Variance reduction estimation for return models with jumps using gamma asymmetric kernels

1 Introduction

In financial modeling and econometrics, diffusion models are widely used to describe uncertainty of changes in native economic variables and asset prices, one of the most famous models is the Black & Scholes asset pricing model. Moreover, as an extension of continuous trajectory process, diffusion model with jumps has a wide range of applications in the field of finance and economics. Jump-diffusion process X_t is represented by the following stochastic differential equation:

where W_t is a standard Brownian motion, μ(⋅) and σ(⋅) are the infinitesimal conditional drift and variance respectively, E=R∖{0}, r(ω,dt,dz)=(p−q)(dt,dz), p(dt,dz) is a time-homogeneous Poisson random measure on R+×R independent of W_t, and q(dt, dz) is its intensity measure, that is, E[p(dt,dz)]=q(dt,dz)=f(z)dzdt, f(z) is its Lévy density. Moreover, jump factor in model (1) is introduced statistically to capture the fatter tail behavior of underlying asset and economically to accommodate impact of sudden and large shocks to financial markets, such as macroeconomic announcements or a dramatic interest rate cut by the Federal Reserve, more details discussed in Johannes (2004). However, on the one hand, because the Brownian motion is not differentiated at any point in probability 1, the mentioned model (1) above can not represent the integrated and differentiated process for integrated economic phenomena, that is, the observation of dynamic process at the discrete time is sometimes not easy to obtain, while the cumulation of the process can be directly observed; on the other hand, model (1) and its subsequent derivative models were mainly involved with the price of asset, while less models characterized returns of asset which is a complete and scale-free summary of the investment opportunity for average investors. Fortunately, the second-order diffusion process with jumps (2) provides a valuable model for the solution of these problems.

Here we extend the continuous autoregression of order two in Nicolau (2007) to a discontinuous and realistic case with jumps for return series based on model (1), which is called as the second-order jump-diffusion model (2),

where X_t represents the continuously compounded return of underlying assets and Y_t denotes the asset price by means of the cumulation of the returns plus initial asset value. It also can model the integrated econometric phenomena, for example, continuously compounded return series X_t of underlying assets can not be directly observed, while its realization Y_t such as the prices of assets can be directly observed.

Despite of the above two advantages, model (2) can be used commonly in empirical financial for another three advantages. Firstly, model (2) can accommodate nonstationary original process and be made stationary by differencing, which is a more widely used technique in empirical financial, which is verified through the Augmented Dickey-Fuller test statistic in the empirical analysis part. Secondly, compared with the model in Nicolau (2007), model (2) accommodate the impact of sudden and large shocks in combination with jump component, which is also testified through the test statistic proposed in Barndorff-Nielsen and Shephard (2006) for empirical financial data. Thirdly, model (2) can also be employed in empirical finance for integrated volatility. In empirical financial, Nicolau (2008) employed the second-order diffusion process to study the return series of three companies and three stock indices in USA and etc. Based on the generalized likelihood ratio statistics, Yan and Mei (2016) validated the parameter forms of the volatility function for returns of stock indices in Europe, America, Asia (Hong Kong, Japan, Korea) and four companies in USA stock market.

For the return model characterized by second-order diffusion process, many scholars considered parametric and nonparametric estimation for the unknown coefficients in model (2). For the case with c(x,z)≡0, Gloter (2001) obtained the parameter estimators for the second-order Ornstein-Uhlenbeck process, and Gloter (2006) gave the minimum deviation parameter estimation for the drift and diffusion coefficients. Not assuming the specific form of the coefficients, Nicolau (2007) systematically studied Nadaraya-Watson estimators, Wang and Lin (2011) presented the local linear estimations using symmetric kernel for bias correction, Hanif (2015) discussed the Nadaraya-Watson estimators using Gamma asymmetric kernel which didn’t coincide with the asymmetric kernel-based method introduced in Chen (2000). For model (2) with c(x,z)≠0, Song (2017) provided the nonparametric estimators for the infinitesimal coefficients μ(x) and σ2(x)+∫Ec2(x,z)f(z)dz in high frequency data based on Gaussian symmetric kernel, Chen and Zhang (2015) discussed the local linear estimators for them based on symmetric kernels, Song, Lin, and Wang (2013) proposed a re-weighted Nadaraya-Watson estimator for the infinitesimal conditional expectation, Funke and Schmisser (2018) constructed adaptive nonparametric estimator for the drift coefficient based on penalized least squares method.

