European Option Pricing under Sub-Fractional Brownian Motion Regime in Discrete Time

: In this paper, the approximate stationarity of the second-order moment increments of the sub-fractional Brownian motion is given. Based on this, the pricing model for European options under the sub-fractional Brownian regime in discrete time is established. Pricing formulas for European options are given under the delta and mixed hedging strategies, respectively. Furthermore, European call option pricing under delta hedging is shown to be larger than under mixed hedging. The hedging error ratio of mixed hedging is shown to be smaller than that of delta hedging via numerical experiments

Recently, researchers have proposed a new stochastic process called sub-fractional Brownian motion (sfBm) as the random driving source of the option pricing model.sfBm B H t = {B H t , t ≥ 0} is a centered Gaussian process with B H 0 = 0 and was proposed by Bojdecki et al. in 2004 [22].The covariance of sfBm is given by where 0 < H < 1 is the Hurst parameter.For H = 1 2 , B H t becomes a standard Brownian motion.From Equation (1), we can find that sfBm has the following properties: (1) selfsimilarity-for any α > 0, (B H αt ) t≥0 has the same distribution as (α H B H t ) t≥0 ; (2) long-term dependence-for 1  2 The above two properties are the same as in fractional Brownian motion.However, differing from fractional Brownian motion, sfBm has non-stationarity in the second moment increments.One can refer to [23][24][25][26][27] for more details of the sfBm.
In recent years, many scholars have studied the pricing problems for options under the sfBm regime.Araneda and Bertschinger [28] proposed the sub-fractional Constant Elasticity of Variance (CEV) model.Based on the transition probability density function of the underlying asset price, they derived the explicit formulas for European options.Wang et al. [29] researched the geometric Asian power option pricing model in the sfBm environment.They derived a closed-form pricing formula for geometric Asian power options based on the It ô formula of sfBm.Wang et al. [30] put forward a new Poisson process based on sfBm.Furthermore, they established the sub-fractional Poisson volatility model for option pricing and obtained the closed-form pricing formulas for European options.Bian and Li [31] considered the European option pricing model in an uncertain environment based on sfBm.Their results indicate that in an uncertain environment, a random source of underlying assets with long-term dependence is more suitable for the financial market.Xu and Li [32] gave the pricing formulas for compound options in the sub-fractional Brownian motion model using the risk neutral valuation method.
However, all the above models are continuous-time models.What about the case of discrete time?As the second moment increments of sfBm are not stationary, it is not easy to build a discrete-time model for options in the sfBm regime.Fortunately, we find that the second moment increments of sfBm are approximately stationary.Based on this, we consider the European option pricing under sfBm in discrete time.Moreover, in the discrete-time model of option pricing, delta hedging is the main hedging method.However, Wang [33] proposed a new hedging strategy called mixed hedging.Under this new hedging method, they obtained a discrete-time pricing formula for European options in the Brownian motion regime.Furthermore, their numerical experiments showed that the hedging error ratio of delta hedging was larger than in the mixed one.Kim et al. [34] considered the European option pricing model in the time-changed mixed fractional Brownian regime using the mixed hedging method.Their numerical results were consistent with those of Wang [32], i.e., the hedging error ratio of delta hedging is larger than that of the mixed one in some situations.
Based on the above, in this paper, we will consider the European option pricing model under the sfBm environment in discrete time.In Section 2, we will give the approximate stationarity of second-order moment increments of sfBm.Based on the approximate stationarity, in Section 3, we will establish the discrete-time model for European option pricing under the delta hedging strategy and mixed hedging strategy, respectively.In Section 4, we will give some numerical analysis to further evaluate the model.In Section 5, we conclude this paper.

Approximate Stationarity of the Second Moment Increments of sfBm
From the covariance of sfBm, we can obtain the following conclusion.

Lemma 1. The sfBm B H
t satisfies the following property: Proof.From the covariance of sfBm, we know that then 2H .In this sense, the second moment increments of sfBm are approximately stationary.

European Option Pricing under the Sub-Fractional Geometric Brownian Motion (sfgBm) Model
In this section, we will derive the pricing formulas for European call options in discrete time under the sfgBm model.We choose the same basic assumptions as Guo et al. [35] except the following.
(1) The dynamics of the underlying asset price S t and a bond price Q t are given by and respectively, where µ, σ, S 0 are constants, B H t is sfBm, and H > 1 2 is the Hurst parameter.(2) The value of the option can be replicated by a portfolio Π with X 1 (t) units of risk asset and X 2 (t) units of risk-less bond.

Pricing Formula for European Call Option in Discrete Time under Delta Hedging Strategy
In this subsection, we will give the discrete-time formulas for European call options under the delta hedging strategy.By C t = C(t, S t ), we denote the European call option price; then, we have the following.Theorem 1.When the underlying asset price S t satisfies Equation (4), under the delta hedging strategy C t , it satisfies the following equation, with the terminal condition C(T, S T ) = (S T − K) + , and K is the strike price.The European call option price at time t is given by where and N(•) is the cumulative normal density function.
Proof.From Equation (4), we know that By the definition of sfBm and Lemma 1, we have By the same token, we can derive It is obvious that and where Moreover, from Assumptions (1)-( 2) and the delta hedging method, we know that Then, from Equations ( 6)-( 9), we have and Subject to E(∆Π t − ∆C) = 0, we have Denote σ2 = σ 2 (∆t) 2H−1 , and we obtain Furthermore, from the Black-Scholes equation [1], we have where The proof is completed.

