Applications Of Fractional Brownian Motion And Related Processes In Backward Stochastic Differential Equations And Financial Derivatives Pricing

In recent years, as an extension of Brownian motion, fractional Brownian motion has drawn more and more people’s attention, it is neither markov process nor martingale, so we cannot use the classical stochastic analysis to discuss it. Fractional Brownian motion has important properties of self-similarity and non-stationary etc., it can more accurately depict some intrinsic characteristics of the things, therefore, the fractional Brownian mo-tion has important applications in the financial, economic, physical, chemical, ecological, aerospace and other fields.This paper mainly studies applications of fractional Brownian motion and related processes in backward stochastic differential equations and financial derivatives pricing. We first study solutions to BSDEs driven by multidimensional fractional Brownian mo-tions, By constructing a compressed image, using the quasi-conditional expectation and fixed point principle, we obtain the existence and uniqueness of the solutions to BSDEs driven by multidimensional fractional Brownian motions. Then we study applications of related processes i.e. mixed fractional Brownian motion, fractional Brownian mo-tion with jumps, mixed fractional Brownian motion with jumps in financial derivatives pricing.We introduce the main research content and innovation of this dissertation.In Chapter 1, we introduce the research background and research questions of chap-ter 2 to chapter 6 in this paper.In Chapter 2, we recall some important results of one-dimemsional fractional Brow-nian motion, and lay a solid foundation for the research of this paper. For example, fractional Ito formula, quasi-conditional expectant, quasi-martingale, Girsonov theorem and so on. Then we study application of fractional Brownian motion in pricing for bond with attached warrant.The innovation of this chapter:we improved the traditional results taking into account the dilution factor of the stock and using firm value volatility instead of the stock price volatility.In Chapter 3, we first recall Ito formula for multidimensional fractional Brownian motions, then we give the concept of the multidimensional fractional (or quasi-) condi-tional expectation, and prove some its properties. Under some basic assumptions, and using the quasi-conditional expectation and a fixed point principle, we obtain the exis-tence and uniqueness of the solutions to BSDEs driven by multidimensional fractional Brownian motions. Finally, solutions to linear fractional backward stochastic differential equations are investigated.The innovation of this chapter:the backward stochastic differential equation consid-ered that is driven by multidimensional fractional Brownian motion, rather than one or two, and the variable of terminal condition is extended one dimension to multi-dimension.This chapter is mainly based on the paper:J. Miao, X. Yang, Solutions to BSDEs driven by multidimensional fractional Brow-nian motions, Mathematical Problems in Engineering, accepted.In Chapter 4, we study the options pricing in a mixed fractional Brownian motion. Assuming that the underlying assets price satisfies the stochastic differential equation driven by mixed fractional Brownian motion, the expected return and volatility are de-terministic continuous function of time t, using the quasi-conditional expectation and probability formula, we obtain pricing model for European option. So we extend the traditional model. Then we study exchange options pricing in the mixed fractional Brownian motion environment. Assuming that the underlying asset price respectively satisfy the stochastic differential equation with constant parameters and time-varying parameter driven by mixed fractional Brownian motion, we get the corresponding ex-change option pricing model. In order to better understand the pricing model, in the case of European option pricing model, the sensitivities of option price with respect to change of various parameters are discussed. Then the comparison of our mixed fractional Brownian motion model and traditional models is undertaken by numerical experiment, and further obtain the relationship between the option price and Hurst parameter and coefficient ε.The innovation of this chapter:(1) We extend the return and volatility of the underlying asset price from constants to the deterministic continuous functions of time, and got a more general European option pricing model.