Fractional Stochastic Volatility Pricing of European Option Based on Self-Adaptive Differential Evolution

The option pricing model can predict the future trend of the financial market. In order to more accurately describe the changing process of the financial market, the Hurst index which can describe the characteristics of long-term memory is introduced into the traditional Heston model. Under the assumption that the underlying asset price follows fractional Brownian motion, the fractional stochastic volatility pricing of European option pricing model (Hurst-Heston model) is constructed, and the closed solution of the model is obtained according to the partial differential equation satisfied by the model. By analyzing the relationship between Hurst index and asset price, it is found that the movement process of asset price under the hypothesis of this model is more consistent with the real market change law, which verifies the rationality of the model. In the process of empirical analysis of SSE 50ETF put option data, Self-adaptive Differential Evolution algorithm is used to estimate parameters. The results showed that the error of Hurst-Heston model is smaller than other models, and the prediction error for consecutive trading days is similar. It showed that the pricing results of Hurst-Heston model are more accurate and stable.


Introduction
The option derived in the late 18th century is a financial tool used by financial institutions and enterprises to avoid risks [1]. It first appeared in the European and American financial markets. As the contract, it gives holders the right to such as the stock price follows a log-normal distribution, the risk-free interest rate is known, the stock price volatility is constant, there is no transaction cost in the hedging portfolio, etc. These conditions cannot be met in the real financial market transactions, so phenomena such as "volatility smile" often occur in the actual application. In order to make up for the deficiency of B-S model, Merton first proposed the lognormal jump diffusion model in 1976 [5], it added a jump process into the B-S model, which was more in line with the real market with sudden risks. Subsequently, [6] proposed a call option replication strategy based on proportional transaction costs, and gave the option pricing formula with transaction costs. In addition, because stock price volatility is random, some scholars introduced stochastic volatility model to improve the option pricing model: [7] constructed Heston model on the basis of traditional B-S model and assumed that volatility of underlying asset followed O-U process; [8] proposed fast mean regression volatility model, which reduced model parameters to facilitate parameter estimation, and at the same time, it better explained the characteristics of volatility smile and return on assets peak thick tail; [9] [10] used non-affine stochastic volatility model for option pricing, which is more accurate than other models; [11] generalized the nonlinear partial differential equation when the underlying asset follows the stochastic single-factor interest rate model, regarded the nonlinear term in the equation as the transaction cost, and proved the existence of the classical solution of the model; [12] proposed the volatility decomposition model, , it divided the stochastic volatility process of the underlying asset into two risk sources based on the Heston model, which could describe the relevant characteristics of the underlying asset more accurately.
The Hurst index H was discovered by British hydrologist H.E. Hurst, when he studied the relationship between water flow and storage capacity [13], and took it as an indicator to judge whether timing data followed the process of biased random walk. 0.5 H = , means that the time series is a standard Brownian motion process; 0. 5  In order to obtain more accurate prediction results on option pricing, we should not only consider the randomness of stock price fluctuations, but also not ignore the long memory of financial markets. However, the existing option pricing models cannot meet these two properties at the same time, which is the original intention of this paper. This model makes up for the deficiency that the existing stochastic volatility option pricing models cannot reflect the characteristics of the fractal market.
2) By analyzing the Hurst index in the model, it is found that the introduction of H into the option pricing model can more accurately reflect the real change process of the financial market, which verifies the rationality of the model.
3) Using the SSE 50ETF put option data for empirical analysis, due to the large number of parameters in this model, the traditional linear least squares parameter estimation method is invalid. Referring to the estimation methods of multi-parameter models in other literatures, this paper will choose the Self-adaptive Differential Evolution algorithm to estimate the parameters of the model. The final analysis results show that, compared with the other pricing model, the fractional stochastic volatility model has more accurate pricing results and less error, which further illustrates the feasibility and accuracy of the pricing model proposed in this paper.

Preliminary Knowledge
When constructing the option pricing model, in order to describe the long-term memory of the financial market more intuitively, it is necessary to introduce and for any t and 0 t ∆ > , the incremental expectation of FBM is 0; 2) For t and s at different moments, their covariance function is: Then the Gaussian process is the Hurst exponent that describes the relationship between the motion increments. If The core property of FBM is that the increment has stationarity, autocorrelation and self-similarity, and the stock price fluctuation process that conforms to fractal characteristics also has similar properties. So FBM can be used to describe the stock price fluctuation process.
Lemma 1 [23]: Assuming that the price of the derivative is , and the underlying asset price S t follows a biased FBM, that is then, at any time t, there is the following relationship: Equation (2) is called fractional Ito Lemma, where µ is the drift rate and σ is the volatility.