However, the current research on the bias corrections for the unknown coefficients in the second-order jump-diffusion process is mainly focused on the simple bias representation (mainly using local linear or re-weighted nonparametric method). According to the asymptotic distribution theorem of the estimators, we can easily find that the mean square error of the estimators is mainly from two parts, one is the mean error of the estimator and the other is the perturbation caused by the asymptotic normal variance. The previous research work does not take variance reduction into full account for these unknown quantities. Based on Gamma asymmetric kernel fully discussed in Chen (2000), Kristensen (2010) considered a kernel-based approach for the realized integrated volatility estimation, Gospodinovy and Hirukawa (2012) constructed nonparametric estimators for scalar diffusion models of spot interest rates. Nadaraya-Watson estimators using Gamma asymmetric kernel have some benefits such as variable bandwidth, variance reduction and resistance to sparse design. Compared with Nadaraya-Watson estimators constructed with Gaussian symmetric kernels, firstly, the curve shapes of Gamma asymmetric kernel vary with the smoothing parameter and the location of the design point similar as the variable bandwidth methods. Secondly, the finite variance of the curve estimation decreases, especially for sparse design points, due to the fact that the smaller asymptotic convergence rate and their variances vary along with the location of the design point x, which is shown in Theorem 2 and Theorem 3. Thirdly, Gamma asymmetric kernels are free from boundary bias (allowing a larger bandwidth to pool more data) and achieve the optimal rate of convergence in mean square error within the class of nonnegative kernel estimators with order two. Fourthly, the estimators constructed with Gamma kernels possesses another advantage: nonnegativity for volatility function compared with local linear estimators. In conclusion, asymmetric kernels is a combination of a boundary correction, variance reduction device and a “variable bandwidth” method.

In this paper, we will propose Nadaraya-Watson estimators for μ(x) and σ2(x)+∫Ec2(x,z)f(z)dz using Gamma asymmetric kernels in model (2) for variance reduction. It is organized as follows. In Section 2, we propose Nadaraya-Watson estimators with asymmetric kernels and some ordinary assumptions for model (2). We present the asymptotic results in Section 3. Section 4 presents the finite sample performance for two models through Monte Carlo simulation study. Estimators are depicted through some empirical financial data in Section 5. Section 6 concludes. Some technical lemmas and the main proofs are explicitly shown in Appendix A.2.

2 Nadaraya-Watson estimators with asymmetric kernels and assumptions

According to the asymmetric kernels used in Chen (2000), the Gamma kernel function is defined as

where Γ(m)=∫0∞ym−1exp⁡(−y)dy, m > 0 is the Gamma function and h is the smoothing parameter. Since the density of Gamma distribution has support [0, ∞), the Gamma kernel function does not generate boundary bias for nonnegative variables or nonnegative part of underlying variables. Here we use modified Gamma kernel function KG(x/h+1,h)(u) instead of KG(x/h,h)(u) due to the fact KG(x/h,h)(u) is unbounded near at x = 0. The Gamma function has shapes varying with the design point x, which is nonnegative and asymmetric, especially at the boundary points. Furthermore, the asymptotic variance of the nonparametric Gamma kernel estimation depends on the design point x, which yields the optimal rate of convergence in mean integrated squared error of nonnegative kernels.

In this paper we consider the Nadaraya-Watson estimators with Gamma asymmetric kernel for the unknown coefficients of the second-order jump-diffusion model. Different from model (1), nonparametric estimations constructed for the coefficients in second-order jump-diffusion model (2) give rise to new challenges for two main reasons.

