Asset Pricing Model Based on Fractional Brownian Motion

Abstract

This paper introduces one unique price motion process with fractional Brownian motion. We introduce the imaginary number into the agent’s subjective probability for the reason of convergence; further, the result similar to Ito Lemma is proved. As an application, this result is applied to Merton’s dynamic asset pricing framework. We find that the four order moment of fractional Brownian motion is entered into the agent’s decision-making. The decomposition of variance of economic indexes supports the possibility of the complex number in price movement.

1. Introduction

Many continuous-time stochastic models have been established after researchers realized the role of Brownian motion in describing prices. Merton (1973) [1] and Merton (1975) [2] initiated a new research framework of the asset pricing model under continuous-time and discussed the portfolio theory under this framework. Davis and Norman (1990) introduced the model with transaction cost in [3]. Zhou 1997 introduced the jump process into the option pricing model based on continuous time and used the Poisson process and normal distribution to describe the discontinuities of company value in [4]. Liu 2007 introduced the form of the solution when the rate of return is a quadratic diffusion process in [5].

Many studies consider using fractional Brownian motion, which was first studied by Kolmogorov 1940 [6], as a substitute for Brownian motion.

Definition 1.

For Hurst parameter $H\in (0,1)$, fractional Brownian motion (FBM) $B^H:=\{B^H(t),t\in R\}$, is the Gaussian process. We take $B^H(0)=0$. Its mean for all t is: $E[B^H(t)]=0$ and its covariance is:

$$\begin{array}{cc}\hfill & E[B^H(t)B^H(s)]=\frac{1}{2}{\{|t|}^{2H}+{\left|s\right|}^{2H}{-|t-s|}^{2H}\}\\ \hfill ⟺& E[{(B^H(t)-B^H(s))}^2]={|t-s|}^{2H}.\end{array}$$

(1)

For $H=1/2$, $B^{\frac{1}{2}}$ is standard Brownian motion.

The use of fractional Brownian motion to characterise asset prices has attracted the attention of many researchers. Mandelbrot and Van Ness 1968 [7] defined the integral transformation from Brownian motion to fractional Brownian motion. Peters 1994 [8] introduced the fractal efficient market hypothesis to challenge the efficient market hypothesis. Rogers 1997 [9] and Cheridito 2001 [10] introduced the regularizing fractional Brownian motion with long memory but no-arbitrage. Cheridito 2001 [10] showed that if there is an upper limit on trading frequency, the arbitrage example in Peters 1994 [9] may be excluded. Sottinen 2001 discussed an approximation that converges weakly to fractional Brownian motion, then discussed a basic market model under this approximation, and discussed the existence of arbitrage opportunities in [11]. Guasoni 2006 discussed how arbitrage opportunities are eliminated with transaction costs in [12]. Hu and Øksendal 2003 [13] introduced the Ito random integral with different fractional Brownian motion between $(1/2,1)$ and showed that there is no arbitrage at this time. Elliott and Van Der Hoek 2003 generalized this result and proposed a framework that combines fractional Brownian motions of different orders between $(0,1)$ in the same process and discussed the option pricing formula under this framework in [14]. Xiao et al. 2010 [15] introduced a currency option pricing model combining the jump process and fractional Brownian motion. Xiao and Yu 2019 [16] developed a class of the Vasicek model with fractional Brownian motion as the random source. Of particular attention is the result made by [17], which is very relevant to us. The core difference is that we use a dynamic optimization approach, the solution process and the final result have more economic implications and we also focus on the possibility of the existence of imaginary numbers. In fact, one of the important reasons why fractional Brownian motion is difficult to shine in economic research is the lack of a matching dynamic optimization framework. A good dynamic optimization framework can help economic researchers understand the agent’s decision-making and the overall movement of the economy. Our main contribution is to give a dynamic optimisation framework that remains feasible under fractional Brownian motion.

In recent years, most of the research on fractional Brownian motion about the economic has focused on option and bond pricing. Miao and Yang 2015 [18] priced convertible bonds involving mixed fractional Brownian motion. Shokrollahi and Kılıçman 2015 [19] priced currency option, when the value of foreign currency option follows the mixed fractional Brownian motion with jumps. Rao 2016 [20] priced the geometric Asian power option. Shokrollahi et al. 2016 priced European options and currency options by time changed mixed fractional Brownian motion in [21]. Zhang et al. 2020 proposed a fuzzy mixed fractional Brownian motion model with jumps in [22]. The article on option pricing is of high applied value. However, the purely pricing equations hardly bring us clearer economic intuition because they are not based on individual decisions. Our core work is to give a solution to the exogenous process as a fractional Brownian motion individual lifetime optimisation solution. A solution based on individual decision making can lead to clearer economic implications.

This paper is arranged as follows. In Section 2, some real data are decomposed to show the necessity of considering $H\le \frac{1}{2}$. The decomposition of real data also supports the idea of introducing imaginary numbers. In Section 3, we show the classical results and discuss the rationality of our independence hypothesis for different stochastic processes. In Section 4, we will discuss the Merton type single asset and multi-asset continuous-time consumption asset pricing models corresponding to multi fractional Brownian motion process. In Section 5, we summarize our results. The proof in this paper is in the Appendix A, Appendix B and Appendix C.

