Arbitrage in the Hermite Binomial Market

Abstract

Much attention has been paid to the arbitrage opportunities in the Black–Scholes model when it is driven by fractional Brownian motions. It is natural to ask whether there exists arbitrage or not when we focus on other fractional processes, such as the Hermite process. We set forth an approximation of the Hermite Black–Scholes model by random walks in the Skorokhod topology, and apply the Donsker type approximation to the Hermite process as the Hurst index is greater than $\frac{1}{2}$. We find that the binary model approximation of the Black–Scholes model driven by Hermite processes also admits arbitrage opportunities. Several numerical examples of the Hermite binomial model are presented as demonstration. Moreover, we provide an option pricing model when the geometric Hermite processes drives the price fluctuations of the underlying asset.

1. Introduction

Since the well-known work of [1], continuous-time models have been widely applied in the field of finance. For instance, [1] used the famous geometric Brownian motion to describe the underlying asset for pricing options. The geometric Brownian motion means that the log-returns of financial assets are independent and Gaussian. However, plenty of works on the statistical properties of log-returns have documented the self-similarity of financial asset returns and the phenomenon of long-range dependence (e.g., [2,3,4]). Thus, using some fractional processes (e.g., [5,6]), econometricians have done a lot of work on describing the long-memory characteristic of financial time-series variables (such as stock prices, commodity prices, market indices, asset volatility and foreign exchange rates). Additionally, stochastic processes with long-range dependence have been considered to depict phenomena in various fields, for example, telecommunication, computer vision (image processing data), traffic networks (ethernet network data), hydrology (noisy water accumulation or rainfall data in river gauges) and so on.

Obviously, fractional Brownian motion (fBm) [7] is one of the most popular and widespread-used stochastic processes that presents long-range dependence. Specifically, behaviors such as self-similar, long-range dependence and the fBm-produced burstiness in the sample path are good examples of financial time-series variables. Therefore, the application of fBm in finance has been discussed extensively from both theoretical and empirical perspectives (e.g., [8,9,10,11,12]). As fBm is not a semimartingale, the basic concepts of mathematical finance offered by [13] has already implied that fBm only allows a certain kind of arbitrage. Hence, the methods of capturing the arbitrage opportunities of the fBm-modulated Black–Scholes model and excluding arbitrage in fBm have been tremendously investigated over the last two decades (e.g., [14,15,16,17,18,19,20,21,22]). In this case, it is complicated to utilize an fBm-modulated Black–Scholes model in finance as fBm can be used to model both self-similar phenomena and long-range dependence for financial time-series variables, and an fBm-modulated Black–Scholes model captures arbitrage opportunities. It has been proved that any arbitrage pricing method cannot be properly used in an fBm-modulated Black–Scholes model. Moreover, the non-Gaussian property and the nonstationary property in finance could not be well described by an fBm-modulated Black–Scholes model (e.g., [23,24,25,26,27,28]). As a result, a few general fractional processes have been introduced, which contain the same properties of fBm (regularity of paths, long-memory, self-similarity) but nonstationary and non-Gaussian properties, such as the Hermite process and the Rosenblatt process. While fBm is extensively applied in finance, some scholars focus on other fractional processes (e.g., [6,28,29]), such as the Hermite process, a self-similar stochastic process with long-range dependence, that fBm is a particular case of. However, other than fBm, the Hermite process is non-normal and can survive the fat-tail situation in practical applications. That is why the Hermite processes is widely used in finance (e.g., [28]).

The Hermite process is worth studying as it fails to be a semimartingale. This is mainly because following the basic theorem of asset pricing (e.g., [13]), the asset pricing is enforced to be a semimartingale. In this situation, two questions arise naturally: first, whether the Black–Scholes model based on the Hermite process admits arbitrage opportunities or not; second, how to create an arbitrage opportunity in the Hermite Black–Scholes model within the binomial setting. Those are exactly what we strive to figure out in this paper; we set forth a binary market model based on the Hermite process and present arbitrage opportunities in this type of model.

For these purposes, following the lines of [21,30], we construct a binary market model, which use random walks in the Skorokhod topology to approximate the Hermite modulated Black–Scholes model. We find that this kind of model admits arbitrage opportunities. Hence, for the purpose of providing a specific case of the arbitrage related to the Hermite process, we subject a binary market model to approximate the Hermite Black–Scholes model, whose randomness of risky asset comes from the Hermite process. It is profitable to work on binary markets because of their simplicity and flexibility to approximate other more complicated models. To build a binary market model driven by the Hermite process, we use a Donsker type theorem to approximate the process through I.I.D random variables in the Skorokhod topology. Fortunately, Ref. [31] found that some disturbed random walks, which came from an I.I.D. sequence of random variables, could approximate the Hermite process. Utilizing this approximation method, we provide a Hermite binary market model under the assumption that the Hurst parameter is greater than one half. We prove that arbitrage opportunities do survive in this approximating “semimartingale” model and create an arbitrage opportunity by capturing the path information beginning at time zero. Furthermore, we show some numerical examples in Hermite binomial models as illustration and discuss the issue of pricing options in the geometric Hermite process environment. The results in this paper are close in spirit to those in [21,30], where the arbitrage in the Black–Scholes model based on fBm and the Rosenblatt processes are investigated, respectively. Nevertheless, our analysis of the Black–Scholes model based on the Hermite process requires analogous but different efforts compared to [21,30].

The rest of this paper is organized as follows. Section 2 describes some fundamental properties of the Hermite process and some facts of the approximation theorem for the Hermite process. Section 3 introduces the definition of a Hermite binary market, which approximates the Hermite modulated Black–Scholes model. Moreover, we show that arbitrage opportunities do survive in such a model. Section 4 exhibits several numerical examples driven by simulated data to demonstrate arbitrage opportunities in the Hermite modulated Black–Scholes model and discuss the question about pricing options when the geometric Hermite process drives the underlying asset pricing. Section 5 offers some conclusions.

2. Hermite Process as a Limit of a Random Walk

In this section, we first recollect several fundamental concepts of the Hermite process. Then, motivated by [21,30], we introduce a Donsker invariance principle for the Hermite process. We now introduce the definition and show the fundamental characteristics of the Hermite process.

Definition 1. 

Let ${\left(W_t\right)}_{t\in \mathbb{R}}$ be a Wiener process. The Hermite process ${\left(X_H^k\left(t\right)\right)}_{t\ge 0}$ of order k and with self-similarity index $H\in \left(\frac{1}{2},1\right)$ is defined as

$$X_H^k\left(t\right)=c(H,k){\int }_{{\mathbb{R}}^k}{\int }_0^t\left(\prod _{j=1}^k{\left(s-y_i\right)}_{+}^{-\left(\frac{1}{2}+\frac{1-H}{k}\right)}\right)dsd{W}_{y_1}\cdots d{W}_{y_k},$$

(1)

where $z_{+}=max(z,0)$, $c(H,k)$ is a normalizing constant so that $\mathbb{E}{\left(X_H^k\left(t\right)\right)}^2=1$ and $c^2(H,k)={\left(\frac{\beta {\left(\frac{1}{2}-\frac{1-H}{k},\frac{2H-2}{k}\right)}^k}{k!H(2H-1)}\right)}^{-1}$. The aforementioned integral is a multiple integral with order k regarding the Brownian motion W.