(2) When we study exchange options pricing, we extend to the financial markets driven by mixed fractional Brownian motion.(3) We combine theoretical research and numerical experiments in this chapter.This chapter is mainly based on the paper:J. Miao, Pricing model for options in a mixed fractional Brownian motion environ-ment, finished.In Chapter 5, We study stochastic calculus for fractional Brownian motion with jumps. Ito formulas for fractional Brownian motion with jumps are obtained. We deduce the Girsanov theorem for fraction Brownian motion with jumps. Finally, By applying fractional Brownian motion with jumps to European option pricing, and assuming that the underlying assets price satisfies the stochastic differential equation driven by frac-tional Brownian motion with jumps, and using the quasi-martingale method, we obtain the European option pricing formula in a jump fractional Brownian motion environment.The innovation of this chapter:(1) We prove the Ito formula for fractional Brownian motion with Poisson jumps, so fractional Ito formula is extended.(2) We Prove the Girsanov theorem for fractional Brownian motion with Poisson jumps, so fractional Girsanov theorem is extended.(3) The options pricing environment is extended from driven by fractional Brownian motion to fractional Brownian motion with jumps.This chapter is mainly based on the paper:J. Miao, X. Yang, Stochastic calculus for fractional Brownian motion with jumps and application, submitted.In Chapter 6, we deal with the problem of pricing convertible bonds in a jump mixed fractional Brownian environment. Ito formulas for mixed fractional Brownian motion with Poisson jumps are obtained. Then assuming that the underlying assets price satisfies the stochastic differential equation driven by mixed fractional Brownian motion with jumps, we obtain the general pricing formula of convertible bonds using the risk neutral pricing principle and quasi-conditional expectant. For the purpose of understanding the pricing model, the sensitivities of convertible bond price with respect to change of various parameters are discussed. Finally, the comparison of our jump mixed fractional Brownian motion model and traditional models is undertaken by numerical experiment, and further obtain the relationship between convertible bond price jump parameters and Hurst parameter.The innovation of this chapter:(1) By changing the pricing environment, we obtain the more general pricing model for convertible bonds.(2) We combine theoretical research and numerical experiments to understand the pricing model.This chapter is mainly based on the paper:J. Miao, X. Yang, Pricing model for convertible bonds in a mixed fractional Brow-nian motion with jumps environment, East Asian Journal on Applied Mathematics, accepted.In Chapter 7, summarize the research of this paper, and put forward the possible research direction in the future,we give the main results of this dissertation。Chapter 2 Stochastic calculus for fractional Brownian motion and ap-plicationIn this chapter, we first recall the stochastic calculus for fractional Brownian motion, On this basis, assuming that the firm value satisfies the stochastic differential equation driven by fractional Brownian motion, we obtain a more realistic pricing model of bond with attached warrant. In this model, firm value process volatility is used.Main results of this chapter:Theorem 2.3.1. Assume that firm value process satisfies the equation then in a risk-niutral world, the price Pt of a bond with attached warrant at any t(t∈ [0,T1]) is given by where whereN(·) is the cumulative normal distribution function. the relationship between firm value volatility and stock price volatility is the following Substituting above equatin to theorem 2.3.1, we obtain the followning theorem.Theorem 2.3.2. In a risk-niutral world, the price Pt of a bond with attached warrant at any t(t € [0,T1]) is given by the following nonlinear equationChapter 3 Solutions to BSDEs Driven by Multidimensional Fractional Brownian MotionsIn this chapter, we study more general backward stochastic differential equations driven by multidimensional fractional Brownian motions. By constructing a compression mapping, and using the quasi-conditional expectation and fixed point principle, we obtain the existence and uniqueness of the solutions to BSDEs driven by multidimensional