2. Decomposition of Real Data

We used monthly U. S. consumption data, S & P 500 daily data and tick level data of the Shanghai Composite Index. The monthly U. S. consumption data and S & P 500 daily data are from the CEIC data set, and the Shanghai composite index data is from Resset data. We show descriptive statistics in

Table 1. Descriptive Statistics.

Variables	Consumption	BP500	Tick_Shanghai
Obs	120	2000	1200
Mean	0.0299677	0.0873281	0.0000137
Std. Dev.	0.0291915	0.0492139	8.78 × 10${}^{-6}$
Min	0.0000297	0.0001219	2.43 × 10${}^{-8}$
Max	0.0988004	0.1808946	0.0000294
Begin	Jan-59	1 January 1976	1 December 2018
End	Jan-19	30 April 2020	31 December 2018

This table is to show the descriptive statistics of the variance. Source: CEIC dataset and Resset data.

.

Among the three data, we find that the variance of the logarithmic growth rate of monthly consumption in the United States has the most obvious nonlinear property with respect to the growth of sampling interval:

It can be seen from Figure 1 that the variance of the logarithmic growth rate of consumption shows an obvious nonlinear growth with respect to the sampling interval, which is contrary to the assumption of random walk.

When an economic stochastic process p satisfies:

$$dp/p=\mu dt+{\displaystyle \sum _{j=1}^{\infty }}{\sigma }_{j}d{B}^{H_j}.$$

(2)

When $B^{H_j}$ independent of each other, then:

$$var(log(p(t+dt))-log(p))={\displaystyle \sum _{j=1}^{\infty }}{\displaystyle {\sigma }_j^2}d{t}^{2H_j}.$$

(3)

On the premise that Gaussian processes can be decomposed orthogonally, we find that the variance of multi fractional Brownian motion can be decomposed of different time increments. This formula also implies that the coefficients should be positive when we do a regression of the power exponents for different time intervals using the variance measured at different frequencies. If the coefficients are negative, then it is possible that the presence of imaginary numbers is implied.

Next, we use the empirical method to try to decompose this relationship in

Table 2. Data Decomposition Results.

Variables	U. S. Consumption	BP500	Tick_Shanghai
$d{t}^2$	2.19 × 10${}^{-5}$ ***	2.83 × 10${}^{-7}$ ***	−5.96 × 10${}^{-11}$ ***
	7.11 × 10${}^{-8}$	1.26 × 10${}^{-8}$	0
$d{t}^{\frac{3}{2}}$	−0.000357 ***	−3.83 × 10${}^{-5}$ ***	5.99 × 10${}^{-9}$ ***
	2.69 × 10${}^{-6}$	1.83 × 10${}^{-6}$	2.69 × 10${}^{-10}$
$dt$	0.00476 ***	0.00310 ***	−2.91 × 10${}^{-7}$ ***
	6.29 × 10${}^{-5}$	0.000162	1.85 × 10${}^{-8}$
$d{t}^{\frac{3}{4}}$	−0.0114 ***	−0.0166 ***	1.37 × 10${}^{-6}$ ***
	0.000196	0.00097	9.79 × 10${}^{-8}$
$d{t}^{\frac{1}{2}}$	0.00887 ***	0.0287 ***	−1.88 × 10${}^{-6}$ ***
	0.000195	0.00184	1.64 × 10${}^{-7}$
Constant	−0.00188 ***	−0.0255 ***	1.01 × 10${}^{-6}$ ***
	6.57 × 10${}^{-5}$	0.00219	1.53 × 10${}^{-7}$
Observations	120	2000	1200
R-squared	1	0.997	1

The explained variable is the variance of the logarithmic growth rate of several economic indicators. $dt$ is the sampling interval of economic data. $variance={\beta }_0+{\beta }_{1}d{t}^2+{\beta }_{2}d{t}^{\frac{3}{2}}+{\beta }_{3}dt+{\beta }_{4}d{t}^{\frac{3}{4}}+{\beta }_{5}d{t}^{\frac{1}{2}}$. Standard errors in parentheses: *** p < 0.01, ** p < 0.05, * p < 0.1. Source: CEIC dataset and Resset data.

.

From the empirical results, we can see that the coefficients of different power of $dt$ are significant, so we think that the economic data is composed of different fractional Brownian motions. Interestingly, the coefficients of some terms are negative, which means that some ${\displaystyle {\sigma }_j^2}$ in (3) are negative, which is consistent with the introduction of the complex number.

We have used this approach for a wider range of data, seeing the Appendix A, Appendix B and Appendix C.