Undoubtedly, the most widely used Hermite process is fBm, which generates in (1) for $k=1$. Moreover, it is the only Gaussian Hermitian process. Indeed, a Hermite process with order two is known as the Rosenblatt process and $X_H^k$ is non-Gaussian if $k\ge 2$. Apart from Gaussianity, the Hermite processes with order $k\ge 2$ and fBm (coinciding with $k=1$) have many common characteristics. Here, we present some basic properties, which can be found in Chapter 3 of [32].

Proposition 1. 

The Hermite process and fBm have many common properties as follows:

• The process shows long-range dependence (the covariance function decays like the power function at zero);
• The process is H-self-similar under the circumstance that for any $c>0,\left(X_H^k\left(ct\right)\right)\stackrel{law}{=}\left(c^H{X}_H^k\left(t\right)\right)$, where $\stackrel{law}{=}$ indicates that entire finite-dimensional distributions are equivalent;
• The increments of the process is stationary, which means that the joint distribution of increments of the process, $\left(X_H^k(t+h)-X_H^k\left(h\right),t\in [0,T]\right)$ is independent with $h>0$;
• The covariance function of the process is

  $$\mathbb{E}\left(X_H^k\left(t\right)X_H^k\left(s\right)\right)=\frac{1}{2}\left(t^{2H}+s^{2H}-{|t-s|}^{2H}\right),s,t\in [0,T].$$

• Accordingly, for every $s,t\in [0,T]$,

$$\mathbb{E}{\left|X_H^k\left(t\right)-X_H^k\left(s\right)\right|}^2={|t-s|}^{2H};$$

(2)

• The process is Hölder continuous with order $\delta <H$.

More characteristics of the Hermite process can be found in [32]. These interesting features make the Hermite process a potential substitute for models that contain self-similarity, long-range dependence and are non-Gaussian and nonstationary. Therefore, it is worthwhile to consider the practice of the Hermite process in various fields. Obviously, one thought-provoking question arises naturally when applying the Hermite process in finance: Does the Hermite modulated Black–Scholes model admit arbitrage opportunities? First, we provide a general case of the arbitrage related to the Hermite modulated Black–Scholes. Subsequently, we look briefly at the method of binomially approximating the Hermite process inspired by [21,30]. To design the approximating binary market model, a Hermite process based on Donsker’s theorem is required. This result is well established for classical Brownian motion and generates instinctive intuition into regarding Brownian motion as the limit of I.I.D. random variables. Additionally, Ref. [21] found that a “disturbed” random walks could approximate fBm. Ref. [30] provided the Rosenblatt process with a Donsker type approximation theorem. Accordingly, one might expect that a comparable method of the Hermite process should also be possibly approximated by I.I.D. random variables. Fortunately, Ref. [31] demonstrated a weak convergence theorem in the Hermite process by using a martingale difference.

It is well known that, for $k=1$ and $H\in (\frac{1}{2},1)$, the process given by (1) is nothing else but a fBm, which conforms to the kernel formula below involving the standard Brownian motion $W_t$

$$\begin{array}{c}\hfill X_H^1\left(t\right)\stackrel{law}{=}{\int }_0^t{K}^H(t,s)d{W}_s,t\in [0,T]\end{array}$$

(3)

with a standard Wiener process $\left(W_t,t\in [0,T]\right)$ and

$$\begin{array}{c}\hfill K^H(t,s)=c_H{s}^{\frac{1}{2}-H}{\int }_s^t{(u-s)}^{H-\frac{3}{2}}{u}^{H-\frac{1}{2}}du\end{array}$$

(4)

where $t>s$, $c_H={\left(\frac{H(2H-1)}{\beta \left(2-2H_{-1}\right)}\right)}^{\frac{1}{2}},\beta$ is the beta function.

Paper [30] offers a resembling representation for the Rosenblatt process. Thus, we get

$$\begin{array}{c}\hfill X_H^2\left(t\right)\stackrel{law}{=}d\left(H\right){\int }_0^t{\int }_0^t\left[{\int }_{y_1\vee y_2}^t\frac{\partial K^{H^{\prime }}}{\partial u}\left(u,y_1\right)\frac{\partial K^{H^{\prime }}}{\partial u}\left(u,y_2\right)du\right]d{W}_{y_1}d{W}_{y_2},\end{array}$$

where $\left(W_t,t\in [0,T]\right)$ is a one-dimensional Brownian motion. The integral here is an iterated double integral about W as $d\left(H\right)=\frac{1}{H+1}{\left(\frac{H}{2(2H-1)}\right)}^{-\frac{1}{2}}$, $H^{\prime }=\frac{H+1}{2}$ and

$$\begin{array}{c}\hfill \frac{\partial K^H}{\partial t}(t,s)=c_H{\left(\frac{s}{t}\right)}^{\frac{1}{2}-H}{(t-s)}^{H-\frac{3}{2}}.\end{array}$$

(5)

Following the inspiration of [21,30], the Hermite process can be described as a multiple integral, with regard to a Wiener process in a finite time interval when using the kernel $K^H$ driven by (4). Let $H\in \left(\frac{1}{2},1\right)$, for $k\ge 1$, we have

$$\begin{array}{cc}\hfill X_H^k\left(t\right)=& d\left(H\right){\int }_0^t\cdots {\int }_0^{t}d{W}_{y_1}\cdots d{W}_{y_k}\hfill \\ \hfill & \times \left({\int }_{y_1\vee \cdots \vee y_k}^t{\partial }_1{K}^{H^{\prime }}\left(u,y_1\right)\cdots {\partial }_1{K}^{H^{\prime }}\left(u,y_k\right)du\right),t\in [0,1],\hfill \end{array}$$

(6)

where $H^{\prime }=1+\frac{H-1}{k}⟺\left(2H^{\prime }-2\right)k=2H-2$ and $d\left(H\right)$ is such that $\mathbb{E}{\left|Y_1^{(q,H)}\right|}^2=1$.

Let us recall some notable facts, which are important for our fundamental analysis. With respect to a series of I.I.D random variables ${\left\{{\xi }_i\right\}}_{i\in \mathbb{N}}$ with $\mathbb{E}{\xi }_i=0$ and $\mathbb{E}{\xi }_i^2=1$. The Donsker invariance principle states that the sequence of processes

$$\begin{array}{c}\hfill W_t^n=\frac{1}{\sqrt{n}}\sum _{i=1}^{\lfloor nt\rfloor }{\xi }_i\end{array}$$

converges weakly to a standard Brownian motion in the Skorokhod topology, where $\lfloor x\rfloor$ defines the greatest integer not exceeding x. The thought of using $W_t^N$ in option pricing was introduced by [33] and extended by [34,35].

