Combined Mutiplicative-Heston Model for Stochastic Volatility
M. Dashti Moghaddam R. A. Serota






mymainaddress]M. Dashti Moghaddam
mymainaddress]R. A. Serotamyfootnote
[myfootnote]serota@ucmail.uc.edu

[mymainaddress]Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221-0011


We consider a model of stochastic volatility which combines features of the multiplicative model for large volatilities and of the Heston model for small volatilities. The steady-state distribution in this model is a Beta Prime and is characterized by the power-law behavior at both large and small volatilities. We discuss the reasoning behind using this model as well as consequences for our recent analyses of distributions of stock returns and realized volatility.



 Volatility Heston Multiplicative Beta Prime Distribution Tails







§ INTRODUCTION

Distributions of stock returns (SR) have long fascinated researchers – see <cit.> for a good summary of earlier works, 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 of fat-tailed distributions, such as stable, Student's t, and generalized t – see <cit.> and references therein and a more recent <cit.>. Studies of intra-day returns also argue in favor of the power-law-tail hypothesis <cit.>. Alternatively, the long multi-day returns seem to be described just as well with an exponentially decaying distribution <cit.>.

Student's t distribution has an appeal of being underpinned by a simple multiplicative stochastic volatility model <cit.>, which leads to an Inverse Gamma (IGa) steady-state distribution for the variance of the volatility<cit.>. Its drawback, however, is that IGa decays exponentially quickly for small values of volatility. Another widely used stochastic volatility model is the Heston model <cit.>, which leads to a Gamma (Ga) steady-state distribution for the variance <cit.>. Ga scales as power law for small volatilities and serves as an underpinning for the exponentially decaying SR distribution that may also be suitable for fitting multi-day returns. In fact, the Kolmogorov-Smirnov (KS) test does not give a clear advantage to either multiplicative or Heston model <cit.>.

In this paper we propose a stochastic volatility model that marries the properties of multiplicative and Heston models and results in a Beta Prime (BP) steady-state distribution that replicates the power-law properties of Ga for small volatilities and of IGa for large volatilities. We discuss what consequences this model has for our previous results on SR distributions <cit.> and realized variance <cit.>. We also consider the question of whether the stochastic equation for SR should be understood in Stratonovich or Ito context <cit.>.

This paper is organized as follows. In Section <ref> we introduce the combined multiplicative-Heston model and discuss its steady-state BP distribution. In Section <ref> we discuss Startonovich versus Ito interpretation of the SR equation and its consequences for SR and leverage. We discuss SR distribution fitting and its moments in light of the new model. In Section <ref> we discuss the theoretical value of the variance of realized variance (RV) <cit.> for this model and compare it with the numerical results from the market data. 




§ MODELS OF VOLATILITY


The two widely used mean-reverting models of stochastic volatility σ_t, expressed in terms of stochastic variance v_t=σ_t^2, are multiplicative (MM) <cit.>

    dv_t = -γ(v_t - θ)dt + κ_M v_tdW_t^(2)

and Heston (HM) <cit.>

    dv_t = -γ(v_t - θ)dt + κ_H √(v_t)dW_t^(2)

where dW_t^(2) is the normally distributed Wiener process, dW_t^(2)∼N(0, dt ). The steady-state distributions for v_t and σ_t are respectively IGa(v_t; α/θ+1, α) and 2σ_t ·IGa(σ_t^2; α/θ+1, α) for MM and αGa(α v_t; α, θ) and 2σ_t ·αGa(ασ_t^2; α, θ) for HM, where 

    α = 2γθ/κ_M,H^2

for both models, with α > 1 for the HM <cit.>. A simple relationship exists between κ_M and κ_H: κ_M θ≈κ_H √(θ) or κ_H^2 / κ_M^2 ≈θ.

Here we introduce a new combination multiplicative-Heston model (MHM), given by 

    dv_t = -γ(v_t - θ)dt + √(κ_M^2 v_t^2 + κ_H^2 v_t )dW_t^(2)

Its steady-state distribution is a BP, 

    BP(v_t; p,q,β)=(1+v_t/β)^-p-q(v_t/β)^-1+p/β B(p,q)

where B(p,q) is the beta function,

    p=2 γθ/κ_H^2

and

    q=1+2 γ/κ_M^2

are the shape parameters and 

    β=κ_H^2/κ_M^2

is the scale parameter and, according to the above, β≈θ = α(θ/α) is the product of the scale parameters of MM and HM. The limiting behaviors of BP is 

    BP(v_t; p,q,β) ∝ (v_t/β)^-q-1,  v_t ≫β

and

    BP(v_t; p,q,β) ∝ (v_t/β)^p-1,  v_t ≪β

that is the same as in MM and HM respectively (p > 1 for the latter). Furthermore, for p ≫ 1, BP can mimic 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^2; p,q,β).