On the one hand, we usually get observations {YiΔn;i=1,2,⋅⋅⋅} rather than {XiΔn;i=1,2,⋅⋅⋅}. The value of Xti cannot be obtained from Yti=Y0+∫0tiXsds in a fixed sample intervals. Additionally, nonparametric estimations of the unknown qualities in model (2) could not be constructed on the observations {YiΔn;i=1,2,⋅⋅⋅} due to the unknown conditional distribution. As Nicolau (2007) showed, with observations {YiΔn;i=1,2,⋅⋅⋅} and given that

we can obtain an approximation value of XiΔn by

On the other hand, the Markov properties for statistical inference of unknown qualities in model (2) based on the samples {X~iΔn;i=1,2,⋅⋅⋅} should be built, which are infinitesimal conditional expectations characterized by infinitesimal operators. Fortunately, under Lemma 1 in subsection 7.2 we can build the following infinitesimal conditional expectations for model (2)

where Ft=σ{Xs,s≤t}. One can refer to Appendix A in Song, Lin, and Wang (2013) for detailed calculations.

We briefly discuss the Gamma Nadaraya-Watson estimators for the coefficients in model (2) based on {X~iΔn;i=0,1,2,⋅⋅⋅}. We firstly construct Gamma Nadaraya-Watson estimators for them based on equations (5) and (6). The Nadaraya-Watson estimators for μ(x) and M(x):=σ2(x)+∫Ec(x,z)f(z)dz based on the Gamma asymmetric kernel are solutions to the following optimal problem:

where KG(x/hn+1,hn)(⋅) is a asymmetric Gamma kernel function and the smoothing parameter h_n → 0 as n → ∞.

The solutions to (7) and (8) are

where p^n(x)=1n∑i=1nKG(x/hn+1,hn)(X~(i−1)Δn), An(x)=1n∑i=1nKG(x/hn+1,hn)(X~(i−1)Δn)X~(i+1)Δn−X~iΔnΔn, Bn(x)=1n∑i=1nKG(x/hn+1,hn)(X~(i−1)Δn)32(X~(i+1)Δn−X~iΔn)2Δn.

According to the property of Gamma kernel function, we can know that the asymptotic variance of Gamma Nadaraya-Watson estimators for the unknown estimators varies with the design point x in the theorems. For convenience of describing the different asymptotic properties, denote by “interior x” and “boundary x” for a design point x that satisfies x/hn→∞ and x/hn→κ for some κ > 0 as n,T→∞, respectively.

We now present some assumptions used in the paper. In what follows, let D=(l,u) with l≥−∞ and u ≤ ∞ denote the admissible range of the process X_t, K denotes KG(x/hn+1,hn).

3 Large sample properties.

Based on the above assumptions and the lemmas in the following proof procedure part, we have the following asymptotic properties. To simplify notations, we define x∈D to be a

4 Monte carlo simulation study

In this section, compared with the existing kernel methods mentioned in Song (2017), Chen and Zhang (2015), and Song, Lin, and Wang (2013) using Gaussian kernels, the benefits of Nadaraya-Watson estimators for the drift function μ(x) and the volatility function M(x) using the asymmetric kernels are demonstrated through the finite-sample performance for various second-order jump-diffusion models. For brevity, the estimator proposed in this paper is denoted as AS, the estimator in Song (2017) as NW, the estimator in Chen and Zhang (2015) as LL and the estimator in Song, Lin, and Wang (2013) as RNW.

Throughout this section, we employ Gamma kernel KG(x/hn+1,hn)(u)=ux/hnexp⁡(−u/hn)hnx/hn+1Γ(x/hn+1) and Gaussian kernel K(x)=12πe−x22. The common bandwidths for the estimators investigated in Song (2017), Chen and Zhang (2015), and Song, Lin, and Wang (2013) using Gaussian kernels are selected as h=cS^(nΔn)−15=cS^T−15, where Ŝ denotes the standard deviation of the data and c represents different constants for different estimators with c = 2.8 for μ^n(x) and c = 1.3 for M^n(x), one can refer to part (4.2) in Xu and Phillips (2011) for more details. In addition, the normal confidence level is assumed to be 95%.

Example 1: Gaussian jumps.

