# Nonparametric Range-Based Double Smoothing Spot Volatility Estimation for Diffusion Models

Authors: Jingwei Cai
Journal: Complexity
 (2020)

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## Abstract

We consider nonparametric spot volatility estimation for diffusion models with discrete high frequency observations. Our estimator is carried out in two steps. First, using the local average of the range-based variance, we propose a crude estimator of the spot volatility. Second, we use usual nonparametric kernel smoothing to reconstruct the volatility function from the crude estimator. By inference, we find such a double smoothing operation can effectively reduce the estimation error.


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## Body

## 1. Introduction

In recent years, stochastic differential equations have been widely studied by many researchers and some interesting results have appeared in the literature (see [1–5] and references therein). In particular, as a kind of stochastic differential equations, diffusion processes are usually involved in many popular models in financial mathematics, statistics, and econometrics. These models are widely used in option pricing, hedging, interest rate (exchange rate) modelling, and engineering management.We consider a class of diffusion process as follows:(1)Xt=X0+∫0tαudu+∫0tβudWu,t∈0,T,with an initial condition X0, where Xtt≥0 solves the equation and is often used to describe the logarithmic price of a security. Wtt≥0 is a standard Brownian motion. The continuous process αtt≥0 and βtt≥0 are the drift term and the diffusion term (spot volatility) of the process Xtt≥0.Consider an arbitrary partition:0=t0<t1<t2<⋯<tn=T. For any nonnegative integer i0≤i≤n, we define δi=ti−ti−1. For the sake of discussion, it is assumed that sample observations are selected at equal time interval, and the time interval between adjacent sample observations can be denoted as δ=T/n; hence, ti=iT/n. For the case of unequal time interval, it is enough to define δ=max0≤i≤nti−ti−1.As it is known to all, volatility is very important in the derivative securities pricing, its correct estimation is the basis of reasonable pricing derivatives. In view of the robustness and accuracy of the nonparametric approach in modelling, many scholars use this approach to estimate volatility (see [6–8]). However, most of the nonparametric volatility estimation works were conducted around integral volatility (see [9, 10]).Constraining a kernel function and choosing appropriate bandwidth parameters, Kristensen [11] proposed a filtered kernel-based spot volatility estimator for time-dependent diffusion models:(2)β^τ2=1h∑i=1nKti−τhXti−Xti−12,where K⋅ and h were the kernel function and its bandwidth, respectively. Under certain conditions, they showed the following central limit theorem:(3)hδβ^τ2−βτ2⟶dN0,2βτ4∫RK2sds.Similar to Kristensen [11], Renault et al. [12] proposed a spot volatility estimator which can approach the nonparametric bound arbitrarily with unevenly spaced data at hand. By using multiple observations, Liu et al. [13] provided an “ideal” preaveraging estimator, established its asymptotic distribution, and considered the asymptotic efficiency of the proposed estimator. Recently, more and more scholars began to consider estimating spot volatility (see [14–16]), and spot volatility inference has become a hot issue in the field of regression dynamic research. Under microstructure noise or Poisson jumps, based on delta sequences, Mancini et al. [17] prevented a nonparametric estimation method of spot volatility. Taking into account fixed time-horizon and infill asymptotics, they obtained the full limit theory under unequally sample observation. Under the coexistence of market microstructure noise and multiple transactions, with high frequency data, Liu et al. [18] gave a spot volatility estimation procedure by taking the sum from the local range increments and discussed the estimator's consistency and asymptotic normality. Similar literatures include [19], among others.Range-based volatility estimation technique is an effective method; Alizadeh et al. [20] revealed this theoretically, practically, and numerically. Christensen and Podolskij [21] defined the range in ti−1,ti as(4)rXti,δ=supti−1≤ς,τ≤tiXς−Xτ.They replaced realized return-based variance with range-based variance, and they found that the latter had higher accuracy. For a scaled Brownian motionXt=βWt, Parkinson [22] proposed the pth moment generating function of its range in ti−1,ti as(5)ErXti,δp=λpδip/2βpp≥1,where λp=ErW1,1p.Christensen and Podolskij [23] provided a range-based multipower variation theory for diffusion models with jumps and showed that their theory was more efficient than returned-based estimators. Recent literatures included but not limited to [24–27].If we substituterXti,δ2/λ2 for the square increment Xti−Xti−12 in equation (2), we may propose a range-based spot volatility:(6)β^τ2=1hλ2∑i=1nKti−τhrXti,δ2.Accordingly, we also may provide the following central limit theorem:(7)hδβ^τ2−βτ2⟶dN0,Λ2βτ4∫RK2sds,where Λ2=λ4−λ22/λ22.Bandi and Phillips [28] used double smoothing technique to estimate drift and diffusion function and proposed their robust and reliable spot volatility estimator in which they took both infill asymptotics and long span one into account. Song [29] expanded this technique to the diffusion models with jumps and illustrated that this two-step smoothing estimator could narrow mean-squared error effectively. It is not difficult to find that the diffusion models in the above three literatures are both homogeneous. As far as we know, so far, few scholars have applied the double smoothing method to time-dependent diffusion models.In this paper, the double smoothing technique is used to estimate the spot volatility of time-dependent diffusion models, and the consistency and asymptotic normality of the estimator are obtained. Compared with single smoothing, the double smoothing method has less error and less information loss. Our consideration is twofold. First, in the first step smoothing, we estimate the spot volatility at any time point by calculating local average of square of the range within a suitable time interval. This allows us to replicate as much as possible the actual features of the volatility function. Using the crude estimator obtained from the first smoothing, we then re-estimate the volatility function by means of usual kernel smoothing. Second, the inferiority on choosing moderate frequency data can be effectively overcome by using the range-based technique. Indeed, the return-based technique is a simple and convenient method to estimate volatility function; however, its disadvantages cannot be ignored either. When the sampling frequency is increased, the high frequency data is easily affected by microscopic noise. With high frequency data at hand, the range-based estimator is more precise than the return-based one.