fractional Brownian motions.Main solutions of this chapter:Definition 3.2.1. The quasi-conditional expectation of some random variable F ∈ L2(Ω, F, P) relative to a fractional Brownian motion with Hurst parameter Hi is defined by where since BtHj,j=1,…,m are independent fractional Brownian motions, by Definition 3.2.1, we haveTheorem 3.2.2.Let and G=g(ηT), where ηT=(η1(T)),…,ηn(T)).Assume that％(s),i=1,…,n,j=1,…,m,(?)s∈[0,T], be measurable and Let At:(Aij(t)) be a n×n-matric,where If F≥G almost surely,then we also have Now we consider the following backward stochastic differential equation where zs(z1,s,…,Zm,s),xt=(x1(t),x2(t),…,xn(t)),where and BsH1,…,BsHm》are independent fractional Brownian motions of Hurst parameters H1,…,Hm.Assume that(H1)xi(0),i:1,…,n are given constants.(H2)For i=1,…,n,bi:[0,T]→R are continuous deterministic functions.(H3)For i,j=1,…,n,k=1,…m,σik:[0,T]→R are continuous deterministic functions,and where(H4)For i:1,…,n,j=1,…,m, where σij(t)=∫t0φj(t,r)σij(r)dr.(H5)9(x)is a continuously differentiable function with respect to x and satisfies polynomial growth.(H6) f(t,x,y,z) is a continuous function with respect to t and twice continuously differentiable with respect to x, y, z, so there exists a constant L> 0, such that for all t e [0,T], x ∈R, y, y∈R, z, z’∈ Rm, we haveTheorem 3.3.1. If (3.25) has a solution u(t,x) which is continuously differentiable with respect to t and twice continuously differentiable with respect to x, thenLemma 3.3.1. Let a(s,x),θj(s,x),j=1,…,m be continuous with respect to t and continuously differentiable with respect to x and let them be of polynomial growth. Assume that (H3), if thenTheorem 3.3.2. Assume that (H3), and let (3.22) have a solution of the form (yt= u(t,xt),z1,t= v1(s,xs),….Zm,t= vm(s,xs)). ThenTheorem 3.3.3. Assume that (H5)-(H6), and let yt=(?){t,xt) and z1,t=ψ1(t,xt), ,Zm,t=ψm(t,xt) ∈C, then the solution (Yt,Z1,t,…Ζ,Zm,t) to (3.3) satisfies Y,Z1, ,Zmt,∈VT.Theorem 3.3.4. Assume that (H1)-(H6), then the BSDE (3.22) has a unique solution in VT.Theorem 3.4.1. Let αt,βt,γi,t, i=1,…,mbe given continuous and adapted processes and ζ ∈.FT.Suppose where bi,r=γi,r+∫0TDtHiβsds. Then the solution to (3.40) exists uniquely. Moreover, we haveChapter 4 Pricing model for options in a mixed fractional Brownian motion environmentIn this chapter, assuming that the underlying assets price satisfies the stochas-tic differential equation driven by mixed fractional Brownian motion, using the quasi-conditional expectation and probability formula, we obtain pricing model for European option and exchange option. The comparison of our mixed fractional Brownian motion model and traditional models is undertaken by numerical experiment.Main results of this chapter:Theorem 4.1.2. (Mixed fractional Ito formula) let 1/2< H< 1, and where X0 is a constant, g satisfies E[sup|<gs|]<∞,f satisfies Let F ∈C1,2([0,T]×R) with bounded second order derivatives, then for (?)t∈[0,T], where DsXs is a Mallivan derivative for the Brownian motion, and DsH Xsis a -derivative for the fractional Brownian motion.In particular, if g and f are deterministic continuous functions of time t, then whereTheorem 4.2.1. For every t E [0,T], and σt is deterministic continuous function of time t, we haveTheorem 4.2.2. Let f be a function such that E[f(BT, BTH)<∞, then for every t ∈ [0, T], and σtis deterministic continuous functions of time t, thenTheorem 4.2.3. Let f is a function such that E[f(BT, BTH))<∞. Let Then for every t∈[0, T], we haveTheorem 4.2.4. The price at any time t ∈ [0, T] of a bounded FT measurable claim F ∈ L2(Ω, FT, Q) is given byTheorem 4.3.1. Assume that the stock price satisfies the following equation then under the risk neutral probability measure, for any t ∈[O,T], the valuation C(t, St) of European option is given by where N(·) is the cumulative normal distribution function.Lemma 4.3.1. In theorem 4.3.1, when rt= r, σt=σ are constants, then for any t ∈ [O, T], the valuation C(t, St) of European option is given by whereTheorem 4.4.1. Assume that the stock price satisfies the following equation under the risk neutral probability measure, for any t∈[0, T], the valuation P(t,St1,St2], of exchange option is given by where N(·) is the cumulative normal distribution function.Theorem 4.4.2. Assume that the stock price satisfies the following equation then under the risk neutral probability measure, for any t E [0, T], the valuation P(t, S1t, St of exchange option is given by where N(-) is the cumulative normal distribution function.Chapter 5 Stochastic calculus for fractional Brownian motion with jumps and applicationIn this chapter, we first prove Ito formulas for fractional Brownian motion with jumps, and deduce the Girsanov theorem for fraction Brownian motion with jumps. By applying fractional Brownian motion with jumps to European option pricing, we obtain the European option pricing formula in a jump fractional Brownian motion environment.Main results of this chapter:Theorem 5.2.1.