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PRICING DERIVATIVES in HERMITE MARKETS Svetlozar T

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PRICING DERIVATIVES in HERMITE MARKETS Svetlozar T PRICING DERIVATIVES IN HERMITE MARKETS Svetlozar T. Rachev Texas Tech University and GlimmAnalytics Stefan Mittnik Ludwig Maximilians University of Munich Frank J. Fabozzi EDHEC Business School Zari’s corrections: All comments of Stefan are taken care of in the text. My changes are in red. December 24, 2014, 6 am Abstract: We introduce Hermite fractional financial markets, where market uncertainties are described by multidimensional Hermite motions. Hermite markets include as particular cases financial markets driven by multivariate fractional Brownian motion and multivariate Rosenblatt motion. Conditions for no-arbitrage and market completeness for Hermite markets are established. Perpetual derivatives, perpetual bonds, forwards, and futures are priced. The corresponding partial and partial-differential equations are derived. Keywords: Hermite processes; fractional Brownian motion; Rosenblatt processes: no-arbitrage and completeness for fractional markets; derivative pricing; term-structure of interest rates; forward and future contracts P a g e 0 | 29 ퟏ.INTRODUCTION We introduce a novel method for pricing derivatives in fractional markets. All existing fractional market models1 assume that the riskless asset has the same dynamics as in the classical option pricing models developed by Black and Scholes (1973) and Merton (1973), hereafter referred to as the BSM model. But that assumption leads to the existence of arbitrage trading strategies (see Rogers (1997) and Shiryayev (1998)). Mixing fractional Brownian motion (FBM) with Brownian motion (BM)2 changes the setting completely as now the overall market driver is no longer a fractional process. Applying Wick integration, Hu and Øksendal (2003) show that an FBM market has no arbitrage opportunities. However, Wick integrations has no economic intuition (see, for example, Björk and Hult (2005)) and the corresponding replicating strategies are very restrictive.3 In this paper, we propose a different approach. We postulate that in arbitrage-free complete fractional markets, exhibiting long-range dependence (LRD), the riskless asset, if publicly traded,4 1 See, for example, Sottinen (2001), Bender, Sottinen, Valkeila (2007), Mishura (2008), and Rostek (2009). 2 See Mishura (2008) , Kozachenko, Melnikov and Mishura (2014). 3 In the FBM-market setup of Elliot and Van der Hoek (2003), the simple buy-and-hold strategy is not self-financing (which is indeed discouraging). 