Then, Ref. [21] extended this result to the example of fBm as the Hurst parameter $H>\frac{1}{2}$. We have

$$K^n(t,s):=n{\int }_{s-\frac{1}{n}}^{s}K(\frac{\lfloor nt\rfloor }{n},u)du,n\ge 1,$$

where $K(·,·)$ is the kernel that transforms the standard Brownian motion into an fBm, i.e.,

$$K^H\left(t,s\right)=\sqrt{\frac{2H\mathsf{\Gamma }(\frac{3}{2}-H)}{\mathsf{\Gamma }(H+\frac{1}{2})\mathsf{\Gamma }(2-2H)}}\left(H-\frac{1}{2}\right)s^{\frac{1}{2}-H}{\int }_s^t{\left(u-s\right)}^{H-\frac{3}{2}}{u}^{H-\frac{1}{2}}du.$$

(7)

Let

$$X_H^{1,n}\left(t\right)={\int }_0^t{K}^n(t,s)d{W}_s^n=\sum _{i=1}^{\lfloor nt\rfloor }n{\int }_{\frac{i-1}{n}}^{\frac{i}{n}}{K}^H(\frac{\lfloor nt\rfloor }{n},s)ds\frac{{\xi }_i}{\sqrt{n}},n\ge 1.$$

The work of [21] confirms that the disturbed random walk $X_H^{1,n}\left(t\right)$ converges weakly to the fBm with $H>\frac{1}{2}$. Moreover, for $k=2$ and $H>\frac{1}{2}$,30] states that the disturbed random walk $X_H^{2,n}\left(t\right)$ converges weakly to the Rosenblatt process, where

$$\begin{array}{ccc}\hfill X_H^{2,n}\left(t\right)& =& \sum _{i,j=1;i\ne j}^{\left[nt\right]}{n}^2{\int }_{\frac{i-1}{n}}^{\frac{i}{n}}{\int }_{\frac{j-1}{n}}^{\frac{j}{n}}F\left(\frac{\left[nt\right]}{n},u,v\right)dvdu\frac{{\xi }_i}{\sqrt{n}}\frac{{\xi }_j}{\sqrt{n}},t\in [0,T],\hfill \\ \hfill F\left(t,y_1,y_2\right)& =& d\left(H\right)1_{[0,t]}\left(y_1\right)1_{[0,t]}\left(y_2\right){\int }_{y_1\vee y_2}^t\frac{\partial K^{H^{\prime }}}{\partial u}\left(u,y_1\right)\frac{\partial K^{H^{\prime }}}{\partial u}\left(u,y_2\right)du.\hfill \end{array}$$

Now, we apply a Donsker invariance principle to the Hermite process. Afterwards, we consider the Hermite process that is given by formula (6). For every $t\in [0,T]$, we denote that

$$\begin{array}{ccc}\hfill F\left(t,y_1,y_2\cdots ,y_k\right)& =& d\left(H\right)1_{[0,t]}\left(y_1\right)1_{[0,t]}\left(y_2\right)\cdots 1_{[0,t]}\left(y_k\right)\hfill \\ & & {\int }_{y_1\vee y_2\cdots \vee y_k}^t\frac{\partial K^{H^{\prime }}}{\partial u}\left(u,y_1\right)\frac{\partial K^{H^{\prime }}}{\partial u}\left(u,y_2\right)\cdots \frac{\partial K^{H^{\prime }}}{\partial u}\left(u,y_k\right)du,\hfill \end{array}$$

(8)

where 1${}_{(0,t)}\left(s\right)$ is an indicator function equal to one iff $s\in (0,t)$ and to zero otherwise.

Then, for $t\in [0,T]$, let

$$\begin{array}{c}\hfill X_t^k\end{array}={\int }_0^T{\int }_0^T\cdots {\int }_0^{T}F\left(t,y_1,y_2\cdots ,y_k\right)d{W}_{y_1}d{W}_{y_2}\cdots d{W}_{y_k}.$$

(9)

From the results of representation (9) and the fBm, for $t\in [0,T]$, we have the representation for the Hermite process

$$\begin{array}{ccc}\hfill X_t^{k,n}& =& {\displaystyle \sum _{{\eta }_1,{\eta }_2,\cdots ,{\eta }_k=1;{\eta }_i\ne {\eta }_j\mathit{for}i\ne j}^{\left[nt\right]}}{n}^k{\int }_{\frac{{\eta }_1-1}{n}}^{\frac{{\eta }_1}{n}}{\int }_{\frac{{\eta }_2-1}{n}}^{\frac{{\eta }_2}{n}}\cdots {\int }_{\frac{{\eta }_k-1}{n}}^{\frac{{\eta }_k}{n}}F\left(\frac{\left[nt\right]}{n},s_1,s_2,\cdots ,s_k\right)\hfill \\ & & d{s}_{1}d{s}_2\cdots d{s}_k\frac{{\xi }_1}{\sqrt{n}}\frac{{\xi }_2}{\sqrt{n}}\cdots \frac{{\xi }_k}{\sqrt{n}}.\hfill \end{array}$$

(10)

Before proceeding with the demonstration of the binary model for the Hermite modulated Black–Scholes model, referring to the idea of [31], we state one proposition, which is employed in the next section.

Remark  1. 

As we can see, the random variables $({\xi }_i;i\ge 1)$ are binary with $P({\xi }_i=1)=P({\xi }_i=-1)=\frac{1}{2}$ for every $i\ge 1$. Consequently, $X_t^{k,n}$ can only take two values since the product term of $X_t^{k,n}$

$$\frac{{\xi }_1}{\sqrt{n}}\frac{{\xi }_2}{\sqrt{n}}\cdots \frac{{\xi }_k}{\sqrt{n}}=\left\{\begin{array}{c}0\mathrm{for}\mathrm{any}{\xi }_i=0,\\ 1\mathrm{for}\mathrm{all}{\xi }_i=1.\end{array}\right.$$

(11)

Remark  2. 

According to (10) and (11), we can see that $X_t^{k,n}$, which is an approximation of the Hermite process, is also a binary process. Thanks to this approximation, we can discuss arbitrage opportunities in the Hermite Black–Scholes model following the idea of [21,30].

Proposition  2. 

The random walk $X_t^{k,n}$ (defined by (10)) converges weakly to the Hermite process $X_t^k$ (obtained from (9)) as the Hurst parameter $H>\frac{1}{2}$ in the Skorokhod space with n goes to infinity.

Proof. 

The proof includes proving that the finite-dimensional distributions of $X_t^{k,n}$ converge to those of $X_t^k$ as well as exhibiting that $X_t^{k,n}$ is tight, which can be found in [31] and hence omitted here. □

Remark  3. 

It is crucial to note that the replacement of $X_t^k$ by $X_t^{k,n}$ is in line with the critical characteristics of the original process. Thus, $X_t^{k,n}$ is long-range dependent and its distribution is asymptotically self-similar.

3. Binary Market of the Hermite Black–Scholes Model

In classical cases (that is, those market models based on a Brownian motion), the binary model is utilized to approximate the notable Black–Scholes model, as is shown in [33]. In this section, inspired by the idea of [21,30], we find that the binary model based on (10) is also capable of approximating the Hermite modulated Black–Scholes model.

Firstly, we focus on two financial assets that are traded constantly in the time period $[0,T]$, where 0 is the current day and the terminal day T is fixed. Let a risk-free asset indicated by B satisfy the following dynamics equation

$$B_t={\int }_0^t{r}_s{B}_{s}ds,$$

(12)

where $r_t$ is a differentiable deterministic function.