## 2. Nonparametric Range-Based Double Smoothing Spot Volatility Estimation

Define(8)ti,1=inft≥0:t−ti≤ε,ti,j+1=inft≥ti,j+δ:t−ti≤ε,where ε satisfies ε=oP1 and δ/ε=oP1. For any i0≤i≤n, define(9)mi=∑j=1nIti,j−ti≤ε=2εδ+1,where I⋅ and ⋅ denote indicative function and rounding function, respectively.Remark 1.
As a matter of fact, wheni<ε/δ or i>n−ε/δ, it has mi<2ε/δ+1. Fortunately, this is only a small number of cases. Therefore, without loss of generality, for any i0≤i≤n, we suppose mi=2ε/δ+1 and 2ε/δ is an integer, then mi=2ε/δ+1 (m=2ε/δ+1 for short).
Some of the technical conditions necessary are given below, which are common in estimating volatility.T1 The processαtt≥0 satisfies(10)sup0<t<Tαt<∞.T2 The processβtt≥0 is differentiable and satisfies(11)supβs−βt,s,t∈0,T,s−t≤ξ=OPξ1/2logξ1/2sup0≤t≤Tβt2<∞.T3 The bandwidthh satisfies(12)h∼δ1/2log1/δ.T4 Kernel functionK⋅ is differentiable with support −1,1 and satisfies(13)i∫−11Kcdc=1,ii∫−11K′cdc<∞,iii∫−11K2cdc<∞,iv∫−11K3cdc<∞.Remark 2.
As we know, the drift parameter is not important in volatility estimation, so we restrict it simply and indispensably in condition T1.Remark 3.
Condition T2 is a common assumption about the diffusion parameter. The same or similar assumptions are used in [16, 30].Remark 4.
Bandwidth selection is very important in kernel-based estimators. Condition T3 chooses the same bandwidth as [30].
Now, we denote the nonparametric range-based double smoothing spot volatility estimator ofβt2 as(14)β^t2=δh∑i=1nKti−th⋅β⌣ti2,where(15)β⌣ti2=1δmλ2∑j=1mrXti,j,δ2.Remark 5.
In equation (15), we take a certain point as the midpoint and set a reasonable period, during which the volatility is only a function of state variables and has nothing to do with time. By calculating a local average of the range square in the time period and supposing it as the initial value of the spot volatility estimator at the time point, the real characteristics of volatility can be reproduced as much as possible. At the same time, the complete sample data are used in ranges to ensure the integrity of information.