(The one-dimensional Ito formula) Let Xt be a process defined by where X0 is a constant, αs,βs and γs(z) are predictable processes, such that for all t∈[O,T],z∈R, Let f:[0, t]× R→R ∈C1,2([O,T]×R) with bounded second order derivatives. ThenTheorem 5.2.2.(The multidimensional Ito formula) In the multidimensional case we are given a J-dimensional fractional Brownian motion BtH= (BtH1,…,BtHJ)T, t∈[O,T], and K independent compensated Poisson random measures N(dt,dz)=(N1(dt,dz1), , NK(dt, dzK))T, t∈[O, T], z=(z1,…zK) ∈RK.令 where α:[O,T]→Rn, β:[O,T]→Rn×J, and γ:[O,T]×RK→Rπ×K RnxK are predictable processes such that the integrals exist. That is and satisfies where Let/:[0,T]×Rn→R∈C1,2([O,T]× Rn) with bounded second order derivatives, then where γs(k) is the column number k of the n×k matrix γ= [γik],γs(ik) is γs(k)is the coordinate number i f andTheorem 5.3.4.(Girsanov theorem). Let T> 0,θ(s,.z)≤ 1, s∈[O,T], z∈R, s∈[O,T] be predictable processes. Let ζ is a continuous function with suppζ,(?) [O, T], u be a function with suppu C [0,T], and (?)f∈S(R) with suppf(?) [0,T], such that for any t∈[O, T], Let and satisfies E[Z(T)]=1. Assume that Z(T) satisfies the Novikov condition Define the probability measure Q on FT by Define the process BQ(t) and the random measure NQ(dt, dz) respectively by andThen BQH(·) is a fractional Brownian motion with respect to Ft, and NQ(·,·) is a (Ft, Q)-compensated Poisson random measure of N(·,·).Theorem 5.4.2. Assume that the price St of the risky asset is assumed to be of the form where Nt=Nt-λt. Let ρ= E[U], then then the valuation V(t) of the European call option at time t(t∈[0, T]) is given by where εn denotes the expectation operator over the distribution of Πin=1(1+Ui); N(·) is the cumulative normal distribution function.Chapter 6 Pricing model for convertible bonds in a mixed fractional Brownian motion with jumps environmentAssuming that the underlying assets price satisfies the stochastic differential equa-tion driven by mixed fractional Brownian motion with Poisson, we obtain a general pricing formula for convertible bonds using the risk neutral pricing principle and quasi-conditional expectation. The sensitivities of convertible bond price with respect to the change of various parameters are discussed. Finally the comparisons of theoretical prices between our jump mixed fractional Brownian motion model and traditional models are undertaken by numerical experiment.Main results of this chapter:Theorem 6.1.1. (Ito formal) Let where Xo is a constant, αs,βs and γs(z) are deterministic continuous functions of time t, such that for all t∈ [0, T], z∈R, Let f:[0,T] x R→R∈C1,2([0,T]×R) with bounded second order derivatives, then whereTheorem 6.2.3. Assume that the stock price St satisfies the following equation then for any t ∈ [0, T], the valuation V(t,St) of a convertible bond with face value F and conversion price Cυ is given by whereTheorem 6.3.1. Let V= V(t,St) be the price of convertible bond at time t∈ [0,T], by Theorem 6.2.3, the influence of the common parameters can be represented asRemark 6.1. From Theorem 6.3.1, we can easily get △≥0, Γ≥0, ▽F≥0, ▽Cυ≤0, vσ≥0 and ρr≤0, therefore the valuation of convertible bond is an increasing function of St, F and σ, and a decreasing function of Cυ and ρr.Γ≥0 reveals the proportion of risky assets increases with the increase of the risky assets price in a portfolio.Theorem 6.3.2. Suppose V=V(t, St) is the price of convertible bond at time t ∈[0, T], by Theorem 6.2.3, The influence of the Hurst parameter can be written asRemark 6.2. From Theorem 6.3.2, we can easily get (?)H/(?)V≥ 0, therefore the valuation of convertible bond will go up with the increase of Hurst parameters.Theorem 6.3.3. Let V=V(t,St) be the price of convertible bond at time t ∈ [0,T], by Theorem 6.2.3, The influence of the jump parameters can be represented as From Figure 6.1 we see that the convertible bond price is a increasing function of λ,μU and σU...