Secondly, the risk asset with price dynamics satisfies the stochastic formula

$$S_t=S_0+{\int }_0^t{a}_s{S}_{s}ds+\sigma {\int }_0^t{S}_{s}d{X}_s^k,$$

(13)

where $X_t^k$ is a Hermite process with $H>\frac{1}{2}$, $\sigma >0$ is a constant and a is a differentiable deterministic function. The integral regarding $X_t^k$ is comprehended from a pathwise perspective. Considering the Hermite process is Hölder continuous of order $\delta <H$ with $H>\frac{1}{2}$, we can define the pathwise integrals in connection with the Hermite process and solve the stochastic formula of (13) in a pathwise sense. More details can be found in [36]. Consequently, analogous to the effort of [36], we adopt the stochastic integral in (13) as Riemann–Stieltjes, then based on the change of variable formula, the result of (13) can be written as

$$S_t=S_0{e}^{{\int }_0^t{a}_{s}ds+\sigma X_t^k},t\in [0,T].$$

(14)

Moreover, it is easy to get the solution of (12) as

$$B_t=B_0{e}^{{\int }_0^t{r}_{s}ds},t\in [0,T].$$

(15)

In the rest of the study, we presume that both the drift of the security $a_t$ and the interest rate $r_t$ are deterministic bounded functions.

Following [21,30], we focus on a specific binary market, an N-period Hermite binary market, which approximates the Hermite modulated Black–Scholes model of (13). In the binary market, we assume that a market where two securities (a stock S and a bond B) are traded over continuous time periods $0=t_0<t_1<t_2<\dots <\dots <t_N=T$ and their dynamics are given by

$$B_n=(1+r_n)B_{n-1}$$

(16)

and

$$S_n=(a_n+(1+Z_n))S_{n-1},$$

(17)

with $n\ge 1$. For each $n\in \{0,1,\dots ,N-1\}$, $B_n$ and $S_n$ denote the values of the bonds and the stocks during the interval between $t_n$ and $t_{n+1}$, respectively. Likewise, $r_n$ and $a_n$ indicate the interest rate and the drift of the stock, respectively. The adapted stochastic process $Z={\left(Z_n\right)}_{n=0}^N$ is binary. Thus, following the idea of [30], given $Z_1,\dots ,Z_{n-1}$, at time n, it can take two possible values defined by $d_n$ (“down”) and $u_n$ (“up”) with $d_n<u_n$. Let us mention that for $k=1$, the market consisting of (16) and (17) is the financial market from [21]. For $k=2$, the market consisting of (16) and (17) is the financial market from [30]. For $k\ge 2$, we can easily see that the market, which consists of (16) and (17), is also a financial market. While $a_n$ from (17) is deterministic, the values of $u_n$ and $d_n$ may be determined by the path of Z until time $n-1$. The parameters $u_n$ and $d_n$ could be regarded as the real-valued functions on ${\{-1,1\}}^{n-1}$ (here $u_1$ and $d_1$ are constants).

From Proposition 3.6.2 in [37] and using similar arguments as in [21,30], we can see that a binary market eliminates arbitrage possibilities iff for all $n\in \{1,\dots ,N\}$,

$$d_n<r_n-a_n<u_n.$$

(18)

Applying Proposition 3.6.2 of [37], we can see that a binary market excludes arbitrage opportunities if and only if for all $n=1,\dots ,N$, the condition $d_n<r_n-a_n<u_n$ is satisfied. Indeed, this is relative to the existence of the equivalent martingale measure. Thus, let $\mathbf{P}$ be the law of X in (17). Then, we have to find a probability measure $\mathbf{Q}$, which equivalent to $\mathbf{P}$, such that $\frac{S}{B}$ is a $\mathbf{Q}$-martingale. It is easy to have that such $\mathbf{Q}$ must satisfy $u_n\mathbf{Q}\left(Z_n=u_n\mid Z_1,\dots ,Z_{n-1}\right)+d_n\mathbf{Q}\left(Z_n=d_n\mid Z_1,\dots ,Z_{n-1}\right)=$$r_n-a_n$, or equivalently $\mathbf{Q}\left(Z_n=u_n\mid Z_1,\dots ,Z_{n-1}\right)=\frac{r_n-an-dn}{u_n-d_n}\in (0,1)$. Obviously, the condition of $\frac{r_n-an-dn}{u_n-d_n}\in (0,1)$ is equivalent to the condition in (18). Moreover, this condition defines a unique martingale measure, i.e., the binary market models are complete. For more details, see Chapter 3 of [37].

We will omit the superscript “k” in $X_t^{k,n}$ and $X_t^n$ without any confusion. Then, the stochastic process Z in (17) is defined by

$$Z_n=\mathsf{\Delta }{X}_{\frac{n}{N}}^N,n=0,\dots \left[NT\right],$$

(19)

where $X_{\frac{n}{N}}^N$ is defined by (10). Moreover, we define

$$r_n=\frac{1}{N}{r}_{\frac{n}{N}}and{a}_n=\frac{1}{N}{a}_{\frac{n}{N}}$$

(20)

where a and r are the functions presenting in (12) and (13), respectively. Given an appropriate selection of the sequence ${\left({\xi }_n\right)}_{n\in \mathbb{N}}$ and the initial value $X_{\frac{1}{N}}^N$, it is fair to say that the model is truly a binary model. In addition, motivated by [21,30], we can prove that this model approximates the Hermite modulated Black–Scholes model under the subsequent circumstances. Although a similar proof can also be found in [21,30], on account of the completeness of our study, the concise derivations are exhibited here.

Theorem  1. 

The binary market model with Z, a and r given by (16) and (17) converges to the Black–Scholes model based on the Hermite process given by (12) and (13) as $N\to \infty$.

Proof. 

First, we focus on the jump component of $\left[X^N\right]$. It is obvious that

$$\mathsf{\Delta }{X}_t^N=X_t^N-X_{t-}^N.$$

Next, the quadratic variation of $\left[X^N\right]$ can be calculated as

$${\left[X^N\right]}_t=\sum _{s\le t}{\left(\mathsf{\Delta }{X}_s^N\right)}^2.$$

Then, we present that the process $\left[X^N\right]$ converges in $L^1([0,T]\times \mathsf{\Omega })$ to zero where $L^1$ denotes the $L^1$ norm. Since the jumps are at time $\frac{k}{N}$, we have

$$\mathbb{E}{\left|\mathsf{\Delta }{X}_t^N\right|}^2\le \mathbb{E}{\left|X_t^N-X_{t-\frac{1}{N}}^N\right|}^2\le \frac{C}{N^{2H}},$$

where the last inequality is driven from (2).

Furthermore, we obtain

$$\mathbb{E}{\left[X^N\right]}_t\le Nt\frac{1}{N^{2H}}=t{N}^{1-2H}.$$

It indicates that

$${\int }_0^T{\left[X^N\right]}_{s}ds\le C\left(T\right)N^{1-2H},$$

which tends to zero as N tends to infinity, where $H>\frac{1}{2}$ and $C\left(T\right)$ denotes a constant depending on T.

