Numerical Methods For American Option Pricing Problems Under Constant And Stochastic Volatility

Options as a kind of important financial derivatives have a wide range of applica-tions. Option pricing problems, especially American option pricing problems, more andmore attract the attention of the financial sector. The model of option pricing problemsdepends on the evolution process of the underlying asset price. In the continuous case,the evolution process of the underlying asset price can be described by a stochastic partialdiferential equation, based on this, as its derivatives——the option price is suitable for adefinite solution problem of a partial diferential equation. Diferent descriptions for thestochastic partial diferential equation of the underlying asset will export diferent optionpricing models. Among the numerous models, Black-Scholes model is the most widelyused one. Under the Black-Scholes model, there are no analytical solutions for Ameri-can option pricing problems, so the study of American option pricing problems under theBlack-Scholes model, especially proposing efective, stable, achievable and fast numericalcomputing methods have become one of the important subjects in financial mathematics.This dissertation mainly concerns American option pricing problems under the Black-Scholes model, and takes American put options for example to study American optionpricing problems under constant and stochastic volatility. For the difculties we meet insolving American options, we present the corresponding techniques, and propose efectiveand fast computing methods. The text mainly finished the following4jobs.1. In Chapter1, we first briefly review the development history of options, and givesome classification methods for options. Then, taking American put options for example,we present Black-Scholes models under constant and stochastic volatility. Later, we givea short review of the current research situation for American options. At last, we collectsome numerical methods related in the subsequent Chapters.2. In Chapter2, we propose a splitting method for solving American options underconstant volatility. This Chapter consists of two divisions: I. The optimal exercise boundary. In this paper, we mainly take American put options for example to study American option pricing problems, for American call options, using call-put symmetry, we can get the same conclusion. The Black-Scholes model for American put options under constant volatility is: where S, t and P=P(S, t) stand for the underlying asset price, time and put option value, respectively. σ, r, q, T and K denote the volatility, interest rate, dividend rate, maturity date and exercise price, respectively. B(t) is the optimal exercise boundary, which divides the solving domain into two parts. The option should be exercised if the underlying asset price S is not exceeding B(t) at time t; otherwise, the option should be held.Considering the Black-Scholes model (1), we can find the option price P=P(S, t) and the optimal exercise boundary B(t) are all unknown, and there is dependence between them, which bring a lot of troubles for solving the option pricing problem. The main idea of the splitting method is completing the whole solving process into two steps. First, we solve a Volterra integral equation to get the optimal exercise boundary B(t), and then obtain the option price by solving a parabolic problem on a bounded domain.The derivative of B(t) has a singularity at the point t=T. Therefore, there is no guarantee on the convergence of standard numerical methods on uniform meshes. On the other hand, the graded meshes works well for this kind of singular Volterra integral equations. In this paper, we use collocation method on graded meshes coupled with Newton’s method to gain the optimal exercise boundary B(t). In section2.1, we describe the solving process of the optimal exercise boundary detailedly. Numerical simulations in section2.3are shown to confirm the efficiency of this method.Ⅱ. The option price. With the calculated optimal exercise boundary, we take the Dirichlet boundary condition as the left boundary condition, and reformulate the Black-Scholes model (1) as a standard parabolic problem. When solving this problem, we are faced with the following three difficulties:(1) The left boundary B(t) is a curve, and the solving domain is irregular;(2) The right boundary at infinity, we cannot adopt numerical methods directly;(3) Choosing an efficient numerical method to get the option price P(S, t). In section2.2, we will present the corresponding techniques for these difficulties. When the boundary is smooth or monotone, the front-fixing method usually is a better pretreatment method for solving varying boundary problems, which can transform the varying boundary into a fixed one. In this paper, we will overcome the first difficulty by the front-fixing transformation. The unbounded domain is truncated to a bounded one by the perfectly matched layer technique, which is a good method for truncating unbounded domain problems, and makes the solving area smaller to accelerate the computation. So the second difficulty