We would like to point out that, obviously, BP has an extra shape parameter relative to IGa and Ga and that it is non-trivial to extract the latter two from BP as limits. We also point out that ordinarily it is assumed that the equations for stochastic variance should be understood in the Ito sense. However, since the term that couples to the Gaussian noise contains powers of v_t it is appropriate to ponder a Stratonovich interpretation as well. We observe, however, that for MHM transition from Stratonovich to Ito involves a simple renormalization of constants γ and θ – or just one of them in its MM and HM limits.





§ STOCK RETURNS DISTRIBUTIONS AND MOMENTS


The standard equation for the stock price reads <cit.>

    dS_t/S_t = μdt + σ_t dW_t^(1)

where dW_t^(1)∼N(0, dt ). As is for volatility, this equation is almost always interpreted as Ito but a Stratonovich interpretation also needs explored. In the latter case, the transformation to Ito yields

    dS_t/S_t = μdt + σ_t^2/2dt+ σ_t dW_t^(1)

Using Ito calculus, eqs. (<ref>) and (<ref>) can be rewritten as 

    dlogS_t = μdt  - σ_t^2/2dt + σ_t dW_t^(1)

and

    dlogS_t = μdt  + σ_t dW_t^(1)

respectively.

The first conclusion that can be drawn is that the equation used to estimate the implied volatility index VIX <cit.>

    dS_t/S_t - dlogS_t =σ_t^2/2dt

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 <cit.> for a detailed analysis.

Denoting r_t = ln (S_t/S_0) and x_t = r_t - μ t, the equations (<ref>) and (<ref>) for log returns become

    dx_t = -σ_t^2/2dt + σ_tdW_t^(1)

and 

    dx_t = σ_tdW_t^(1)

respectively. 

It should be pointed out that in general dW_t^(1) and dW_t^(2) are correlated as

    dW_t^(2) = ρdW_t^(1) + √(1-ρ^2)dZ_t

Where dZ_t is independent of dW_t^(1), and ρ∈ [-1,  1] is the correlation coefficient. The latter can be evaluated from leverage correlations <cit.>. We showed, howewver, that SR distributions are not effected by these correlations and that one can set ρ=0 <cit.>. 

The SR distribution in (<ref>) can be evaluated as the product distribution (PD) of volatility and normal distribution <cit.>. We also showed that in (<ref>) the first term in the r.h.s. does not yield significant corrections to the SR distribution until very long periods of returns  <cit.>. Nonetheless, we evaluated the distribution in (<ref>) as a joint probability (JP) distribution and found that PD fit of the market data had lower KS values than JP <cit.>, which points to that (<ref>) should be interpreted as Startonovich and reduced to Ito as according to (<ref>). Furthermore, comparing (<ref>) and (<ref>), if the former is interpreted as Stratonovich, it is the latter that should be used for evaluations of leverage. In <cit.>, we show that indeed this approach gives a better statistical fit of the market data.

Using now 2 σ_t BP(σ_t^2; p,q,β) as the distribution of stochastic volatility σ_t and taking a PD with the normal distribution in (<ref>) in a manner explained in <cit.>, we obtain the following SR distribution:

    ψ_MH(z) = Γ(q+1/2) U(q+1/2, 3/2-p, z^2/2 βτ)/√(2πβτ) B (p,q )

where U is the confluent hypergeometric function, dx_t was replaced with z and and dt was replaced with τ – the number of days over which the returns are calculated.

We use the same fitting procedures as in <cit.> to find the parameters p, q and β. Figs. <ref> - <ref> are plotted as a function of τ, the number of days over which the returns are calculated. Fig. <ref> shows q of MHM vis-a-vis α/θ+1 of MM, which reflects long tails of the stochastic variance per (<ref>). Fig. <ref> shows p of MHM vis-a-vis α of HM, which reflects small stochastic variance behavior per (<ref>). Fig. <ref> shows θ, the mean value of the stochastic variance, for all three models; for BP it is determined using (<ref>) below and for IGa and Ga from direct fitting <cit.>. We also show θ calculated directly from the variance of SR, z^2=θτ – see (<ref>) below – and parameter β to confirm (see above) that β≈θ. Fig. <ref> gives KS values for SR fits, the only new element relative to <cit.> being the fit using (<ref>).