The following demonstration resembles the proof of Theorem 3 in [21]. However, our result is slightly different from that in [21], so we give the brief derivations here. By (17) it suffices to show that

$$S_t^N:=S_0^N\prod _{s\le t}(1+\mathsf{\Delta }{X}_s^N),$$

(21)

which converges weakly to the “geometric Hermite process” $S_0^N{e}^X$. We divide the process $X^{H,N}$ into two parts $X_t^{1,N}={\sum }_{s\le t}\mathsf{\Delta }{X}^H{\mathbf{1}}_{\{\mathsf{\Delta }{X}_s^N<\frac{1}{2}\}}$ and $X_t^{2,N}={\sum }_{s\le t}\mathsf{\Delta }{X}^N{\mathbf{1}}_{\{\mathsf{\Delta }{X}_s^N\ge \frac{1}{2}\}}$. In this way, we have

$$S^N=S^{1,N}{S}^{2,N},$$

where $S_t^{i,N}={\prod }_{s\le t}(1+\mathsf{\Delta }{X}_s^{i,N})(i=1,2).$ Resembling the proof of Theorem 3 in [21], we can easily obtain that $S^{1,N}$ converges weakly to $e^X$ while $S^{2,N}$ goes to one in probability. □

What is more, we show that the binary market based on (16) and (17) is not free of arbitrage. Specifically, motivated by [21,30], utilizing (10) and (19), for every $n\ge 2$ we have

$$\begin{array}{ccc}\hfill Z_n& =& \sigma \sqrt[k]{N}\sum _{{\eta }_1,{\eta }_2,\cdots ,{\eta }_k=1;{\eta }_i\ne {\eta }_j\mathit{for}i\ne j}^n({\int }_{\frac{{\eta }_1-1}{N}}^{\frac{{\eta }_1}{N}}{\int }_{\frac{{\eta }_2-1}{N}}^{\frac{{\eta }_2}{N}}\cdots {\int }_{\frac{{\eta }_k-1}{N}}^{\frac{{\eta }_k}{N}}(F\left(\frac{n}{N},s_1,s_2,\cdots ,s_k\right)\hfill \\ & & -F\left(\frac{n-1}{N},s_1,s_2,\cdots ,s_k\right))d{s}_{1}d{s}_2\cdots d{s}_k){\xi }_1{\xi }_2\cdots {\xi }_k.\hfill \end{array}$$

The random variables $({\xi }_i;i\ge 1)$ are binary, so

$$P({\xi }_i=1)=P({\xi }_i=-1)=\frac{1}{2}$$

for every $i\ge 1$.

For every $n\ge 2$, isolating the part relating to ${\xi }_n$, we obtain

$$Z_n=f_{n-1}({\xi }_1,\dots ,{\xi }_{n-1})+{\xi }_n{g}_{n-1}({\xi }_1,\dots ,{\xi }_{n-1}).$$

where

$$\begin{array}{ccc}\hfill f_{n-1}(x_1,\dots ,x_{n-1})& =& \sigma \sqrt[k]{N}\sum _{{\eta }_1,{\eta }_2,\cdots ,{\eta }_k=1;{\eta }_i\ne {\eta }_j\mathit{for}i\ne j}^{n-1}({\int }_{\frac{{\eta }_1-1}{N}}^{\frac{{\eta }_1}{N}}{\int }_{\frac{{\eta }_2-1}{N}}^{\frac{{\eta }_2}{N}}\cdots {\int }_{\frac{{\eta }_k-1}{N}}^{\frac{{\eta }_k}{N}}\hfill \\ & & \left(F\left(\frac{n}{N},s_1,s_2,\cdots ,s_k\right)-F\left(\frac{n-1}{N},s_1,s_2,\cdots ,s_k\right)\right)\hfill \\ & & d{s}_{1}d{s}_2\cdots d{s}_k)x_1{x}_2\cdots x_k,\hfill \end{array}$$

(22)

and

$$\begin{array}{ccc}\hfill g_{n-1}(x_1,\dots ,x_{n-1})& =& 2\sigma \sqrt[k]{N}\sum _{i=1}^{n-1}({\int }_{\frac{i-1}{N}}^{\frac{i}{N}}{\int }_{\frac{n}{N}}^{\frac{n-1}{N}}\cdots {\int }_{\frac{n}{N}}^{\frac{n-1}{N}}\hfill \\ & & F\left(\frac{n}{N},s_1,s_2,\cdots ,s_k\right)d{s}_{1}d{s}_2\cdots d{s}_k)x_i.\hfill \end{array}$$

(23)

Note that ${\xi }_i$’s are binary, we have

$$u_n=f_{n-1}({\xi }_1,\dots ,{\xi }_{n-1})+g_{n-1}({\xi }_1,\dots ,{\xi }_{n-1})$$

(24)

and

$$d_n=f_{n-1}({\xi }_1,\dots ,{\xi }_{n-1})-g_{n-1}({\xi }_1,\dots ,{\xi }_{n-1}).$$

(25)

Now, motivated by [21,30], let us show the major result of our work.

Theorem 2. 

The Hermite binary market with (16) and (17) admits arbitrage opportunities.

Proof. 

Throughout this demonstration, most estimates involve unspecified constants. In order to have a clear notation, let $C\left(H\right)$ be a positive generic constant whose value is determined by H only and that varies from appearance to appearance. Then, we find that the condition (18) is invalid for some $n\ge 2$. Particularly, by symmetry we have

$$f_{n-1}(1,1,\dots ,1)-g_{n-1}(1,1,\dots ,1)⟶\infty ,$$

(26)

as $n\to \infty$.

Paying attention to $f_{n-1}(1,1,\dots ,1)$, applying (22), we have

$$\begin{array}{ccc}\hfill f_{n-1}(1,1,\dots ,1)& =& \sigma \sqrt[k]{N}\sum _{{\eta }_1,{\eta }_2,\cdots ,{\eta }_k=1}^{n-1}({\int }_{\frac{{\eta }_1-1}{N}}^{\frac{{\eta }_1}{N}}{\int }_{\frac{{\eta }_2-1}{N}}^{\frac{{\eta }_2}{N}}\cdots {\int }_{\frac{{\eta }_k-1}{N}}^{\frac{{\eta }_k}{N}}\hfill \\ & & \left(F\left(\frac{n}{N},s_1,s_2,\cdots ,s_k\right)-F\left(\frac{n-1}{N},s_1,s_2,\cdots ,s_k\right)\right)\hfill \\ & & d{s}_{1}d{s}_2\cdots d{s}_k)x_1{x}_2\cdots x_k\hfill \\ & & -\sigma \sqrt[k]{N}\sum _{{\eta }_i=1}^{n-1}({\int }_{\frac{{\eta }_i-1}{N}}^{\frac{{\eta }_i}{N}}{\int }_{\frac{{\eta }_i-1}{N}}^{\frac{{\eta }_i}{N}}\cdots {\int }_{\frac{{\eta }_i-1}{N}}^{\frac{{\eta }_i}{N}}\hfill \\ & & \left(F\left(\frac{n}{N},s_1,s_2,\cdots ,s_k\right)-F\left(\frac{n-1}{N},s_1,s_2,\cdots ,s_k\right)\right)\hfill \\ & & d{s}_{1}d{s}_2\cdots d{s}_k)x_1{x}_2\cdots x_k\hfill \\ & :=& \sigma \sqrt[k]{N}\left(A-B\right).\hfill \end{array}$$