can be solved easily.Using the above transformations, the American option pricing problem could be changed into a definite solution problem on a rectangular domain. At last, we consider the numerical method for solving the American option pricing problem. In section2.2.3, we will use a finite element method (FEM) to solve problem (2.16). By numerical experiments in section2.3, we can find, the proposed method in this paper can get more accurate numerical results compared with other methods.Theorem1Assume u and uh are the exact solution and the semi-discrete approximation of the exact solution for the PML problem (2.16), respectively. If the initial approximation Uh(x,0) satisfiesthen the following error estimate holdswhere C is a constant independent of h.In section2.2.3, we will give a detailed proof of the convergence for using the finite element method to solve the PML problem (2.16). Arrive here, we have solved all the difficulties for numerical solving the American option pricing problem.In Chapter2, we also study equations corresponding to some important financial parameters (Greeks) and numerical methods for solving them, and verify the effectiveness of our algorithm through numerical experiments. At the end of this chapter, we present3-D images for American options and Greeks by finite element and discontinuous Galerkin methods respectively.3. In Chapter3. we propose a coupling method for solving American options under constant volatility. This Chapter consists of four divisions: I. Problems on a bounded domain. Let’s consider the American put option pricing problem (1), B(t) is an unknown curve, P=P(S,t) is the option price to be solved, and it’s not hard to see there is dependence between them. Obviously, the problem (1) is a very complicated nonlinear system, for solving which numerically, we must face the following difficulties:(1) The left boundary B(t) is an unknown curve, the solving domain is irregular;(2) The right boundary at infinity, we cannot adopt numerical methods directly;(3) Choosing an efficient numerical method to get B(t) and P(S, t) simultaneously.The main idea of the coupling method is considering the original problem (1) directly, and giving an efficient numerical method to solve the optimal exercise boundary B(t) and the option price P(S, t) simultaneously. Of course, we must solve the first two difficulties firstly.Using the standard variable transforms (3.1) and the front-fixing transforms (3.5), we can reformulate the variable domain problem (1) into a problem on a semi-infinite strip domain. Then, we can truncate this semi-infinite problem into the following problem on a rectangular domain by PML technique (3.9): the coefficients, right hand function and free boundary after transformation will be given in Chapter3. Through a series of transformations, the American option pricing problem has been change into a parabolic problem on a bounded domain.In the process of the option and stock trading, practitioners are not only concerned about the movement of the option price itself, but also pay attention to some risk statistic parameters. We denote these parameters by Greeks, including Rho(R), Vega(V) and Delta(△), which are partial derivatives of the option price with respect to the interest rate, volatility and stock price, respectively. In this division, we will also study the equations corresponding to Greeks.Now, let’s consider Rho, Vega and Delta. When the optimal excise boundary B(t) known, differentiating (1) with respect to r, σ and S on the both sides (For simplify, we only take the Dirichlet boundary condition), we can get the problems corresponding to Greeks. Using the same method as for the problem (1),we can get the approximating problems for Greeks on a bounded domain.So far,we have transferred the American option pricing problem and the problems for Greeks into parabolic problems on a bouded domain. In the following part,we will discuss numerical methhods for these problems.We will combine the standard finite element method(FEM)with Newton’s method to solve the problem (4)to get the optiOn price P(S,t) and the optimal excise boundary B(t).Given the optimal excise boundary, we Can apply FEM to solve Rho and Vega. Since Delta has a singularity at the point (T,B(T)), FEM can not reach a high accuracy, so we shall introduce a discontinuous Galerkin method (DGM) and a weak Galekin method(WGM)to solve Delta.Ⅱ.A coupling method for solving the option price.In this division,we mainly solve the problem (4) by a finite element method coupled with Newton’s method,and prove the stability and nonnegativity of the finite element solution.When dealing with the problem(4)numerically, we get u(x,T)by solving a variational problem, which is formulated by equation (4)with the Neumann boundary condition ux(O,T)=gx(b)(T),T).The Dirichlet boundary condition at the left boundary x=O can be viewed as an implicit function of the free boundary b(T),which is used ti solve b(T). The main idea of the coupling method is to solve these two steps alternatively.In section3.2.1,we will illustrate the details for solving the PML problem (4)by applying the finite element method coupled with Newton’s method. Now,we give the stability and nonnegativity of the finite element solution.Theorem2Assume-C11+C2(?)