## 3. The Consistency and the Asymptotic Normality of the Estimator

Theorem 1.
Assume that the processXtt≥0 satisfies the model in equation (1) and conditions T1–T4 hold; then, as δ⟶0,(16)β^t2⟶Pβt2.Proof.
β^t2 can be decomposed into A+B, where(17)A=δh∑i=1nKti−th⋅β⌣ti2−βti2,B=δh∑i=1nKti−th⋅βti2.
We discuss the termB first. It can be further decomposed into B1+B2+B3, where(18)B1=1h∑i=1nβti2∫ti−1tiKti−th−Ku−thdu,B2=1h∑i=1n∫ti−1tiKu−thβti2−βu2du,B3=1h∑i=1n∫ti−1tiKu−thβu2du.
ForB1, using Tylor’s formula, we can obtain(19)B1=1h∑i=1nβti2∫ti−1tiK′u−th⋅ti−uh+oti−uhdu=OPδh2∫0TK′u−thdu=OPδh∫−11K′sds.
By the conditions of T3 and T4 (ii), we can getB1=OPδ/h=OP=1. From the conditions of T2 and T4 (i), it is obvious that B2=δ1/2logδ1/2=oP1. Consider B3 term below. Let(20)u−th=s.
Then,(21)B3=∫−11Ksβsh+t2ds=∫−11Ksβt2+OPh1/2logh1/2ds=βt2+OPh1/2logh1/2.
CombiningB1,B2 with B3, we have B⟶Pβt2. In order to prove Theorem 1, we just have to prove A⟶P0, namely, β⌣ti2⟶Pβti2. Let(22)ωi,j=1δmλ2βti2rWi,j,δ2.
It can be known from equation (5) that(23)∑j=1mEωi,j=βti2.
Now, suppose(24)θi,j=ωi,j−Eωi,j,we can obtain(25)Eθi,j2=Λm2βti4,where Λ=λ4−λ22/λ22. Furthermore,(26)∑j=1mEθi,j2⟶P0.
Therefore,(27)∑j=1mωi,j⟶Pβti2.
Next, we proveβ⌣ti2⟶P∑j=1mωi,j. Notice that(28)β⌣ti2−∑j=1mωi,j=1δmλ2∑j=1mrXti,j,δ+βtirWti,j,δrXti,j,δ−βtirWti,j,δ=D1+D2,where(29)D1=1δmλ2∑j=1mrXti,j,δ−βtirWti,j,δ2,D2=2δmλ2∑j=1mrXti,j,δ−βtirWti,j,δβtirWti,j,δ.
ForD1, we further decompose it as follows:(30)D1≤1δmλ2∑j=1msupti,j−1≤ς,τ≤ti,j∫ςταudu+∫ςτβu−βtidWu2≤2δmλ2∑j=1msupti,j−1≤ς,τ≤ti,j∫ςταudu2+2δmλ2∑j=1msupti,j−1≤ς,τ≤ti,j∫ςτβu−βtidWu2=D1,1+D1,2.
From condition T1, it is obvious thatD1,1=OPδ. For D1,2, by the Burkholder–Davis–Gundy inequality (BDG inequality for short), there exists a constant C>0 to make(31)ED1,2≤2Cδmλ2∑j=1mE∫ti,j−1ti,jβu−βti2du.
By the condition of T2 and the definition ofti,j, we can obtain(32)ED1,2=OPεlogε.
Therefore,D1=oP1. For D2, using the decomposition similar to D1, it holds that(33)D2≤2δmλ2∑j=1mβtirWti,j,δsupti,j−1≤ς,τ≤ti,j∫ςταudu+∫ςτβu−βtidWu.
Using Hölder’s inequality, we obtain(34)D2≤21δmλ2∑j=1mβtirWti,j,δ21/2⋅1δmλ2∑j=1msupti,j−1≤ς,τ≤ti,j∫ςταudu+∫ςτβu−βtidWu21/2.
Using Hölder’s inequality again, we may further obtain(35)ED2≤2E1δmλ2∑j=1mβtirWti,j,δ21/2E1δmλ2∑j=1msupti,j−1≤ς,τ≤ti,j∫ςταudu+∫ςτβu−βtidWu21/2.
From the discussion ofD1, it is true that ED2=OPε1/2logε1/2. So, β⌣ti2−∑j=1mωi,j⟶P0, and these complete the proof.Theorem 2.