(27)

Using (5) and (8), we have

$$\begin{array}{ccc}\hfill A& =& C\left(H\right)({\int }_0^{\frac{{\eta }_1}{N}}{\int }_0^{\frac{{\eta }_2}{N}}\cdots {\int }_0^{\frac{{\eta }_k}{N}}·{\int }_{\frac{n-1}{N}}^{\frac{n}{N}}{(a-s_1)}^{H^{\prime }-\frac{3}{2}}{(a-s_2)}^{H^{\prime }-\frac{3}{2}}\dots {(a-s_k)}^{H^{\prime }-\frac{3}{2}}\hfill \\ & & a^{k{H}^{\prime }-\frac{k}{2}}{s}_1^{\frac{1}{2}-H^{\prime }}{s}_2^{\frac{1}{2}-H^{\prime }}\dots s_k^{\frac{1}{2}-H^{\prime }}da·d{s}_{1}d{s}_2\cdots d{s}_k)\hfill \\ & \ge & C\left(H\right){\left(\frac{n-1}{N}\right)}^{\frac{k}{2}-k{H}^{\prime }}({\int }_0^{\frac{{\eta }_1}{N}}{\int }_0^{\frac{{\eta }_2}{N}}\cdots {\int }_0^{\frac{{\eta }_k}{N}}·{\int }_{\frac{n-1}{N}}^{\frac{n}{N}}{(a-s_1)}^{H^{\prime }-\frac{3}{2}}\hfill \\ & & {(a-s_2)}^{H^{\prime }-\frac{3}{2}}\dots {(a-s_k)}^{H^{\prime }-\frac{3}{2}}{a}^{k{H}^{\prime }-\frac{k}{2}}da·d{s}_{1}d{s}_2\cdots d{s}_k)\hfill \\ & \ge & C\left(H\right){\left(\frac{n-1}{N}\right)}^{\frac{k}{2}-k{H}^{\prime }}{\int }_{\frac{n-1}{N}}^{\frac{n}{N}}{\left(a^{H^{\prime }-\frac{1}{2}}-{\left(a-\frac{n-1}{N}\right)}^{H^{\prime }-\frac{1}{2}}\right)}^k{a}^{k{H}^{\prime }-\frac{k}{2}}da\hfill \\ & \ge & C\left(H\right){\left(\frac{n-1}{N}\right)}^{\frac{k}{2}-k{H}^{\prime }}{\int }_{\frac{n-1}{N}}^{\frac{n}{N}}{\left(\frac{{(n-1)}^{H^{\prime }-\frac{1}{2}}-1}{N^{H^{\prime }-\frac{1}{2}}}\right)}^k{a}^{k{H}^{\prime }-\frac{k}{2}}da\hfill \\ & =& \frac{C\left(H\right)}{N^{k{H}^{\prime }-\frac{k}{2}+1}}o\left(n^{k{H}^{\prime }-\frac{k}{2}}\right).\hfill \end{array}$$

(28)

Next, we deal with B. We denote that

$$\begin{array}{ccc}\hfill B& =& C\left(H\right)\sum _{{\eta }_i=1}^{n-1}{\int }_{\frac{{\eta }_i-1}{N}}^{\frac{{\eta }_i}{N}}{\int }_{\frac{{\eta }_i-1}{N}}^{\frac{{\eta }_i}{N}}\cdots {\int }_{\frac{{\eta }_i-1}{N}}^{\frac{{\eta }_i}{N}}·{\int }_{\frac{n-1}{N}}^{\frac{n}{N}}{(a-s_1)}^{H^{\prime }-\frac{3}{2}}{(a-s_2)}^{H^{\prime }-\frac{3}{2}}\hfill \\ & & \dots {(a-s_k)}^{H^{\prime }-\frac{3}{2}}{a}^{k{H}^{\prime }-\frac{k}{2}}{s}_1^{\frac{1}{2}-H^{\prime }}{s}_2^{\frac{1}{2}-H^{\prime }}\dots s_k^{\frac{1}{2}-H^{\prime }}da·d{s}_{1}d{s}_2\cdots d{s}_k\hfill \\ & \le & C\left(H\right){\left(\frac{n}{N}\right)}^{k{H}^{\prime }-\frac{k}{2}}\sum _{{\eta }_i=1}^{n-1}{\int }_{\frac{{\eta }_i-1}{N}}^{\frac{{\eta }_i}{N}}{\int }_{\frac{{\eta }_i-1}{N}}^{\frac{{\eta }_i}{N}}\cdots {\int }_{\frac{{\eta }_i-1}{N}}^{\frac{{\eta }_i}{N}}·{\int }_{\frac{n-1}{N}}^{\frac{n}{N}}{(a-\frac{n}{N})}^{k{H}^{\prime }-\frac{3k}{2}}\hfill \\ & & s_1^{\frac{1}{2}-H^{\prime }}{s}_2^{\frac{1}{2}-H^{\prime }}\dots s_k^{\frac{1}{2}-H^{\prime }}da·d{s}_{1}d{s}_2\cdots d{s}_k\hfill \\ & \le & C\left(H\right){\left(\frac{n}{N}\right)}^{k{H}^{\prime }-\frac{k}{2}}\sum _{{\eta }_i=1}^{n-1}{\int }_{\frac{{\eta }_i-1}{N}}^{\frac{{\eta }_i}{N}}{\int }_{\frac{{\eta }_i-1}{N}}^{\frac{{\eta }_i}{N}}\cdots {\int }_{\frac{{\eta }_i-1}{N}}^{\frac{{\eta }_i}{N}}{s}_1^{\frac{1}{2}-H^{\prime }}{s}_2^{\frac{1}{2}-H^{\prime }}\dots s_k^{\frac{1}{2}-H^{\prime }}\hfill \\ & & \times \frac{{(n-i)}^{k{H}^{\prime }-\frac{3k}{2}+1}-{(n-(i-1))}^{k{H}^{\prime }-\frac{3k}{2}+1}}{N^{k{H}^{\prime }-\frac{3k}{2}+1}}d{s}_{1}d{s}_2\cdots d{s}_k\hfill \\ & \le & C\left(H\right){\left(\frac{n}{N}\right)}^{k{H}^{\prime }-\frac{k}{2}}\sum _{{\eta }_i=1}^{n-1}{\left({\left(\frac{i}{N}\right)}^{\frac{3}{2}-H^{\prime }}-{\left(\frac{i-1}{N}\right)}^{\frac{3}{2}-H^{\prime }}\right)}^k\hfill \\ & & \times \frac{{(n-i)}^{k{H}^{\prime }-\frac{3k}{2}+1}-{(n-(i-1))}^{k{H}^{\prime }-\frac{3k}{2}+1}}{N^{k{H}^{\prime }-\frac{3k}{2}+1}}\hfill \\ & \le & C\left(H\right){\left(\frac{n}{N}\right)}^{k{H}^{\prime }-\frac{k}{2}}\frac{1}{N^{\frac{3k}{2}-k{H}^{\prime }}}{\left(\frac{n}{N}\right)}^{k{H}^{\prime }-\frac{3k}{2}+1}\hfill \\ & \le & C\left(H\right)\frac{1}{N^{k{H}^{\prime }-\frac{k}{2}+1}}o\left(n^{2k{H}^{\prime }-2k+1}\right).\hfill \end{array}$$

(29)

Substituting (28) and (29) into (27), we can see

$$\begin{array}{c}\hfill f_{n-1}(1,1,\dots ,1)\to \infty ,\end{array}$$

(30)

as $n\to \infty$.