The real economic time series characteristics we find are derived from autocorrelation. In the past, most studies have been anchored on the Hurst parameter to investigate autocorrelation in economic time series. If an economic time series is a stationary process, its Hurst parameter is $1/2$. For the definition of the Hurst parameter, see Definition 1. Eom et al. 2007 [23] demonstrated the role of the Hurst parameter in measuring market efficiency. Many real data support the existence of chaos in real economic laws. Brock and Sayers 1988 [24] tested the U. S. macroeconomic data and confirmed the existence of two-dimensional chaos. Ashley and Patterson 1989 [25] found that nonlinear dynamics is an important feature of the macroeconomic index. Karuppiah and Los 2005 [26] pointed out that the Hurst index of foreign exchange is in the range of $(0,0.5)$. Wang et al. 2009 found that the Hurst parameters of stocks in the U.S. market obey normal distribution in [27]. Hassani and Thomakos 2010 introduced the successful application of single spectrum analysis in financial time series, which confirmed the non-stationarity of financial time series in [28]. Onali and Goddard 2011 analyzed the European market and found that the Italian stock market has a long memory in [29]. Garnier and Sølna 2017 introduces the option pricing model with volatility correlation in [30]. Caraiani 2012 ([31]) found evidence of the existence of chaos in the central and Eastern European stock markets. Bal and Rath 2015 ([32]) found that there is a nonlinear Granger causality between crude oil and exchange rate in India. Jin 2016 found that during the financial crisis in 2008, the return of the Asian market showed long memory, and the Hurst parameter sometimes was above $0.5$, sometimes below $0.5$ in [33]. Bariviera 2017 found that the Hurst parameter results of the bitcoin market measured by different methods are different, but overall, they are greater than $0.5$ in [34]. Wei 2018 ([35]) found that among 456 cryptocurrencies, the increase of liquidity will reduce the predictability of prices and found that the mean value of Hurst parameters of cryptocurrencies with stronger predictability is less than $0.5$. Takaishi 2020 thought that the generalized Hurst parameter of logarithmic fluctuation increment of bitcoin price is less than $0.5$ in [36].

Of course, some scholars are against it. For example, Shintani and Linton (2003) [37] believed that there may be no chaos in the macroeconomy, or it is difficult to identify because of the small amount of data.

From the past research, we can get the chaotic system from the general equilibrium theory. Brock and Hommes 1997 [38] discussed the local steady-state instability in a heterogeneous belief model with agent learning. Hommes et al. 2005 [39] discussed the chaos of asset prices when heterogeneous traders are allowed to update their beliefs asynchronously. Chiarella et al. 2006 [40] discussed the survival of traders in different chaotic sequences in a heterogeneous belief pricing model that allows traders to update their beliefs. Sommervoll et al. 2010 [41] discussed in a complex dynamic system that if the house is set as the mortgage of the loan, the price will fluctuate sharply.

3. Classical Model and Orthogonalization

In Section 3.1, we recall a classical continuous-time asset pricing model from [2] as a basis for further exploration. In order to give a basis for the assumption that fractional Brownian motions in asset price movements are independent of each other, in Section 3.2 we give an orthogonalization of stochastic processes. The existence of an orthogonalization implies that although stochastic processes in reality may be correlated with each other, we can transform the stochastic processes in reality into stochastic processes that are independent of each other.

3.1. Classical Model and Result

In the classical model, it is assumed that the price of risky asset p satisfies the following conditions: $dp/p=rdt+\sigma dB$, and B satisfies the standard Brownian motion: $dB\sim N(0,dt)$, $E_t(dBdB)=dt$. r is the expected return rate of risk assets, $\sigma$ is the risk exposure. Risk-free asset price $p_f$ satisfies $d{p}_f/p_f=r_{f}dt$, where $r_f$ is the return rate of risk-free assets. w is the process of wealth movement. x is the weight of risk assets. c is the consumption. Individual investors need to solve the optimization problem as follows:

$$\begin{array}{cc}\hfill ma{x}_{c,w}& E{\displaystyle {\int }_0^{+\infty }}{e}^{-\rho t}(\frac{c^{\alpha }}{\alpha })dt\hfill \\ \hfill s.t.& dw=(w(r_f(1-x)+rx)-c)dt+xw\sigma dB\hfill \\ \hfill & w(0)=w_0\hfill \end{array}$$

Finally, it can be transformed into a system of differential Equations (where V is the Bellman Equation):

$$\begin{array}{c}\hfill c^{\alpha -1}-V^{\prime }=0\\ \hfill V^{\prime }w(r-r_f)+V^{″}x{(w\sigma )}^2=0.\end{array}$$

The final solution is obtained: $c=\left(\frac{1}{1-\alpha }\left(\rho -\alpha r_f-\frac{\alpha }{2(1-\alpha )}{\left(\frac{r-r_f}{\sigma }\right)}^2\right)\right)w$, $x=\frac{r-r_f}{(1-\alpha ){\sigma }^2}$.

3.2. Orthogonalization of Stochastic Processes

In this paper, in order to make the solution easier, some stochastic processes are assumed to be independent. The following describes the orthogonalization process of stochastic processes support our assumption.

Proposition 1.

For a group of stochastic processes: $B_i,i\in \{1,2,\cdots ,I\}$, which have $d{B}_i(t)\sim N({\mu }_{B_i(t)},{\displaystyle {\sigma }_{B_i(t)}^2}d{t}^{\alpha })$, and $cov(d{B}_i(t),d{B}_j(t))={\sigma }_{i,j,t}d{t}^{\alpha }$. Then there exist $z_i,i\in \{1,2,\cdots ,I\}$ which have $d{z}_i(t)\sim N({\mu }_{z_i(t)},{\displaystyle {\sigma }_{z_i(t)}^2}d{t}^{\alpha })$, are independent of each other. $d{w}_i,i\in \{1,2,\cdots ,I\}$ can be expressed linearly by $d{z}_i,i\in \{1,2,\cdots ,I\}$.