|log(Tm)|≤δTbm<0,0<Tm≤T, Where C1and C2positive constant, if α≥-1/2+r-q/2-r,β≤r+α(r-q)-γα(1+α)-γc2, then the numerical sclution uhm(m=1,...,M) of system (4) is stability when θ=0or0.5, we havewhere‖·‖denotes the norm of L2(Ω).Theorem3Under assumption of Theorem2, if α≤-1/2+r-q/2r+(r-q-r)2+4ry/2r andare small enough, then the numerical solution umh(m=1,...,M) of system (4) is non-negative, that is At the end of this division, let us explain how to compute the approximation of the option price at a given point (S, t). For m=1,2,..., M, i=1,2,...,N1, using the inverse transforms of (3.1) and (3.5), the approximation of P(Smi,Tm) is given bywhere Smi=Kexi+bm, bm=b (Tm).III. A discontinuous Galerkin method. In this division, we introduce the meth-ods of calculating Greeks. We shall apply the finite element method to solve Rho and Vega respectively. Since Delta has a singularity at the point (T, B(T)), the standard finite element method is not an efficient solver, so we mainly introduce a numerical method for solving Delta——the discontinuous Galerkin method (DGM).In section3.2.2, we will illustrate the details for solving Delta by using the discontin-uous Galerkin method. Adopting the backward Euler method for time discretization, and the piecewise polynomial for space discretization, then we can get the DGM approximation solution for Delta. We also obtain the error estimate for Delta by using the discontinuous Galerkin method in section3.2.2.Theorem4Let w(T, x) and wh be the exact solution and DGM solution of problem (3.17) respectively. Assume w(T,x)∈H1(0,T; Hs(Ω)) with s>3/2, and take∈=1in (3.30). Then‖w(·, T)-Wh(·,T)‖L2(Ω)≤Chmin(k+1,s)-where C is a constant independent of h.Through numerical experiments in section3.3.1, we find the DGM is more efficient to capture the discontinuity of Delta meanwhile keeping the accuracy compared to the finite element method.IV. A weak Galerkin method. In this part, we will introduce a weak Galerkin method for solving Delta. This method is proposed by professor Wang, which is very effec-tive for dealling with discontinuous or singularity problems. The weak Galerkin method is a generalization of the traditional finite element method, which mainly replace the tradi-tional derivative with a weak derivative. It doesn’t require the continuation of the original function space and the derivative function space.In order to ensure the weak derivative operator is well defined, it needs to add some conditions for the degree of polynomials used to approximate the function spaces. Theorem5With the restriction r=j+1, the discrete derivative satisfiesIn section3.2.3, we will introduce the weak Galerkin method for solving Delta indetail. Through numerical experiments in section3.3.2, we obtain the weak Galerkinmethod can approximate Delta accurately. At the end of this part, we give the approxi-mation forms of Greeks. If wh represents the approximation of w, which is solved by thefinite element method (FEM) or weak Galerkin method (WGM), then the approximationsof Rho, Vega and Delta arehere y=log(S/K). At the end of this chapter, we use several numerical examples to verify the effective-ness of the algorithm in this dissertation, and give3-D images of option pricing problemsat the same time.4. In Chapter4. We mainly study American option pricing problems under stochasticvolatility, and propose a method for solving the optimal exercise surface and option pricesimultaneously. To simplify the model, we only consider the case q=0, the result aboutthe case q≠0is similar. The model of American put option pricing problems understochastic volatility is given as follows: From the structure of the problem (7), we can see that the solving domain is a three-dimensional domain. which is one dimensional in the direction of time, and two dimen-sional in space direction. The left boundary of (7) is an unknown surface, which is theoptimal exercise surface B(t, y). The boundary conditions in the positive directions of axisS and y are both at infinite, so we will have several difculties when solving the problem(7) numerically:(1) The left boundary B(t, y) is an unknown surface, the solving domain is irregular;(2) The solving domain is unbounded, we cannot adopt numerical methods directly;(3) Choosing an efcient numerical method to get the optimal exercise surface B(t, y)and the option price P (S, y, t) simultaneously.For the first difculty, we apply the front-fixing transformation to change the leftboundary into a plane of x=0; The second difculty can be solved by using PMLtechnique, this dissertation adopts the truncation method directly to simplify the solvingprocess; Finally, we apply the finite element method coupled with Newton’s method tosolve the third difculty. In chapter4, we get the approximate results of the option priceP (S, y, t) and the optimal exercise surface B(t, y) through numerical experiments, whichverify the algorithm in this dissertation can fit the option price and the optimal exercisesurface well.