4 In this paper, we do not discuss the existence of a riskless asset. This has been subject of considerable debate recently with Fisher (2013) arguing: ”The idea of risk-free sovereign bonds 1 should have a cumulative rate which increases in time as a regularly varying function (RVF)5 of order greater that one. This is a necessary for no-arbitrage in fractional markets. Indeed, in the BSM-model the variances of the increments of the BM increase linearly in the increments’ durations. In sharp contrast, in fractional markets, such as an FBM-market, the variance of the increments of the market driver increases to infinity as RVF of order greater than one6. In fractional markets exhibiting LRD, the stock traders are clairvoyant.7 Thus, the deterministic cumulative return process of the riskless asset should grow faster in time than the linear growth of the riskless- asset return process in the BSM-model. Suppose that in a fractal market exhibiting LRD, a trader can trade a riskless asset which has the same dynamics as the riskless asset in the BSM-model. The trader also can trade a risky asset whose fractal-LRD price dynamics exhibit a stochastic trend with a mean path above the deterministic path of the riskless asset price. Then, the trader will is best thought of as an oxymoron or as an anomaly of recent history. It is not a useful, necessary or an enduring feature of the financial landscape.” 5 See Seneta (1976). 6 In our setting, a monotone transformation of the time (a market timing transformation, or a time-subordination) can transform the dynamics of the riskless asset in the Hermite market to become identical to the dynamics of the riskless asset in the BSM market. 7 This is because the risky assets (the stocks) are driven by a fractional process with LRD and the increments of the process are positively correlated. 2 easily realize various arbitrage trades.8 This observation is at the heart of our method outlined below. The paper is organized as follows. In Section 2 we introduce Hermite markets, in which the market uncertainty is modeled by Hermit motion.9 Our choice for using Hermite markets to model a general fractional market is motivated by the flexibility10 of Hermit motion as a self- similar process with stationary increments with rich autocorrelation dynamics. In Section 3 we establish the conditions for guaranteeing that a Hermite market is arbitrage-free and complete. The valuation of perpetual derivatives, perpetual bonds, forward and future contracts in a Hermite market is derived in Section 4. Concluding remarks are in Section 5. The proofs are given in the Appendix. ퟐ.THE PRIMITIVES OF A HERMIT MARKET In this section we state the main properties of a Hermit motion market. 2.1 The Hermite Motion as a Model For Market Uncertainty 8 Examples of such arbitrage strategies are well-known, see Shiryayev (1998) and Rostek (2009). 1 1 9 The parameters of Hermit motions ℋ(퐻,퓀)(푡), 푡 ≥ 0, 퐻 ∈ ( , 1) are ∈ ℕ, 퐻 ∈ ( , 1). H is the 2 2 Hurst index, also known as the self-similarity index, see Torres and Tudor (2009). We denote ℕ = {1,2, … }. For 퓀 = 1, HM is FBM. For 퓀 > 1, ℋ(퐻,퓀) is a non-Gaussian process, and ℋ(퐻,2) is known as Rosenblatt motion, see Taqqu (2011). 10 This flexibility will be advantageous fitting models empirically. 