Assume that the processXtt≥0 satisfies the model in equation (1) and the condition T1–T4 hold, given δ⟶0, such that(36)εh2loghδ2=oP1.(37)hε2logεδ2=oP1.Then,(38)εhδβ^t2−βt2⟶PN0,Λ2βt42∫−11K2sds,where Λ2=λ4−λ22/λ22.Proof.
Decomposeβ^t2−βt2 as follows:(39)β^t2−βt2=β^t2−δh∑i=1nKti−th⋅∑j=1mωi,j+δh∑i=1nKti−th⋅∑j=1mθi,j+δh∑i=1nKti−th⋅∑j=1mEωi,j−βt2=F1+F2+F3.
By the proof of Theorem1, we know(40)F1=OPε1/2logε1/2,F3=OPh1/2logh1/2.
It can be seen from equations (36) and (37) that(41)εhδF1=oP1,εhδF3=oP1.
Now, we discuss(42)εhδF2=εh∑i=1nKti−th∑j=1mθi,j.
Let(43)ρi=εhKti−th∑j=1mθi,j.
For∀i=1,2,…,n,j,k=1,2,…,m, as j≠k,θi,j and θi,k are independent of each other, so(44)Eρi2=εhK2ti−th∑j=1mEθi,j2=εΛmhK2ti−thβti4.
Furthermore,(45)∑i=1nEρi2=εΛmh∑i=1nK2ti−thβti4=εΛmh∑i=1nK2ti−thβti4−βt4+βt4=εΛmδβt4∫−11K2sds+OPhlogh⟶PΛβt42∫−11K2sds.
Similarly,(46)∑i=1nEρi3=ε3/2h3/2∑i=1nK3ti−th∑j=1mEθi,j3=ε3/2h3/2∑i=1nK3ti−thβti6m2λ6−3λ4λ2+2λ23λ23=λ6−3λ4λ2+2λ23λ23⋅ε3/2m2h1/2δβt6∫−11K3sds+OPhlogh.
From Remark1 and condition T4 (iv), it holds that(47)∑i=1nEρi3=Oδε1/2h1/2=oP1.
It can be obtained from Lyapunov's central limit theorem that(48)εhδF2⟶dN0,Λβt42∫−11K2sds.Remark 6.
When we chooseε=δlogδ, equations (36) and (37) are both satisfied. Then, it holds that m=2logδ+1. From equation (38), we can find that the greater the ε you choose, the higher the accuracy of the estimator you will obtain. Therefore, the optimal ε is its maximum, satisfying equations (36) and (37).Remark 7.
In equation (3), the constant in front of βτ4∫RK2sds is 2 while the constant at the corresponding position in equation (7) is Λ2≈0.4 (see [21]), so the precision of the range-based single smoothing estimator is better than the return-based single smoothing estimator. By comparing equations (7) and (38), we find that the range-based double smoothing estimator is more accurate than the single one because δ/εh=oPδ/h.


## 4. Conclusions

Using the entire sample data, range-based estimator not only guarantees the integrity of information but also reduces influence of microstructure noise. Compared with single smoothing, the double smoothing method can replicate the actual features of the volatility function more. In this paper, combining the range technique with the double smoothing method, we propose a range-based double smoothing spot volatility estimator and prove its consistency and asymptotic normality. We also find that the range-based single smoothing estimator is better than the return-based single smoothing one, while the range-based double smoothing one is the best.



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