Proof. 

We only need to use the Schmidt orthogonalization method and make the coefficient time-varying. Let

$$d{z}_1=d{B}_1,$$

$$d{z}_2=d{B}_2-\frac{cov(d{B}_2,d{z}_1)}{var(d{z}_1)}d{z}_1=d{B}_2-\frac{cov(d{B}_2,d{B}_1)}{var(d{B}_1)}d{B}_1=d{B}_2-\frac{{\sigma }_{1,2,t}}{{\displaystyle {\sigma }_{B_1(t)}^2}}d{B}_1,$$

$$d{z}_3=d{B}_3-\frac{cov(d{B}_3,d{z}_1)}{var(d{z}_1)}d{z}_1-\frac{cov(d{B}_3,d{z}_2)}{var(d{z}_2)}d{z}_2.$$

It is easy to verify: $d{B}_i,i\in \{1,2,\cdots ,I\}$ and $d{z}_i,i\in \{1,2,\cdots ,I\}$ are equivalent groups of random variables. □

4. Modelling of Fractional Brownian Motion

In this section, we consider a multi fractional Brownian motion process for price p ( $\mu$ is the expected return and $\sigma$ is the risk exposure)

$$dp=\mu (p)dt+{\displaystyle \sum _{j=1}^{\infty }}{\sigma }_{j}d{B}^{H_j}$$

(4)

where $H_j\in [\frac{1}{4},1)$. Our current solution scheme can only cope with the case $H_j\in [\frac{1}{4},1)$. If $H_j\in [\frac{1}{6},\frac{1}{4})$, we need to deal with Taylor expansions of order 6. Our current proof for the Ito Lemma type lemma is already very difficult. On the other hand, it is difficult to extend the range of values of $H_j$ further to obtain more specific economic implications. Therefore, we choose $H_j\in [\frac{1}{4},1)$.

If $B^{H_j}$ are independent of each other. We can talk about more concise issues. Since we plan to prove a framework similar to Ito lemma, we only need to focus on $H_j=\frac{1}{2}$ and $H_j=\frac{1}{4}$ when a function $f(x)$ is expanded by Taylor. The rest will be $o(dt)$, which will affect the expected return of the price but will not affect the higher-order terms in Taylor expansion.

To sum up, we will simplify the process of (4) and further discuss the following one types of price movements:

$$\begin{array}{cc}\hfill dp/p& =rdt+{\sigma }_{0}d{B}^{\frac{1}{2}}+({\sigma }_1+i{\sigma }_1)d{B}^{\frac{1}{4}}.\end{array}$$

(5)

When the fractional Brownian motion with Hurst exponent less than $0.5$ is considered directly, its variance is a low order infinitesimal of $dt$ when the time interval $dt$ tends to zero. Therefore, when agents make decisions, they will get some strange asset allocation because of the fear of violent fluctuations in small time intervals (This can be proved by considering the way similar to Ito lemma. After retaining the $\sqrt{dt}$ brought by fractional Brownian motion variance, the Bellman equation will change $dt$ into $o(\sqrt{dt})$. ). Therefore, we use the method of introducing imaginary number to let the agent ignore the influence of fractional Brownian motion on the square difference. This scheme is also the best one we think of at present, introducing fractional Brownian motion while maintaining mathematical rigour under the consumer utility framework. The introduction of imaginary numbers is a common scheme in quantum mechanics, such as seeing [42] p. 20. From another point of view, the introduction of an imaginary number shows that the agent ignores the extreme fluctuation caused by fractional Brownian motion in a very short time and has subjectivity.

4.1. Solution of Single Risk Asset Model

In this subsection, we assumed that the price of risky asset p satisfies the following conditions:$dp/p=rdt+{\sigma }_{0}d{z}_0+({\sigma }_1+i{\sigma }_1)d{z}_1$, and $z_0$ satisfies the standard Brownian motion: $d{z}_0\sim N(0,dt)$, $E_t(d{z}_{0}d{z}_0)=dt$. r is the expected return rate of risk assets, $\sigma$ is the risk exposure. $z_0,z_1$ are independent of each other. i is the imaginary root: $i^2=-1$. $d{z}_1$ is subject to $N(0,\sqrt{dt})$. In fact, it may be problematic to assume that the expectation of fractional Brownian motion increment is zero. However, in reality, traders cannot observe the value of fractional Brownian motion. Therefore, it is reasonable to assume that the increment of fractional Brownian motion in traders’ subjective probability is zero. Risk-free asset price $p_f$ satisfies $d{p}_f/p_f=r_{f}dt$, where $r_f$ is the return rate of risk-free assets. w is the process of wealth movement. x is the weight of risk assets. c is the consumption. Individual investors need to solve the optimization problem as follows:

$$\begin{array}{cc}\hfill max& ERe({\displaystyle {\int }_0^{+\infty }}{e}^{-\rho t}(\frac{c^{\alpha }}{\alpha })dt\hfill \\ \hfill s.t.& dw=(w(r_f(1-x)+rx)-c)dt+wx({\sigma }_{0}d{z}_0+({\sigma }_1+i{\sigma }_1)d{z}_1)\hfill \\ \hfill & w(0)=w_0.\hfill \end{array}$$

The method of Bellman equation is used to solve the problem:

$$\begin{array}{c}\hfill V(t,w)=ma{x}_{x,c}{E}_{t}Re\left(\left[{\displaystyle {\int }_t^{t+dt}}{e}^{-\rho t}(\frac{c^{\alpha }}{\alpha })dt+V(t+dt,w+dw)\right]\right).\end{array}$$

It should be noted that Ito lemma (seeing p. 93 in [43]) can not be used directly here, so the following lemma is used:

Lemma 1.