Pricing Formula for European Call Option in Discrete Time under Mixed Hedging Strategy
In this subsection, we will obtain the pricing formulas for European call options in discrete time under the sfBm model by using the mixed hedging strategy.Theorem 2. When the price of underlying asset S t satisfies Equation (4), the mixed hedging strategy under the sfBm model is given by Proof.From [33], a mixed hedging strategy is the solution of the following problem: subject to and Denote and Then, we have where G 2 (∆t) = o((∆t) 2H ); subject to Equation ( 16), we can obtain Let then, from Lemma 1 we have and Furthermore, Selecting X 1 (t) satisfies the following equation: and by calculation, we have Remark 1.The expression of the mixed hedging strategy under the sfBm model is the same as that under the Brownian motion model [33].This is consistent with the result in [34].
Based on the expression of the mixed hedging strategy, the pricing formula for European call options in discrete time under the sfBm model is given by the following.
Theorem 3. When we use the mixed hedging strategy, the European call option price C(t, S t ) satisfies the following equation, and the pricing formula is given by where Proof.Substituting Equation ( 28) into Equations ( 17)-( 21), we can obtain Then, C(t, S t ) satisfies the following final value problem of the partial differential equation: It is easy to see that Equation ( 32) is a Black-Scholes-type equation.Therefore, from [1], the European call option price can be given by Equations ( 29)- (31).
Table 1 shows that the European call option price under delta hedging is larger than under mixed hedging.As the exercise price K increases, the differences in the European call option price between delta hedging and mixed hedging are decreased.
From Figure 1, we can see that the parameter H has an important influence on the European call option price.Moreover, as the parameter H increases, the difference in the European call option price between delta hedging and mixed hedging gradually becomes larger.

Numerical Analysis 4.1. Price of European Call Option in Discrete Time under sfBm Model
In this subsection, we will compare the European call option price between delta hedging and mixed hedging.We set S 0 = 49, r = 0.05, T = 1, µ = 0.11, σ = 0.2, ∆t = 0.02, H = 0.8.

Comparison of Delta Hedging Method and Mixed Hedging Method in sfBm Model
In this subsection, we will compare the delta hedging and the mixed hedging strategies in the sfBm model across the hedging error ratio.
From Table 2, we can see that the European call option price under delta hedging at week 0 is $71,778.710.The discount of the total cost of writing the option and hedging to week 0 is equal to $106,355.902.The hedging error of delta hedging is (106,355.902− 71,778.710)/71,778.710,which is close to 0.48172932.
From Table 3, we can see that the European call option price under mixed hedging is $69,141.684.The discount of the total cost of writing the option and hedging to week 0 is equal to $97,938.092.The hedging error of mixed hedging is (97,938.092− 69,141.684)/69,141.684,which is approximately 0.41648404.From Tables 2 and 3, we can see that error ratio of mixed hedging is less than that of delta hedging.Table 4 and Figure 2 show the effects of Hurst parameter H on the hedging cost and hedging error ratio when H only varies from 0.65 to 0.9 (S 0 = 49, K = 50, µ = 0.11, σ = 0.2).
From Table 4 and Figure 2, we can see that as Hurst parameter H increases, the hedging error ratios of the two hedging methods are both decreased, but the hedging error ratio of mixed hedging decreases faster.

Conclusions
This paper deals with the European call option pricing in discrete time under the sfBm model.The numerical results show that the European call option price of delta hedging is larger than the price of mixed hedging.The hedging error ratio of the mixed hedging strategy is less than that of the delta hedging strategy in some situations.Moreover, the Hurst parameter H plays an important role in the European call option price and hedging error ratio.Based on the results of this study, we can study option pricing under sfBm in discrete time; the future research directions mainly include the following.
(i) The real financial market is not smooth.The trading of the risk asset (like stock) always incurs transaction costs and dividends.Therefore, the study of the option pricing model with transaction costs or dividends in the sfBm regime is of great significance.
(ii) The changes in the risk asset price often accompany jumps.Both Brownian motion and sfBm cannot describe this situation.Thus, one can generalize the sfBm model to the jump-diffusion model in discrete time.

Figure 2 .
Figure 2. Hedging error ratio across Hurst parameter H.

Table 1 .
European call option price under delta hedging and mixed hedging across strike price K.
Figure 1.European call option price across Hurst parameter H.

Table 2 .
Simulation of delta hedging per week with T = 20/52 years.

Table 3 .
Simulation of mixed hedging per week with T = 20/52 years.

Table 4 .
Hedging error ratios under different Hurst parameters H.