Bull. Korean Math. Soc. 55 (2018), No. 4, pp. 1241{1261 https://doi.org/10.4134/BKMS.b170719 pISSN: 1015-8634 / eISSN: 2234-3016 GENERATING SAMPLE PATHS AND THEIR CONVERGENCE OF THE GEOMETRIC FRACTIONAL BROWNIAN MOTION Hi Jun Choe, Jeong Ho Chu, and Jongeun Kim Abstract. We derive discrete time model of the geometric fractional Brownian motion. It provides numerical pricing scheme of financial deri- vatives when the market is driven by geometric fractional Brownian mo- tion. With the convergence analysis, we guarantee the convergence of Monte Carlo simulations. The strong convergence rate of our scheme has order H which is Hurst parameter. To obtain our model we need to convert Wick product term of stochastic differential equation into Wick free discrete equation through Malliavin calculus but ours does not in- clude Malliavin derivative term. Finally, we include several numerical experiments for the option pricing. 1. Introduction Most models of financial processes are assuming Brownian motion of the asset prices. Gaussian and Markovian properties of Brownian motion facilitate greatly pricing of many financial derivatives. But it is found that empirical asset returns may have memory about past history. Empirical asset prices are affected not only by their most current values, but also by their history (see Mandelbrot [18] and Shiryaev [25]). It implies that market is not exactly Markovian, too. Therefore most classical models need to be modified so that they may describe the dynamics of markets more accurately. One alternative is to use the fractional Brownian motion (fBm) instead of the Brownian mo- tion. Then the classical Black-Scholes model can be extended to the fractional Black-Scholes model. However, Bj¨orkand Hult [8] pointed out the difficulties of interpretation based on Wick calculus. Also the pricing models based on geometric fBm can give explicit arbitrage strategies in [25]. However, to exclude arbitrage several modifications of the fractional mar- ket setting have been suggested (Hu and Økesendal [15], Elliott and van der Received August 14, 2017; Accepted November 3, 2017. 2010 Mathematics Subject Classification. Primary 60G22. Key words and phrases. discrete asset model, Monte Carlo, geometric fractional Brownian motion, Malliavin calculus, Euler-Maruyama scheme, Black-Scholes model. This work was supported by NRF under grant 2015R1A5A1009350. c 2018 Korean Mathematical Society 1241 1242 H. CHOE, J. CHU, AND J. KIM Hoek [12] and Rostek [23]). In case that market prices move at least slightly faster than any market participant can react, arbitrage can be excluded. Re- nouncement of continuous trability can obtain a reasonable financial model where no arbitrage occurs (Rostek [22]). In the more realistic market models which includes transaction costs, the ideal continuous time trading strategies turn out to be of bounded variation. In this case Guasoni [13], Guasoni et al., [14] show that geometric fBm models can be economically meaningful. In this respect, Valkeila [28] studied the approximation of geometric fBm in the sense of weak convergence. The fractional Brownian motion can be represented as a limit of a random walk (see Sottinen [26]). There are many methods of simulation for fractional Brownian motion like Hosking method, Cholesky method, Davies and Harte method, FFT method as is reviewed in [16]. The stochastic differential equa- tion describing fractional Brownian market includes stochastic terms consisting of Wick product of fractional Brownian motion. In the continuous time stochas- tic differential equation, even when such a solution can be found, it may be only in implicit form or too complicated to visualize and evaluate numerically. The time discrete approximation or discretisation of stochastic differential equation is the method that generate values Sb∆t; Sb2∆t; Sb3∆t;:::; Sbn∆t;::: at given dis- cretization times 0 < ∆t < 2∆t < ··· < n∆t < ··· . The main difficulty of sampling geometric fractional Brownian motion is that Wick product cannot be evaluated pathwise, but depends on all possible other paths as is stated in Bender [5]. We suggest a recursive method for generating sample paths of geometric fractional Brownian motion in the context of Wick-It^ointegral. The main idea is to replace Wick products by ordinary products plus expectations of possible other paths. Our scheme has convergence rate H which is Hurst parameter. Theorem 1.1. Suppose St; t 2 [0;T ] is geometric fractional Brownian motion T and Sbk∆t; k 2 f0; 1;:::; ∆t g is discrete geometric fractional Brownian motion. There is a constant C depending only on H such that T 2H H E[jST − SbT j] ≤ C(T + 1)e ∆t : Through numerical experiments we compare sample paths from our algo- rithm with exact paths of the geometric fractional Brownian motion. We also evaluate European option price when underlying asset is governed by geometric fractional Brownian motion by Monte Carlo method. 