If the random process w satisfies:

$$dw=f(t,w)dt+g_1(t,w)d{z}_0+(g_2(t,w)+i{g}_2(t,w))d{z}_1.$$

(6)

Furthermore, $z_0$ satisfies the standard Brownian motion, there will be $d{z}_0\sim N(0,dt)$. $z_0,z_1$ are independent. i is the imaginary root: $i^2=-1$.$z_0,z_1$ obey $N(0,\sqrt{dt})$. For $u=F(t,w)$, if F is resolved in a neighbourhood of $(t,w)$, then u satisfies:

$$\begin{array}{cc}\hfill du=& \left(F_t+f{F}_w+\frac{1}{2}{\displaystyle g_1^2}{F}_{ww}-\frac{1}{2}{\displaystyle g_2^4}{F}_{wwww}\right)dt+2i{\displaystyle g_2^2}\sqrt{dt}\\ \hfill & +F_w(g_{1}d{z}_0+g_{2}d{z}_1+i{g}_{2}d{z}_2).\hfill \end{array}$$

Next, we solve the optimization problem. Bellman equation is,

$$\begin{array}{cc}\hfill \rho V=& ma{x}_{c,x}\left(\frac{c^{\alpha }}{\alpha }+(w(r_f(1-x)+rx)-c)V_w+\frac{{(wx{\sigma }_0)}^2}{2}{V}_{ww}\right.\\ \hfill & \left.-\frac{{(wx{\sigma }_1)}^4}{2}{V}_{wwww}\right).\hfill \end{array}$$

(7)

It should be noted that the Bellman equation is a quartic equation with respect to x. According to our later derivation, we can find that the coefficient of $x^4$ may be positive, which means that there is no suitable solution less than infinity for this equation. For the properties of the fourth derivative of utility function, see [44]. Therefore, when the fourth derivative of the indirect utility function is less than 0, only when there are short selling and borrowing constraints or investors only rely on the first-order conditions for decision-making can there be a finite solution. Next, we will show possible solutions with loan constraints.

The derivation of c yields $c^{\alpha -1}-V_w=0$, and the derivative of x can be obtained as follows:

$$V_{w}w(r-r_f)+V_{ww}x{(w{\sigma }_0)}^2-2V_{wwww}{x}^3{(w{\sigma }_1)}^4=0.$$

(8)

The guess solution is as follows:$V=A{w}^{\alpha }$. (8) can be reduced to:

$$(r-r_f)+(\alpha -1){\displaystyle {\sigma }_0^2}w-2(\alpha -1)(\alpha -2)(\alpha -3){\displaystyle {\sigma }_1^4}{w}^3=0.$$

(9)

Let $q_1=\frac{(\alpha -1){\displaystyle {\sigma }_0^2}}{2(\alpha -1)(\alpha -2)(\alpha -3){\displaystyle {\sigma }_1^4}}$, $q_2=\frac{(r-r_f)}{2(\alpha -1)(\alpha -2)(\alpha -3){\displaystyle {\sigma }_1^4}}$, Then (9) can be reduced to $x^3+q_{1}x+q_2=0$. There must be a real number solution by using the method of Cardano formula (In solving practical problems, we will explore the corresponding properties of the three points and select the best one. This is just an example.):

$$x^{*}=\sqrt[3]{-\frac{q_2}{2}+\sqrt{{\left(\frac{q_2}{2}\right)}^2+{\left(\frac{q_1}{3}\right)}^3}}+\sqrt[3]{-\frac{q_2}{2}-\sqrt{{\left(\frac{q_2}{2}\right)}^2+{\left(\frac{q_1}{3}\right)}^3}}.$$

(10)

Put $c^{\alpha -1}=V_w$ and $V=A{w}^{\alpha }$ in Bellman equation:

$$\begin{array}{cc}\hfill A=& \frac{1}{\alpha }\left[\frac{1}{1-\alpha }\left(\rho -\alpha (r_f(1-x^{*})+r{x}^{*})-\frac{1}{2}{(x^{*}{\sigma }_0)}^2\alpha (\alpha -1)\right.\right.\\ \hfill & {\left.\left.+\frac{1}{2}{(x^{*}{\sigma }_1)}^4\alpha (\alpha -1)(\alpha -2)(\alpha -3)\right)\right]}^{\alpha -1}.\hfill \end{array}$$

(11)

Finally, we have $c={(A\alpha )}^{1/(\alpha -1)}w$. It can be seen that the central result of this subsection is the introduction of ${\displaystyle {\sigma }_1^4}$ into the decision result, as opposed to the results of Section 3.1. Due to the existence of ${\displaystyle {\sigma }_1^4}$, the decision equation for x has the cubic and ${\displaystyle {\sigma }_1^4}$ appears in the final solution. This means that one of the components of the fourth order moment of the stochastic process of the price of a risky asset, ${\displaystyle {\sigma }_1^4}$, affects the proportion of the agent’s investment in the risky asset. The ratio of the agent’s consumption decision to the asset is also only one additional term corresponding to ${\displaystyle {\sigma }_1^4}$ compared to the result in Section 3.1.