Regarding the term $g_{n-1}(1,1,\dots ,1)$, a standard calculation shows that

$$\begin{array}{ccc}\hfill g_{n-1}(1,1,\dots ,1)& \le & 2\sigma \sqrt[k]{N}\sum _{i=1}^{n-1}\left({\int }_{\frac{i-1}{N}}^{\frac{i}{N}}{\int }_{\frac{n}{N}}^{\frac{n-1}{N}}\cdots {\int }_{\frac{n}{N}}^{\frac{n-1}{N}}F\left(\frac{n}{N},s_1,s_2,\cdots ,s_k\right)d{s}_{1}d{s}_2\cdots d{s}_k\right)\hfill \\ & \le & 2\sigma \sqrt[k]{N}C\left(H\right){\left(\frac{n}{N}\right)}^{k{H}^{\prime }-\frac{k}{2}}\sum _{i=1}^{n-1}{\int }_{\frac{i-1}{N}}^{\frac{i}{N}}{\int }_{\frac{n}{N}}^{\frac{n-1}{N}}\cdots {\int }_{\frac{n}{N}}^{\frac{n-1}{N}}{\int }_{s_k}^{\frac{n}{N}}\hfill \\ & & s_1^{\frac{1}{2}-H^{\prime }}{s}_2^{\frac{1}{2}-H^{\prime }}\dots s_k^{\frac{1}{2}-H^{\prime }}{\left(\frac{n-1}{N}-s_1\right)}^{H^{\prime }-\frac{3}{2}}{(a-s_2)}^{H^{\prime }-\frac{3}{2}}\hfill \\ & & {(a-s_3)}^{H^{\prime }-\frac{3}{2}}\dots {(a-s_k)}^{H^{\prime }-\frac{3}{2}}da·d{s}_{1}d{s}_2\cdots d{s}_k\hfill \\ & \le & 2\sigma \sqrt[k]{N}C\left(H\right){\left(\frac{n}{N}\right)}^{k{H}^{\prime }-\frac{k}{2}}{\left(\frac{n-1}{N}\right)}^{\frac{k-1}{2}-\left(k-1\right)H^{\prime }}\hfill \\ & & \sum _{i=1}^{n-1}{\int }_{\frac{i-1}{N}}^{\frac{i}{N}}{\int }_{\frac{n}{N}}^{\frac{n-1}{N}}\cdots {\int }_{\frac{n}{N}}^{\frac{n-1}{N}}{s}_1^{\frac{1}{2}-H^{\prime }}{\left(\frac{n-1}{N}-s_1\right)}^{H^{\prime }-\frac{3}{2}}\hfill \\ & & {\left(\frac{n}{N}-s_2\right)}^{H^{\prime }-\frac{1}{2}}{\left(\frac{n}{N}-s_3\right)}^{H^{\prime }-\frac{1}{2}}\dots {\left(\frac{n}{N}-s_k\right)}^{H^{\prime }-\frac{1}{2}}d{s}_{1}d{s}_2\cdots d{s}_k\hfill \\ & \le & 2\sigma \sqrt[k]{N}C\left(H\right){\left(\frac{n}{N}\right)}^{k{H}^{\prime }-\frac{k}{2}}{\left(\frac{n-1}{N}\right)}^{\frac{k-1}{2}-\left(k-1\right)H^{\prime }}{\left(\frac{1}{N}\right)}^{\frac{k-1}{2}+\left(k-1\right)H^{\prime }}\hfill \\ & & {\int }_0^{\frac{n-1}{N}}{s}_1^{\frac{1}{2}-H^{\prime }}{\left(\frac{n-1}{N}-s_1\right)}^{H^{\prime }-\frac{3}{2}}d{s}_1\hfill \\ & \le & 2\sigma \sqrt[k]{N}C\left(H\right){\left(\frac{1}{N}\right)}^{k{H}^{\prime }+\frac{k-2}{2}}o\left(n^{H^{\prime }-1/2}\right).\hfill \end{array}$$

(31)

Ultimately, using (30) and (31), we can achieve (26). □

Remark 4. 

Consistent with the fBm case ([21]) and the Rosenblatt process case ([30]), the analogous binary models also admit arbitrage opportunities when $H>\frac{1}{2}$.

Remark 5. 

As we can see in this section, the Black–Scholes market driven by the pure Hermite process may admit arbitrage opportunities in the discrete-time situation. However, this does not mean that we rule out the application of the Hermite process in finance. In fact, several approaches can be employed for excluding arbitrage in the Hermite process. For example, the absence of arbitrage possibilities may be achieved by taking into account the peculiarities of trading in a real market (see [38,39]), modifying the stochastic process of the stock price in order to get a semimartingale [40], replacing the weighting kernel of the integral representation of the Hermite process with a related one [19], using the fractional model to describe the volatility [41,42,43,44,45,46] or the interest rate [47,48,49,50] and so on.

Remark 6. 

It would be interesting to see what happens in the Hermite process case when the self-similarity order H tends to $\frac{1}{2}$ and $k=1$. We can expect to have no-arbitrage when $H=\frac{1}{2}$ as in the standard Brownian motion case because now the Hermite process at $H=\frac{1}{2}$ and $k=1$ collapses to a martingale.

Remark 7. 

It should be mentioned that our paper extends the result of [21] with $k=1$ and [30] with $k=2$. Thus, our paper provides a general method to detect the financial arbitrage for any stochastic process, whose representation is the integral with respect to Wiener processes on a finite interval.

4. Simulation Analysis

This section reports several numerical cases to demonstrate the violation of applying the Hermite modulated Black–Scholes model. First, we discuss the arbitrage opportunities in the Black–Scholes model based on the Hermite process. Then, we handle the issue about pricing options in the Hermite modulated Black–Scholes model.

Now, we discuss the violation of the assumption that arbitrage opportunities cannot survive in the Hermite modulated Black–Scholes model. Under the circumstance, we simulated data of a Hermite process utilizing the approximation given by (10), with ${\left({\xi }_n\right)}_{n\in \mathbb{N}}$ a series of I.I.D. binary random variables for different values of H. Then, formula (19) made it easy for us to obtain (21), which converges weakly to the geometric Hermite process. Let ${\xi }_1,{\xi }_2\dots ,{\xi }_k$ take values $\{-1,1\}$ in (10). Now, we can analyze the arbitrage in Hermite binary markets, which are denoted by (16) and (17) with a series of binary models approximating the Hermite modulated Black–Scholes model. Figure 1, Figure 2, Figure 3 and Figure 4 delineate some binomial trees generated by the described random walk with n = 10,000, $t=4$ and $S_0^4=1$ for different values of the Hurst parameter H and k.