3 1 The Hermite Motion (HM),11 ℋ(퐻,퓀)(푡), 푡 ≥ 0, 퓀 ∈ ℕ, 퐻 ∈ ( , 1) is defined by 2 (2.1) ℋ(퐻,퓀)(푡) = 퐶(퐻,퓀) 퐾(퐻,퓀)(푣(1), … , 푣(퓀)) 푑퐵(푣(1)) … 푑퐵(푣(퓀)), ∫풟퓀 푡 푡 ≥ 012, where (푯푴풊) 풟퓀 ≔ {핧 = (푣(1), … , 푣(퓀)) ∈ 푅퓀: 푣(푖) ≠ 푣(푗), 푖, 푗 = 1, … , 퓀, 푖 ≠ 푗}; (퐻,퓀) (1) (퓀) 퓀 13 (푯푴풊풊) For a given 푡 ≥ 0, the kernel 퐾푡 (핧), 푡 ≥ 0, 핧 = (푣 , … , 푣 ) ∈ 푅 , is defined by 퐻−1 1 푡 − (2.2) 퐾(퐻,퓀)(핧) ≔ ∫ [∏퓀 (푠 − 푣(푗)) 퓀 2] 푑푠 푡 0 푗=1 + 11 See Taqqu (1979), Dobrushin (1979), Dobrushin and Major P. (1979) , Dehling and Taqqu (1989), Lacey (1991), Lavancier F.(2006) , Maejima and Tudor (2007),Tudor (2008),Torres and Tudor (2009), Pipiras and Taqqu (2010), Chronopoulou, Tudor and Viens (2011), Tudor (2013), Marty (2013), Bai and Taqqu (2014), Sun and Cheng (2014), Clausel et al (2014), and Fauth and Tudor (2016). 12 The integral is understood as Winer-Itô multiple integral, cf. Dobrushin (1979), Nualart (2006), and Clausel et al (2014). 푏 푎 , 푖푓 푎 ≥ 0 (퐻,퓀) 13푎푏 : = { , 푏 ∈ 푅. For every given 푡 ≥ 0, the kernel 퐾 (핧), is symmetric and has + 0, 푖푓 푎 < 0 푡 2 a finite ℒ (푅퓀)-norm: ‖퐾(퐻,퓀)‖ = √∫ (퐾(퐻,퓀)(핧)) 푑핧 < ∞ . Thus, ℋ(퐻,퓀)(푡), 푡 ≥ 0 is 2 푡 퓀 푅퓀 푡 ℒ2(푅 ) well-defined process. 4 1 (푯푴풊풊풊) 퐻 ∈ ( , 1) is the Hurst index (index of self-similarity);14 2 2 (퐻,퓀) (퐻,퓀) (푯푴풊풗) 퐶 > 0 is a normalizing constant such that 피 (ℋ (1)) = 1. 15 16 (푯푴풗) 퐵(푣), 푣 ∈ 푅 is a two-sided BM defined on (Ω, ℱ, {ℱ푡}푡≥0, ℙ). 17 (퐻,퓀) An alternative representation, ℋ (푡), 푡 ≥ 0, is given by 퓀 (푗) 푖푡 ∑푗=1 푢 (퐻,퓀) (퐻,퓀) 푒 −1 (ℂ) (1) (ℂ) (퓀) (2.3) ℋ (푡) = 푐 퐵 (푑푢 ) … 퐵 (푑푢 ), ∫푅퓀 2퐻−2+퓀 퓀 (푗) 퓀 (푗) 2퓀 푖[∑푗=1 푢 ]|∏푗=1 푢 | 2 where 푐(퐻,퓀) is a normalizing constant, so that 피 (ℋ(퐻,퓀)(1)) = 1, and 퐵(ℂ)(푑푢) is a complex random measure generated by a standard Brownian motion. 14 See, for example, Samorodnitsky (2016) for an extensive study of LRD processes. −1 3 2퐻Γ( −퐻) 15 (퐻,퓀) (퐻,퓀) (퐻,1) 2 (퐻,2) 퐶 = (√퓀! ‖퐾 (핧)‖ ) , 퐶 = √ 1 and 퐶 = 1 퓀 Γ( +퐻)Γ(2−2퐻) ℒ2(푅 ) 2 퐻 퐻 Γ(1+ )√ (2퐻−1) 2 2 퐻 , see Clausel et al (2014). Γ( )Γ(1−퐻) 2 16 The two-sided Brownian motion 퐵(푣), 푣 ∈ 푅 is defined as follows: 퐵(1)(푣), 푓표푟 푣 ≥ 0 퐵(푣) = { , 퐵(2)(−푣), 푓표푟 푣 < 0 where 퐵(1)(푡), 푡 ≥ 0 and 퐵(2)(푡), 푡 ≥ 0 are two independent standard Brownian motions. 17 See Taqqu (1979). 5 18 The existence of the HM follows from the non-central invariance principle: 1 ⌊푛푡⌋ (퓀) (푗) (푤푒푎푘푙푦) (퐻,퓀) ∑푗=1 하 (휉 ) → (ℋ (푡)) , 퐻 0≤푡≤푇 2 where (푖) 휉(푗), 푗 = 0, ± 1, ±2 …, is a stationary Gaussian sequence, with 피휉(푗) = 0, 피(휉(푗)) = 1 and covariance function (휉)(푛) = 피[휉(0)휉(푛)] having power decay for some slowly varying function19 퐿(푛), 푛 = 1,2, …, (휉)(푛) lim푛↑∞ 2퐻−2 < ∞; 퐿(푛)푛 퓀 2 (푖푖) 하(퓀): 푅 → 푅, where 피하(퓀)(휉(0)) = 0, 피 (하(퓀)(휉(0))) < ∞, and 하(퓀) has Hermite rank 퓀. 20 (퐻,퓀) (퐻,퓀) The basic properties of the HM, ℋ = {ℋ (푡), 푡 ≥ 0}, are: 18 See Dobrushin and Major (1979), Taqqu (1979), and Torres and Tudor (2009). 19 Seneta (1976). 푥2 푥2 ( ) 푑 − 20 Let 푔 푚 (푥) = (−1)푚푒 2 푒 2 , 푥 ∈ 푅, be a Hermite polynomial of order 푚 = 0,1, … The 푑푥 function 하(퓀): 푅 → 푅 has Hermite rank 퓀, if 하(퓀) has the following representation: 1 하(퓀) = ∑ 푐(푚) 푔(푚)(푥), 푐(푚) ≔ 피 {하(퓀) (휉(0)푔(푚)(휉(0)))}, 푚! 푚≥0 with 퓀 = min{푚: 푐(푚) ≠ 0}, see Torres and Tudor (2009).
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