIT Press, Cambridge, MA, 1964.[15] K. Demeterfi, E. Derman, M. Kamal, J. Zou, A guide to volatility and variance swaps, The Journal of Derivatives 6 (4)(1999) 9–32.[16] K. Demeterfi, E. Derman, M. Kamal, J. Zou, More than you ever wanted to know about volatility swaps, Tech. rep.,Goldman Sachs (1999).[17] J.-P. Bouchaud, A. Matacz, M. Potters, Leverage effect in financial markets The retarded volatility model, PhysicalReview Letters 87 (22) (2001) 228701.[18] J. Perello, J. Masoliver, Stochastic volatility and leverage effect, arXiv:cond-mat/0202203 (2002).[19] M. Dashti Moghaddam, Z. Liu, R. Serota, Distributions of historic market data – relaxation and correlations, to besubmitted to arXiv.[20] M. Dashti Moghaddam, Z. Liu, R. Serota, Realized versus implied volatility and their distributions, to be submitted toarXiv.[1] D. F. Harris, C. C. Kucukozmen, The empirical distribution of uk and us stock returns, Journal of Business Finance &Accounting 28 (5-6) (2001) 715–740.[2] P. N. Rathie, M. Coutinho, T. R. Sousa, G. S. Rodrigues, T. B. Carrijo, Stable and generalized-t distributions andapplications, Communications in Nonlinear Science and Numerical Simulation 17 (12) (2012) 5088–5096.[3] A. Gerig, J. Vicente, M. A. Fuentes, Model for non-gaussian intraday stock returns, Physical Review E 80 (2009)065102R.[4] S. K. Behfar, Long memory behavior of returns after intraday financial jumps, Physica A: Statistical Mechanics and itsApplications 461 (2016) 716–725.[5] A. A. Dragulescu, V. M. Yakovenko, Probability distribution of returns in the heston model with stochastic volatility,Quantitative Finance 2 (2002) 445–455.[6] Z. Liu, M. Dashti Moghaddam, R. Serota, Distributions of historic market data – stock returns, arXiv:1711.11003 (2017).[7] D. Nelson, Arch models as diffusion approximations, Journal of Econometrics 45 (1990) 7.[8] P. D. Praetz, The distribution of share price changes, Journal of Business (1972) 49–55.[9] M. A. Fuentes, A. Gerig, J. Vicente, Universal behvior of extreme price movements in stock markets, PLoS ONE 4 (12)(2009) 1.[10] T. Ma, R. Serota, A model for stock returns and volatility, Physica A: Statistical Mechanics and its Applications 398(2014) 89–115.[11] S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options,The Review of Financial Studies 6 (2) (1993) 327–343.[12] M. Dashti Moghaddam, Z. Liu, R. Serota, Distributions of historic market data – implied and realized volatility,arXiv 1804.05279.[13] J. Perello, J. M. Porraa, M. Monteroa, J. Masoliver, Black–scholes option pricing within itˆo and stratonovich conventions,Physica A: Statistical Mechanics and its Applications 278 (1-2) (2000) 260–274.[14] M. F. Osborne, in The Random Character of Stock Market Prices, MIT Press, Cambridge, MA, 1964.[15] K. Demeterfi, E. Derman, M. Kamal, J. Zou, A guide to volatility and variance swaps, The Journal of Derivatives 6 (4)(1999) 9–32.[16] K. Demeterfi, E. Derman, M. Kamal, J. Zou, More than you ever wanted to know about volatility swaps, Tech. rep.,Goldman Sachs (1999).[17] J.-P. Bouchaud, A. Matacz, M. Potters, Leverage effect in financial markets The retarded volatility model, PhysicalReview Letters 87 (22) (2001) 228701.[18] J. Perello, J. Masoliver, Stochastic volatility and leverage effect, arXiv:cond-mat/0202203 (2002).[19] M. Dashti Moghaddam, Z. Liu, R. Serota, Distributions of historic market data – relaxation and correlations, to besubmitted to arXiv.[20] M. Dashti Moghaddam, Z. Liu, R. Serota, Realized versus implied volatility and their distributions, to be submitted toarXiv.