Pricing Model Construction and Parameter Optimization
The option pricing model constructed in this paper is an improvement on the basis of the Heston model. It not only considers the dynamic process of the financial market as a stochastic fluctuation, but also combines its characteristics of long memory, which makes up for the defects of previous models that only consider a single influence.

Hurst-Heston Option Pricing Model
In the financial market, stock market volatility is a dynamic process with long memory characteristics. In order to reflect the nature of the financial market more comprehensively in the option pricing model, this paper improves Heston model and introduces FBM on the basis of this model to obtain the Hurst-Heston model as shown in Definition 2.
Definition 2: It is assumed that under the neutral probability measure Q, the underlying asset S t follows fractional Brownian motion and the volatility V t follows the O-U process, that is, the price and volatility of the underlying asset satisfy the following differential equations respectively: Equation (3) and Equation (4) used to solve option price is called Hurst-Heston option pricing model. Among them µ is the drift rate, V t is the volatility, θ is the long-term mean value of volatility, κ is the mean reversion rate, σ is the coefficient of variance variation, that is, the volatility of volatility. Both In this model, it is assumed that there are risk-free assets S t which can be freely traded in the financial market and meet the following conditions: 1) The price of stock index options in the fractal market only depends on the stock price and the risk-free interest rate; 2) There are no taxes and transaction costs in the market, and no dividends are paid during the transaction; 3) The underlying asset price S t satisfies the fractional Brownian motion; 4) The underlying asset fluctuates randomly, and its volatility V t follows the O-U process.

Analytical Pricing of European Options under the Hurst-Heston Model
Theorem 1: The partial differential equation satisfied by European option price under Hurst-Heston model is: In Equation (5), t means any moment, V is the volatility of stock price, S is the stock value, H is the Hurst index, σ is the volatility of volatility, ρ is the r is the risk-free interest rate, κ is the mean regression rate, θ is the long-term mean of volatility, λ is the ratio of the market price of volatility to volatility.
Prove: According to the standard arbitrage theory, assuming that there is no arbitrage opportunity, there is a risk-free investment portfolio I, which contains an option with a value of ( ) According to differential Equation ( Since the assumption is under the risk-neutral measure, the dV and terms in Equation (7) are eliminated, so that: Then we can get: where r is the risk-free interest rate, put it together with Equation (9) into Equation (7), then Equation (7) can be written as follows: According to the research of Heston [6], it is known that the function is the market price of volatility risk. Meanwhile, according to Brecden's assumption that the market price of volatility is its linear function, that is ( ) , then Equation (5) can be obtained from Equation (10), which is proved.
According to Theorem 1, analogy with the solution process of Heston model, the option price is simulated by numerical method under the condition of Using Gaussian quadrature and fast Fourier transform [24], the following Theorem 2 can be obtained by making Theorem 2: The Pricing formula of European call option in the Hurst-Heston model is: Ht The put option pricing formula is: Note: The parity formula between the European call option price and the put option price is:

Parameter Estimation for Pricing Model
Because the option pricing model proposed in this paper is a multi-parameter model, the common parameter estimation methods such as the least squares method and the maximum likelihood estimation method are not applicable, so the Self-adaptive Differential Evolution (SaDE) algorithm will be used to estimate the parameters of this model. The evolution process of the SaDE algorithm is equivalent to the Differential Evolution algorithm [25] [26] [27], which is im-  2) Mutation operation The basic mutation vector is generated according to formula (14): In Equation (14), is the scaling factor, also known as the mutation operator, which is used to control the amplification of the deviation vector, and this factor can reflect the global optimization ability of the algorithm. The smaller the F value, the better the local searching ability of the algorithm. The larger the F value, the more the fitness function value can jump out of the local minimum point, and the slower the convergence speed. 1 2 3 , , G G G X X X are the three different individuals in the population whose fitness function value is optimal.
In the basic differential evolution algorithm, the mutation operator often takes a constant, but its value is difficult to determine accurately. If the mutation rate is too large, the global optimal solution will be low; If the value of mutation rate is too small, the diversity of the population will decline, and it is easy to appear the phenomenon of "premature". Therefore, the improved adaptive mutation operator λ is adopted in this paper: In Equation (15), F 0 is the initial mutation operator, G m is the current evolution algebra, and G is the maximum evolution algebra. The mutation operator of this mutation form is 2F 0 at the beginning, which can maintain the diversity of the population and prevent premature maturity. With the development of evolution, the mutation operator is gradually reduced to F 0 , which can effectively avoid the destruction of the optimal solution.