Fig. <ref> contains reduced moments for n=1 and n=2

    (z^2n/E(z^2n))^1/2 n

where z^2n is numerically calculated average from the market data and E(z^2n) is its analytical value calculated from all three models. E_M(z^2n) and E_H(z^2n) are given in <cit.> so here we only list the MHM values:

    E_MH(z^2)=p βτ/q-1 = θτ


    E_MH(z^4)=3 p (p+1)β^2 τ^2/(q-1)(q-2)=3(2γθ^2+κ_H^2θ) τ^2/2γ-κ_M^2

We point out that one must have 2γ>κ_M^2. We recall that in MM 2γ/κ_M^2=α/θ defines the exponent of the power-law tail. This parameter is greater than one <cit.> – see also (<ref>) and Fig. <ref>.

JP results in Fig. <ref> is shown to illustrate Stratonovich versus Ito discussion in this Section. While KS test does not give an advantage to either of the three models – MM, HM, and MHM – the latter describes the moments in Fig. <ref> clearly better than the other two. Finally, using (<ref>) and (<ref>), 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. <ref>; κ_M^2 and κ_H^2 are plotted in Fig. <ref>.















§ APPLICATION TO REALIZED VARIANCE 
 
The correlation function of stochastic variance <cit.>

    E[v_t v_t+τ]=E[v_t]^2+var[v_t] e^-γτ

can be used (along with leverage <cit.>, with minor differences in the result <cit.>), to determine γ. Here

    E[v_t]=θ

for the mean-reverting models (for BP it can be obtained by integration with (<ref>), and 
    var[v_t]=E[v_t^2]-(E[v_t])^2

To find E[v_t v_t+τ] we must use dx_t^2 = v_tdt, which follows from (<ref>). We observe that 

    E[dx_t^2 dx_t+τ^2 ]  = (E[v_t v_t+τ] + 2 E[v_t ^2]) dt^2

and in particular,

    E[dx_t^4] = 3 E[v_t ^2] dt^2

The factor of 3 is purely combinatorial and is model-independent. It can be verified for any of the mentioned models. For instance, integrating v_t^2 with BP in (<ref>), we find

    E[v_t^2]=2γθ^2+κ_H^2θ/2γ-κ_M^2

in agreement with (<ref>) and (<ref>). When higher moments exist, we can similarly obtain E[dx_t^2n] = (2n-1)!! E[v_t ^n] dt^n – see for instance those for HM in <cit.>.

Using (<ref>), we obtain the following expression for the theoretical variance of RV:

    E[(1/T∫_0^T v_tdt-θ)^2] = var[v_t] f(γ T)

where f(γ T) describes the time dependence of the variance of RV:

    f(γ T) = 2(-1+e^-γ T +γ T)/(γ T)^2≈
    1  ,γ T ≪ 1
    
    2 (γ T)^-1,γ T ≫ 1

We evaluate the variance of RV and var[v_t] from the market data and plot their ratio, together with f(γ T), in Fig. <ref>. We should mention that theoretical plots with γ found from correlations (see values in Fig. <ref>) and leverage, γ_DJIA = 0.049, γ_S&P = 0.047, are virtually indistinguishable to an eye; still leverage data give about 0.6% better fit to the market data.





§ CONCLUSIONS 

Multiplicative and Heston model are simple stochastic volatility models, which successfully explain many features of stock returns, particularly over multiple days of accumulation. However they suffer from some shortcomings: multiplicative model seems to underestimate the effects of volatility for small volatilities and Heston the effects for large volatilities. The combined multiplicative-Heston model studied here breeches the two models and reproduces the power-law tails of the multiplicative model for large volatilities and Heston model behavior at small volatilities.

We also examined the even moments of the stock returns vis-a-vis the theoretical predictions of this model and found a good agreement. We discussed the fact that the theoretical moments can be derived alternatively 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 of stochastic volatility distribution function and normal distribution, indicating that the stock returns equations should be interpreted in the Stratonovich sense.

Finally, we examined the correlation function of stochastic variance and used it to determine the relaxation parameter and to calculate the time dependence of the variance of realized variance. We will address the distribution of realized variance, as well as various measures of comparing it to implied variance, in a future publication <cit.>.