In fact many studies have provided a basis for the importance of higher order moments. Peiro 1999 [45] found that the asymmetry of returns can not be rejected when using other distributions of non-lognormal distribution to characterize returns. Patton 2004 [46] discussed the impact of asset return skewness and asymmetric dependence on the portfolio and used the information of the fourth moment to analyze in the process of analysis. Aït-sahali and Brandt 2001 [47] mentioned the importance of skewness when introducing the selection of investment decision variables corresponding to different utility functions. In relation, Kimball 1993 [48] studied the properties of higher derivative of utility functions.

4.2. Comparison of Results and Numerical Simulations

In this Subsection, we will systematically compare the relationship and differences between our work in Section 4.1 and Merton 1975 [2], Miao and Yang 2015 [18], Shokrollahi and Kılıçman 2015 [19], Rao 2016 [20], Shokrollahi et al. 2016 [21], Zhang et al. [22]. In order to be able to illustrate the value of our work, we conducted numerical simulations using BP500 data and observed whether our decision options improved lifetime utility. Based on the conventions of general economic research, we let $1-\alpha =3$. Let $\rho =0.08$ and $r_f=0.03$, see Kyle and Xiong 2001 [49]. Let r,${\sigma }_0$ and ${\sigma }_1$ be the moment estimates for the past 1000 trading days. After rolling, we computed lifetime utility values under different decisions in

Table 3. Comparison of Results and Numerical Simulations.

	[2]	[18,19,20,21,22]	This Paper
Research Issues
Subject of Study	Any Financial Asset	Options	Any Financial Asset
Consider Personal Decisions	YES	NO	YES
Number of Assets	Multi-Asset	Single Asset	Multi-Asset
Stochastic Process	Brownian Motion	FBM	FBM
Solution Method
Solving Methods	Stochastic Optimal Control	Arbitrage Free Pricing	Stochastic Optimal Control
Core Lemma	Ito Lemma	Ito Integral	New Lemma
Expandability
DSGE Model	YES	NO	YES
Heterogeneous Agents Model	YES	NO	YES
Study Results
Difference from Reality	No Consideration of FBM	No Microfoundations	Imaginary Numbers
FBM is Included	NO	YES	YES
Economic Variable Correlation is Given	YES	NO	YES
Agent Welfare
Utility Value	−2.9227	-	−2.1076

The data is daily data from 1 May 1976 to 19 January 2022 of BP500. BM is Brownian Motion. FBM is Fractional Brownian Motion. Source: CEIC dataset and Resset data.

.

An important point about our model is that it has the same microfoundations of individual choice as [2]. The specific form of individual choice allows us to see two important facts. The first aspect is what are the mechanisms that influence individual choice when exogenous shocks occur. The second aspect is the linking of risky asset prices to macro variables such as consumption and savings rates, which can help us to better analyse the impact of macro policies.

Further, our model with its micro-foundations can be extended to a general equilibrium model. Dynamic stochastic general equilibrium models as a central research tool in modern economics allow for the study of economic laws in a broader context. At the meantime, both our model and [2] can be used to study economics under heterogeneous agents. Studying heterogeneous traders can help us to better understand how an economic policy affects different groups of people.

Our central starting point with [2] is the maximisation of investor utility. We therefore consider how our decision making approach and the decision making approach in [2] behave in realistic data.The decision approach in [2] derives a welfare discount of $-2.9227$. Our decision approach derives a welfare discount of $-2.1076$. The welfare improvement is $0.8151$. Realistic data suggest that our decision scheme does improve relative to the classical work of Merton 1975 ([2]). The reason for the comparison with [2] is that the main aim of our work is to propose a new framework for solving economics problems based on fractional Brownian motion and the example we give is the individual choice problem in [2].

4.3. Multidimensional Situation

Here we assume a variety of risky asset prices of $p_l$ satisfies: $d{p}_l/p_l=r_{l}dt+{\displaystyle {\sum }_{k=1}^K}{\sigma }_{0,l,k}d{z}_{0,k}+{\displaystyle {\sum }_{j=1}^J}{\sigma }_{1,l,j}(1+i)d{z}_{1,j}$. $d{z}_{0,k}$ satisfies the standard Brownian motion: $d{z}_{0,k}\sim N(0,dt)$, $E_t(d{z}_{0,k}d{z}_{0,k})=dt$. r is the expected return rate of risk assets, $\sigma$ is the risk exposure. $z_{0,k},z_{1,j}$ are independent of each other. i is the imaginary root: $i^2=-1$. $d{z}_{1,j}$ are subject to $N(0,\sqrt{dt})$. Risk-free asset price $p_f$ satisfies $d{p}_f/p_f=r_{f}dt$, where $r_f$ is the return rate of risk-free assets. w is the process of wealth movement. $\overrightarrow{x}$ is the weight of risk assets. c is the consumption. Individual investors need to solve the optimization problem as follows:

$$\begin{array}{cc}\hfill max& ERe\left({\displaystyle {\int }_0^{+\infty }}{e}^{-\rho t}(\frac{c^{\alpha }}{\alpha })dt\right)\hfill \\ \hfill s.t.& d{p}_l/p_l=r_{l}dt+{\displaystyle \sum _{k=1}^K}{\sigma }_{0,l,k}d{z}_{0,k}+{\displaystyle \sum _{j=1}^J}{\sigma }_{1,l,j}(1+i)d{z}_{1,j}\hfill \\ \hfill & dw=w{\overrightarrow{x}}^{\prime }\overrightarrow{dlog(p)}+[w{r}_f(1-{\overrightarrow{x}}^{\prime }\overrightarrow{1})-c]dt\hfill \\ \hfill & w(0)=w_0.\hfill \end{array}$$

The method of Bellman equation is used to solve the problem:

$$\begin{array}{c}\hfill V(t,w)=ma{x}_{x,c}{E}_{t}Re\left(\left[{\displaystyle {\int }_t^{t+dt}}{e}^{-\rho t}(\frac{c^{\alpha }}{\alpha })dt+V(t+dt,w+dw)\right]\right).\end{array}$$

It should be noted that Ito lemma can not be used directly here, so the following lemma is used:

Lemma 2.

If the random process $w_l$ satisfies:

$$\begin{array}{c}\hfill d{w}_l=f(t,w_l)dt+{\displaystyle \sum _{k=1}^K}{g}_{1,l,k}(t,w)d{z}_{0,k}+{\displaystyle \sum _{j=1}^J}{g}_{2,l,j}(t,w)(1+i)d{z}_{1,j}.\end{array}$$

(12)

Furthermore, $d{z}_{0,k}$ satisfies the standard Brownian motion, there will be

$d{z}_{0,k}\sim N(0,dt)$, $E_t(d{z}_{0,k}d{z}_{0,k})=dt$. $z_{0,k},z_{1,j}$ are independent. i is the imaginary root: $i^2=-1$.$z_{1,j}$ obey $N(0,\sqrt{dt})$. For $u=F(t,w)$, if F is resolved in a neighborhood of $(t,w)$, then u satisfies:

$$\begin{array}{cc}\hfill Re{E}_t(du)=& (F_t+{\overrightarrow{f}}^{\prime }\overrightarrow{F_w}+\frac{1}{2}{\overrightarrow{1}}^{\prime }{\displaystyle {\overrightarrow{g}}_1^{\prime }}{F}_{ww}{\overrightarrow{g}}_1\overrightarrow{1}\hfill \\ \hfill & +F_{w_{l_1}{w}_{l_2}{w}_{l_3}{w}_{l_4}}({\displaystyle \sum _{j=1}^J}{\displaystyle \sum _{l_1=1}^L}{\displaystyle \sum _{l_2=1}^L}{\displaystyle \sum _{l_3=1}^L}{\displaystyle \sum _{l_4=1}^L}{g}_{2,l_1,j}{g}_{2,l_2,j}{g}_{2,l_3,j}{g}_{2,l_4,j}\hfill \\ \hfill & -\frac{3}{2}{\displaystyle \sum _{j_1=1}^J}{\displaystyle \sum _{j_2=1}^J}{\displaystyle \sum _{l_1=1}^L}{\displaystyle \sum _{l_2=1}^L}{\displaystyle \sum _{l_3=1}^L}{\displaystyle \sum _{l_4=1}^L}{g}_{2,l_1,j_1}{g}_{2,l_2,j_1}{g}_{2,l_3,j_2}{g}_{2,l_4,j_2}))dt+o(dt).\hfill \end{array}$$

Next, we solve the optimization problem. Bellman equation is,

$$\begin{array}{cc}\hfill \rho V=ma{x}_{c,x}& [\frac{c^{\alpha }}{\alpha }+(w(r_f(1-{\overrightarrow{x}}^{\prime }\overrightarrow{1})+{\overrightarrow{x}}^{\prime }\overrightarrow{r})-c)V_w+\frac{w^2{\overrightarrow{x}}^{\prime }{\sigma }_0\overrightarrow{x}}{2}{V}_{ww}\\ \hfill & -w^4{V}_{wwww}(\frac{3}{2}{\displaystyle \sum _{j_1=1}^J}{\displaystyle \sum _{j_2=1}^J}{\displaystyle \sum _{l_1=1}^L}{\displaystyle \sum _{l_2=1}^L}{\displaystyle \sum _{l_3=1}^L}{\displaystyle \sum _{l_4=1}^L}\\ \hfill & {\sigma }_{2,l_1,j_1}{\sigma }_{2,l_2,j_1}{\sigma }_{2,l_3,j_2}{\sigma }_{2,l_4,j_2}{x}_{l_1}{x}_{l_2}{x}_{l_3}{x}_{l_4}\\ \hfill & -{\displaystyle \sum _{j_1=1}^J}{\displaystyle \sum _{l_1=1}^L}{\displaystyle \sum _{l_2=1}^L}{\displaystyle \sum _{l_3=1}^L}{\displaystyle \sum _{l_4=1}^L}{\sigma }_{2,l_1,j_1}{\sigma }_{2,l_2,j_1}{\sigma }_{2,l_3,j_1}{\sigma }_{2,l_4,j_1}{x}_{l_1}{x}_{l_2}{x}_{l_3}{x}_{l_4})].\end{array}$$