3) Crossover operation
In order to increase the diversity of vectors in the population, the following crossover operation is introduced: In Equation (16) is a crossover operator. The larger the value of CR, the faster the convergence speed of the algorithm. This paper adopts the crossover operator of random range as follows: This method can keep the mean value of the crossover operator at about 0.75, which ensures the diversity of the population.

4) Selection operation
In order to determine whether the vector in the population can become a member of the next generation, we need to compare the test vector with the current target vector, and calculate the fitness function ( ) F k value of each vector, then the following selection operation is performed: The fitness function selected in this paper is: In Equation (19), n is the sample size,

5) Boundary condition processing
The vectors beyond the bounds are replaced by randomly generated parameter vectors in the feasible region.

The Relationship between Hurst Index and Asset Price
In this paper, the Hurst index H, which measures the degree of long memory, is ( ) Prove: According to Equation (2), Equation (3) and theorem 1, we can get the following formula: According to Equation (3) f According to Equation (21) and Equation (22), it can be deduced that the underlying asset price S t approximately obeys the following distribution (Theorem that the yield series with long memory will show a peak distribution, and as the value of H gradually increases, the peak phenomenon observed in the yield distribution will become more and more obvious. In the real financial market, the yield of the underlying assets has the characteristics of peak and thick tail. In the "peak" period, the market is stable as a whole, and the price trend conforms to the technical analysis and statistical laws; but in the "thick tail" period, the market changed dramatically due to the impact of special events. It can be seen that the larger the value of the Hurst index, which measures long-term memory, the more it can indicate that the market is stable at this stage, and the "peak" state of the stock return density curve is more obvious. In conclusion, it can be verified that the change process of the underlying assets under the assumption of the Hurst-Heston model is closer to the real situation, which can better describe the changing trend of stock prices in the real financial market.

Empirical Analysis
Before pricing options using the pricing model, it is necessary to estimate the parameters in the model, and the implicit parameter estimation method will be adopted in this paper [28]. This method uses the actual price of the market to infer the model parameter value, and the parameter value obtained is the parameter value under the risk neutral probability measure. This method requires a relatively small amount of data, and the estimated value is more effective.
In order to estimate the parameters of the Hurst-Heston model, this paper se-  the results were shown in Table 1.
As can be seen from the results in Table 1 Table 3, and the absolute error comparison diagram of the three models is shown in Figure 4.
In Table 2, HH represents the Hurst-Heston model proposed in this paper, and BSH represents the fractional B-S model. In Table 3  proposed in this paper is smaller than that of other comparison models, indicating that this model has better pricing effect and higher accuracy.
It can be observed from Figure

Conclusion
In this paper, the Hurst index H is introduced on the basis of the Heston option pricing model, and the Hurst-Heston European option pricing model with long memory characteristics is constructed, then obtained Hurst-Heston option pricing formula by solving the partial differential equation under the model by Fourier transform method. In addition, through the distribution function of the underlying asset price, the influence of H on the asset price is analyzed. When the value of H is between 0.5 and 1, as the value of H increases, the probability density curve of S t gradually presents the phenomenon of peak and thick tail, which is more in line with the reality. In the empirical analysis, the sample data is used to estimate the model parameters, and then the option prices in the next but also reflect the long-memory characteristics of the financial market. The change process of the underlying asset depicted by this model is closer to the real situation and can better describe the stock price change trend in the real financial market. In this paper, we assumed that the stock price fluctuates randomly with a single factor in the process of research, but in practice, the stock price fluctuation can be affected by a variety of factors, so we can consider the situation that the stock price change is a multi-factor random fluctuation in the follow-up research. In the empirical analysis, stock option data are used to verify the model, and the model has not been applied to other financial markets, so we can try to extend the model to real option pricing in the future.