In this subsection, a simple simulation experiment is considered for the model (10) aimed at evaluating the better finite-sampling performance of nonparametric estimators (9) constructed with Gamma asymmetric kernels

where the coefficients of continuous part are similar as the ones used in Nicolau (2007) and J_t is a compound Poisson jump process, that is, Jt=∑n=1NtZtn with arrival intensity λ⋅T=20 and jump size Zn∼N(0,0.0362) corresponding to Bandi and Nguyen (2003), where t_n is the nth jump of the Poisson process N_t. Taking the integral from 0 to t in the second expression of (10), we obtain

By (11) we have

X_t can be sampled by the Euler-Maruyama scheme according to (11), which will be detailed in the following algorithm (one can refer to Cont and Tankov (2004)). One sample trajectory of the differentiated process X_t and integrated process Y_t with T = 10, n = 1000, X₀ = 0 and Y₀ = 100 using Algorithm 1 is shown in Figure 1. Through the observation of Figure 1(B), we can find the features of the integrated process Y_t: absent mean-reversion, persistent shocks, time-dependent mean and variance, nonnormality, etc.

Figure 1:

Sample Paths of X_t and Y_t for Model (10).

(A) The differentiated process X_t. (B)The integrated process Y_t.

Here we will select the optimal smoothing parameter hnAS for Nadaraya-Watson estimation using asymmetric kernels based on the mean square error (MSE) according to Theorem 2 and Theorem 3 as shown in Remark 6. Similarly as the common bandwidths selected for estimators constructed with Gaussian kernels, the data-driven bandwidth is represented as hnAS=cS^T−25 for “interior x” or h=cS^T−13 for “boundary x”, where c is an unknown parameter to be selected. Curve of MSE(c) for the unknown coefficients in model (10) based on Gamma kernels with λ = 2, T = 10, n = 1000 and jump size Zn∼N(0,0.0362) is depicted in Figure 2, from which we can select the optimal parameter c_opt = 1.44 for drift function and c_opt = 1.11 for volatility function.

Figure 2:

Curve of MSE(c) for the unknown coefficients in model (10) based on Gamma kernels with λ = 2, T = 10, n = 1000 and jump size Zn∼N(0,0.0362).

(A) Curve of MSE(c) for drift function. (B) Curve of MSE(c) for volatility function.

The Nadaraya-Watson estimators for drift function μ(x) = −x and volatility function M(x)=0.01+0.01×x2+2×0.0362 in model (10) constructed with Gaussian and Gamma kernels from a sample with λ = 2, T = 10, n = 1000 and jump size Zn∼N(0,0.0362) are shown in Figure 3. We can observe that AS performs better than the others such as NW, LL and RNW, especially at the boundary points, which is also verified through the biases calculated at various points of the sample X_t, in Table 2. Additionally, Figure 4 represents 95% Monte Carlo confidence intervals for these estimators of the unknown coefficients based on Remark 8, and Table 3 shows the length of confidence bands of various estimators for drift and volatility functions. They show that the lengths of 95% Monte Carlo confidence intervals for μ(x) constructed with Gamma asymmetric kernels are almost shorter than those constructed with Gaussian symmetric kernels and those for M(x) using Gamma asymmetric kernels are shorter than the others with Gaussian symmetric kernels especially at the sparse design boundary point, which coincides with the conclusion in Remark 9 and reveals variance reduction. In addition, at the sparse design boundary point, the true value of μ(x) or M(x) cannot fall in the 95% Monte Carlo confidence bands of some nonparametric estimators using Gaussian symmetric kernels, but the 95% Monte Carlo confidence bands constructed with Nadaraya-Watson estimator with Gamma asymmetric kernel.

Figure 3:

Various Estimators for unknown coefficients in model (10) with λ = 2, T = 10, n = 1000 and jump size Zn∼N(0,0.0362).

(A) Estimators for drift function μ(x) = −x. (B) Estimators for volatility function M(x)=0.01+0.01×x2+2×0.0362.

Figure 4:

95% Monte Carlo confidence intervals for unknown coefficients based on various estimators with λ = 2, T = 10, n = 1000 and jump size Zn∼N(0,0.0362).