It is worth mentioning that for $H=\frac{1}{2},k=1$ and the standard Brownian motion case, we obtain the highly recombining binomial tree. This is mainly because for a classical Brownian motion, we can approximate it by raising the amount of identically and independently distributed random variables. However, based on Figure 1, Figure 2, Figure 3 and Figure 4, we cannot acquire a recombinant tree in Hermite binary markets when the Hermite process relates to an infinite past. Thus, for any time t, the moving average of the historic awareness must be completely re-estimated based on weights of the entire history. Consequently, an independent symmetric modeling is deficient (please see [20] for a similar discussion of fBm). As we can see from Figure 1, Figure 2, Figure 3 and Figure 4, the four trees are no longer recombining, and the outer branches are no longer steady lines and symmetric trajectory. Similar results were obtained previously in fractional binary market (e.g., [18,20]). Thereby, the envelopes of Figure 1, Figure 2, Figure 3 and Figure 4 illustrate a convex shape of nature. The main reason for this is due to the occurrence of persistence. In this case, it enforces the selected path and exacerbates the deviation from the average. Moreover, the terminal nodes are not equidistant at all.

Now, let us take a closer look at Figure 1, Figure 2, Figure 3 and Figure 4 again. At the fourth step of each figure, arbitrage opportunities exist. Thus, associating with the nodes before last, it shows that—beginning with the top node—both descending branches appear to trend upward. This means due to the lack of a risk-free rate, a one-step buy-and-hold strategy would always pay and yield risk-free returns. For example, money can be borrowed to purchase the risk asset if the price reaches 1.1851, so that it generates a risk-free profit (see Figure 1). It is perhaps worth pointing out that the representation of an arbitrage opportunity implies a considerable potential earning. However, it can be quite deceptive, since the arbitrage “opportunity” could be an opportunity against investors. In other words, an arbitrage opportunity for one investor may be a risk that another investor cannot hedge against. Figure 1, Figure 2, Figure 3 and Figure 4 illustrate this clearly, where investors holding assets at 0.8343 (see Figure 1), at 0.8495 (see Figure 2), at 0.8666 (see Figure 3) and at 0.8866 (see Figure 4), face certain losses and should sell. Furthermore, using a similar argument as above, we can achieve that starting at any point in the binary trees in Figure 1, Figure 2, Figure 3 and Figure 4, we can reach the point of arbitrage by only going up or down enough times. As a result, we come to the conclusion that it exists the possibility of arbitrage in Hermite binary markets.

Lastly, we handle the issue about pricing options in the Hermite modulated Black–Scholes environment. Obviously, we cannot directly use the binomial trees shown in Figure 1, Figure 2, Figure 3 and Figure 4 to price options due to the absence of a martingale measure. However, when the set of permissible strategies is limited, arbitrage can be eliminated by prohibiting the trading strategies with arbitrage profits. Hence, the pricing formula of the Black–Scholes European options under the continuous-time model with the Hermite process driving noise is given. We do not discuss it further here, since the work has been done by [28] in continuous-time models. In fact, when one uses the Hermite modulated Black–Scholes model to price options, the information of the underlying asset is not simply transferred by volatility and spot price—as in the standard Black–Scholes framework—but rather in a more complex way. Moreover, we can see form [28] that when the driving noise is the Hermite process, the price of an option with maturity T relies on $T-t$ as well as the value of t. In the Black–Scholes market, where an investor observes the market at time $t_1$ and an investor enters it at time $t_2>t_1$, just be consistent with the volatility $\sigma$ (possibly by estimating its implied value or realized values) so that options are priced in the same way at time $t\ge t_2$. In the Hermite modulated Black–Scholes environment, it is insufficient to only consider the value of the coefficient $\sigma$ in Equation (13). For the purpose of replacing the first investor’s option pricing function, the second investor—in the Hermite market circumstance—should rely on the value of $t_1$ as well. This is unusual but expected, as the Hermite market rules out (except the particular one when $H=\frac{1}{2}$) the independence of increments. We expect that our discoveries offer a new perspective to observe arbitrage in a binary market.

5. Conclusions

The traditional stochastic model used to capture the fluctuations in financial markets is the Markov process. However, the ongoing financial crisis has shown that most of the Markov models used do not fit some properties of the observed financial data. One reason may be that Markov models fail to capture the long-range dependence exhibited by financial asset returns. Based on the existence of long-range dependence and non-Markovianity, scholars have introduced many stochastic processes with long-range dependence in the financial field. FBm is the most wildly applied process with long-range dependence in a continuous period with its Hurst parameter $H\in (\frac{1}{2},1)$. To be specific, fBm is extensively used in financial economics. However, some research suggest that arbitrage opportunities exist in the fBm-driven Black–Scholes market. Similar arbitrage possibilities can also be found in the Black–Scholes market based on the Rosenblatt process. In this case, many authors have proposed some new stochastic processes with long-range dependence, such as Hermite processes, bi-fBm, sub-fBm, weighted fBm, etc.

In this paper, we introduced the approximation of the Hermite process driven by a series of I.I.D random variables in Skorokhod topology and discussed arbitrage opportunities in the Hermite binomial model. For illustrating the arbitrage opportunities in the Hermite binomial model, we also provided some numerical examples. Our results showed that there were arbitrage opportunities for the binary approximation of the Hermite modulated Black–Scholes model when $H\in (\frac{1}{2},1)$, which is attributable to the long-range dependence in the Hermite process as $H>\frac{1}{2}$. Inspiringly, if a security’s price has a raising tendency long enough, it will keep going up for a while. This result shows that no arbitrage pricing method can be reasonably utilized in the Hermite market model. Such arbitrage opportunities within a discrete period do not preclude the utilization of the Hermite processes as a model in finance. Whether or not arbitrage opportunities exist in continuous time relies on the interpretation of the permissible trading strategies, just like transaction costs or expensive information. Thus, setting appropriate constraints on trading strategies, a reasonably predictable interest rate and introducing a so-called mixed model can incorporate the Hermite model with an arbitrage-free framework.

Furthermore, the paper introduces some different perspectives for future research. Firstly, we can present the approximations of bi-fBm, sub-fBm, weighted fBm or a fractional Brownian sheet based on random walks and prove the presence of arbitrage possibilities in the binomial models. The discussions give some intriguing extended content of the work on derivatives pricing in a long-range dependence circumstance. The second suggestion for future research is to apply pathwise or $L^2$ integration theory to the corresponding continuous framework and use long-range dependence processes other than fBm to obtain derivatives pricing models. Thirdly, we can further give the proof that at any node of the Hermite binomial tree, one can arrive at an arbitrage spot by only tracking up or down for enough time. We can also show that when H is near one, the probability of a path in the binary tree crossing infinitely many arbitrage points is one. This implies that, within the limits, there is an infinite number of arbitrage points. This is why we just provide Hermite binomial trees when H is large. Moreover, we can propose a different description of the permissible trading strategy to avoid the presence of arbitrage strategies and may offer insights into examining the robustness of the results in this paper.