The derivation of c yields $c^{\alpha -1}-V_w=0$, and the derivative of $\overrightarrow{x}$ can be obtained as a cubic equation system of l (the coefficients are all known constants), and the solution can be obtained when there are specific values. After assuming that the solution is obtained, the value of A can be solved. The guess solution is as follows: $V=A{w}^{\alpha }$.

Put $c^{\alpha -1}=V_w$ and $V=A{w}^{\alpha }$ in Bellman equation:

$$\begin{array}{cc}\hfill A=& \frac{1}{\alpha }\left[\frac{1}{1-\alpha }\left(\rho -\alpha (w(r_f(1-{\overrightarrow{x^{*}}}^{\prime }\overrightarrow{1})+{\overrightarrow{x^{*}}}^{\prime }\overrightarrow{r})-c)-\frac{1}{2}{\overrightarrow{x^{*}}}^{\prime }{\sigma }_0\overrightarrow{x^{*}}\alpha (\alpha -1)\right.\right.\\ \hfill & -\alpha (\alpha -1)(\alpha -2)(\alpha -3)\left(\frac{3}{2}{\displaystyle \sum _{j_1=1}^J}{\displaystyle \sum _{j_2=1}^J}{\displaystyle \sum _{l_1=1}^L}{\displaystyle \sum _{l_2=1}^L}{\displaystyle \sum _{l_3=1}^L}{\displaystyle \sum _{l_4=1}^L}\right.\\ \hfill & {\sigma }_{2,l_1,j_1}{\sigma }_{2,l_2,j_1}{\sigma }_{2,l_3,j_2}{\sigma }_{2,l_4,j_2}{\displaystyle x_{l_1}^{*}}{\displaystyle x_{l_2}^{*}}{\displaystyle x_{l_3}^{*}}{\displaystyle x_{l_4}^{*}}\\ \hfill & {\left.\left.\left.-{\displaystyle \sum _{j_1=1}^J}{\displaystyle \sum _{l_1=1}^L}{\displaystyle \sum _{l_2=1}^L}{\displaystyle \sum _{l_3=1}^L}{\displaystyle \sum _{l_4=1}^L}{\sigma }_{2,l_1,j_1}{\sigma }_{2,l_2,j_1}{\sigma }_{2,l_3,j_1}{\sigma }_{2,l_4,j_1}{\displaystyle x_{l_1}^{*}}{\displaystyle x_{l_2}^{*}}{\displaystyle x_{l_3}^{*}}{\displaystyle x_{l_4}^{*}}\right)\right)\right]}^{\alpha -1}.\end{array}$$

Finally, we have $c={(A\alpha )}^{1/(\alpha -1)}w$. In contrast to Section 4.1, this subsection shows the results for multi-asset decisions. We find that the generalisation of the single-asset to multi-asset does not affect the form of the solution. Relative to the results of Section 3.1, the final result for the consumption and investment ratios after the model is multi-asset and is added to the fractional Brownian motion only increases the fourth order moments of the stochastic process for the price of the risky asset.

Aït-Sahalia and Lo 1998 [50] found negative skewness and excess kurtosis for asset returns are important features that affects the pricing of options. Figure 7 in [50] illustrated the effect of skewness and kurtosis on term prices. If we use expectation theory to explain price formation, i.e., investors form expectations which lead to decisions and ultimately determine prices, then we can see that skewness and kurtosis do influence investors’ decisions through. This result from classical observations supports the results of our model. Ahmed and Al Mafrachi 2021 [51] found that kurtosis has a predictive effect on returns in the cryptocurrency market. In our model investors’ decisions are influenced by the kurtosis of an asset. If we still believe in the path: kurtosis affects decisions and hence prices, the kurtosis of an asset will have an impact on future returns. Our model provides an easily understood explanation for [51]. A similar observation is [52]. The role of kurtosis on returns strongly supports our results.

5. Conclusions

We brought the stochastic process with fractional Brownian motion random source into the economic research framework of utility maximization with the complex number. Surprisingly, the decomposition of variance of real economic data in Section 2 supported the scheme of introducing the complex number.

In Section 3, if motions in the random source group are Brownian motion, we proved that the random sources can be set to be independent of each other. In this section, we also reviewed Merton’s classical continuous-time asset pricing model.

Furthermore, we introduced a stochastic process in Section 4 by introducing complex numbers. It had been found that the introduction of fractional Brownian motion affected the decision-making of the agent but did not destroy the original structure. Compared to the results of Section 3.1, the form of the return and variance of the risky asset return did not change in the results of Section 4. The fourth order moments of the stochastic process for the price of the risky asset appeared in the decision result of Section 4.