(A) Confidence bands for drift function μ(x) = −x. (B) Confidence bands for volatility function M(x)=0.01+0.01×x2+2×0.0362.

We will assess the small-sampling performance of Nadaraya-Watson estimators constructed with Gamma asymmetric kernels and those constructed with Gaussian symmetric kernels, for drift function μ(x) and M(x) via the Mean Square Errors (MSE)

and

where μ^(x) or M^(x) are the various estimators of μ(x) or M(x), respectively and {xk}1m are chosen uniformly to cover the range of sample path of X_t.

The mean square error (MSE) of various estimators for the drift function μ(x) or volatility function M(x) are calculated under various lengths of observation time intervals T (= 50, 100, 500) and sample sizes n (= 500, 1000, 5000) with Δn=Tn are listed in Table 4 and Table 7. Table 5 gives the results on these MSE under different jump sizes Z_n and sample sizes n (= 500, 1000, 5000). Table 6 and Table 8 show these values of MSE under different jump arrival intensity λ and sample sizes T (= 10, 50, 100) over 500 replicates.

From Table 4–Table 6 or Table 7–Table 8, we can get the following findings.

• The Nadaraya-Watson estimator for μ(x) or M(x) constructed with Gamma asymmetric kernels performs a little better than that constructed with Gaussian symmetric kernels in terms of MSE under different time spans, different sample sizes, different jump sizes and different arrival intensity. This is mainly due to the fact that the smaller coverage rate for “interior x” and the smaller asymptotic variance for “boundary x”;

• From Table 4 and Table 7, for the same time interval T, as the sample sizes n tends larger, the performances of the estimators constructed with Gamma asymmetric kernels or Gaussian symmetric kernels for μ(x) or M(x) are improved due to the fact that more information for estimation procedure is sampled as Δ_n → 0. However, for the same sample sizes n, as the time interval T expands larger, the performance of the estimators for μ(x) or M(x) gets worse due to the fact that more jumps happens in larger time interval T in steps 3 of Algorithm 1;

• From Table 5, Table 6 or Table 8, for the same jump size or the same jump arrival intensity, as the sample sizes n tends larger, the performances of the estimators for μ(x) or M(x) are also improved due to the fact that more jump information for estimation procedure is collected as Δ_n → 0. However, for the same sample sizes n, as the amplitude or frequency of jump becomes larger, the MSE of the estimators gradually becomes larger because of larger and more jumps which is more likely to produce outliers;

• To some extent, the previous remark confirms that the drift and volatility functions cannot be identified in a fixed time span, which corresponds to the results of Theorem 2 and Theorem 3.

Figure 5 and Figure 6 give the QQ plots of Nadaraya-Watson estimators for the drift function μ(x) and conditional variance function M(x) constructed with Gamma asymmetric kernels at boundary and interior points, that is 10%, 50% and 90% quantile points of X_t, with λ = 2, T = 10, n = 1000 and jump size Zn∼N(0,0.0362). This reveals the normality of the estimators of the drift function μ(x) and conditional variance function M(x) constructed with Gamma asymmetric kernels, which confirms the results in Theorem 2 and Theorem 3.

Figure 5:

QQ plots of Nadaraya-Watson estimator for μ(x) = −x using Gamma kernels at boundary and interior points, that is 10%, 50% and 90% quantile points of X_t, with λ = 2, T = 10, n = 1000 and jump size Zn∼N(0,0.0362).

(A) QQ plot for left point. (B) QQ plot for median point. (C) QQ plot for right point.

Figure 6:

QQ plots of Nadaraya-Watson estimator for M(x)=0.01+0.01×x2+2×0.0362 using Gamma kernels at boundary and interior points, that is 10%, 50% and 90% quantile points of X_t, with λ=2, T=10, n=1000 and jump size Zn∼N(0,0.0362).

(A) QQ plot for left point. (B) QQ plot for median point. (C) QQ plot for right point.

Example 2: Mixed Gaussian jumps (The Variance Gamma model).

In this subsection, a repeated experiment is conducted for another model (15) aimed at evaluating the better finite-sampling performance of nonparametric estimators (9) constructed with Gamma asymmetric kernels