2. Discrete model of the geometric fractional Brownian motion 2.1. Fractional Brownian motion H The fractional Brownian motion Bt is Gaussian process which satisfies the following condition: H E(Bt ) = 0 8t 2 R; THE GEOMETRIC FRACTIONAL BROWNIAN MOTION 1243 1 E(BH BH ) = (s2H + t2H − js − tj2H ); t s 2 1 H where H is the Hurst parameter and in case H = 2 , Bt is the classical Brow- 1 nian motion. For the case 0 < H < 2 , the covariance of the increments is negative, i.e., the process is mean-reverting and the increments of the process are negatively correlated. So the processes are called anti-persistent. For the 1 case 2 < H < 1 , the covariance of the increments is positive, i.e., increments of the process are positively correlated. So the processes are called persistent and 1 have long-range dependency property. The Case H > 2 makes the fractional Brownian motion a plausible model in mathematical finance. Several empir- ical studies of financial time series say that the log-returns have long-range 1 dependence. In this paper, we consider the case H > 2 in our financial model. The fractional Brownian increment H H H ∆Bt;s = Bt − Bs has the following moment properties; H H H H H E(∆Bt;s) = E(Bt − Bs ) = E(Bt ) − E(Bs ) = 0; 8t; s 2 R; H 2 H H H H 2H E((∆Bt;s) ) = E((Bt − Bs )(Bt − Bs )) = jt − sj ; 8t; s 2 R: The covariance of two non-overlapping increments is H H H H H H E(∆Bt;s∆Bs;0) = E((Bt − Bs )(Bs − B0 ))(1) H H H H H 2 H H = E(Bt Bs ) − E(Bt B0 ) − E((Bs ) ) + E(Bs B0 ) 1 = [t2H + s2H − (t − s)2H ] − s2H 2 1 = [t2H − s2H − (t − s)2H ]: 2 1 Therefore the increments of fBm are correlated except for H = 2 . Consider partitions πn = f0 = t0 < t1 < ··· < tn = T g of the interval [0;T ] such that jπnj −! 0 for n −! 1. Then we obtain the following: Lemma 2.1. For p ≥ 1, 8 1 if pH < 1, n p <> X H H H p (2) lim Bt − Bt = EjBT j if pH = 1, jπj!0 k k−1 k=1 :>0 if pH > 1: Proof can be referred to Rogers [21]. We can conclude quadratic variation 1 of fractional Brownian motion is zero in case H > 2 . We can obtained the following corollary: 1 Corollary 2.2. For the fBm quadratic variation is zero if H > 2 and does not 1 H exist if H < 2 . Moreover B has unbounded variation P-a.s. 1244 H. CHOE, J. CHU, AND J. KIM 2.2. Wick-It^ointegration The stochastic integration with respect to fBm is based on a renormalization operator Wick product introduced by Duncan, Hu and Pasik-Duncan [11] and Hu and Øksendal [15]. 1 2 Let H 2 ( 2 ; 1). We define the fractional kernel φ : R ! R by φ(s; t) = H(2H − 1)jt − sj2H−2: We endow the space of Borel measurable functions f; g : [0;T ] ! R with the 2 norm jj · jjφ and inner product h·; ·iφ : Z TZ T 2 jjfjjφ := f(s)f(t)φ(s; t)dsdt; 0 0 Z TZ T hf; giφ := f(s)g(t)φ(s; t)dsdt 0 0 and we define 2 2 Lφ([0;T ]) = ff : [0;T ] ! R : f is Borel measurable, and jjfjjφ < +1g: 2 For a deterministic function f 2 Lφ([0;T ]) we define its Wick integral in the following way. Let πn = f0 = t0 < ··· < tn = T g be a sequence of partitions of [0;T ] such that jπnj ! 0, and fn be the step functions approximating f : X n 2 fn(t) = ai 1[ti;ti+1)(t) −! f(t) in Lφ([0;T ]): i Then we define Z T X f (t)dBH := an(BH − BH ) n t i ti+1 ti 0 i and Z T Z T H H f(t)dBt := lim fn(t)dBt : 0 n!1 0 For a more detailed discussion of stochastic integral of fractional Brownian motion, we refer [7]. Then the following property can be obtained [11]: Z T (3) E[ fdBH ] = 0; 0 Z T Z T H H E[ fdB gdB ] = hf; giφ; 0 0 Z T H 2 2 E[ fdB ] = jjfjjφ (Wick-It^oisometry): 0 R T H 2 We denote by I(f) := 0 f(s)dBs for every f 2 Lφ([0;T ]). Random variables of Lp(Ω; F; P) can be approximated with arbitrary exactness by linear combi- nations of so-called Wick exponentials exp(I(f)) that are defined by fractional THE GEOMETRIC FRACTIONAL BROWNIAN MOTION 1245 integrals with deterministic integrands f's: 1 exp(I(f)) := exp(I(f) − jjfjj2 ) 2 φ (cf. [11], [20]). Duncan et al. [11] defined the Wick product implicitly on these Wick exponentials by exp(I(f)) exp(I(g)) = exp(I(f + g)): Furthermore, X Y , that is, the Wick product can be extended to random variables X; Y in Lp.