Numerical Methods For Pricing Several Different Types Of American Options

Options as an important financial tool for hedging, which can be effectively avoided risk, and guided the market participants to invest, has been very popular in European and American countries. Recently, with the continuous improvement of the financial markets, and the requirements of the financial risk control, the option productions have been developed in China. In addition, various types of options have been generated for the requirement of the market participants, and become main product of financial market transactions. Compared with the European options, American options has more flexibility which allows the owner to select the any time prior to a maturity date, according to the change of market prices and the practical demand. Therefore, American options has been attracting more attention. Since American options has the characteristics of both advancing the implementation and the price changed with market supply, thus how to choose the optimal exercise boundary, as well as the reasonable option price, will have a direct impact on the sale of both profit and loss situations. Therefore, the optimal exercise boundary and pricing problems for American options become the kernel issue of options trading.This dissertation is devoted to the valuation of American option systematically, which starts from the American put option. Based on the standard American options, we study more complex options, and analysis the difficulties we meet in solving pricing problems.For these difficulties, we present the corresponding techniques, and propose effective and fast numerical methods. The rest of this thesis is organized as follows:1. The overview of American optionsIn this part, we first introduce the development history of options briefly, and give some classification methods for options. Then, taking American put options for example,we review the main work of Black and Scholes, including classical Black-Scholes modelassumptions, model establishment and derivation process of European options pricing formula. Later on, we give a short summary of the current research situation for several types of typical American option pricing problem. At the end of this chapter, we briefly introduce the main work of this paper.2. Pricing problem of standard American optionsIn Chapter 2, we concern on the valuation of standard American options, which satisfies the free boundary problem, and present a valid, fast and efficient numerical method.Because American options satisfies call-put symmetry formula, we only consider American put option pricing problem. The value V(S, t) satisfies the following free boundary problem: where B(t) stands for the optimal exercise boundary, let σ, r, q, K and T be the volatility,interest rate, dividend rate, exercise price and date of expiration, respectively.In this chapter, we give a short review of the current research results for model(1),and analysis the difficulties when using numerical method to solve the problem. For these difficulties, we present the corresponding techniques, which consists of two divisions:(a)Consider the model(1), we can find that the solving region is irregular and unbounded, i.e.left boundary B(t) is unknown and the right boundary is positive infinity. Fortunately, a front-fixing transformation can be an efficient method to handle left boundary problem,which can transform the curved domain into semi infinite regular area.(b) The unbounded domain is truncated to a bounded one by the perfectly matched layer(PML) technique,which is a good method for truncating unbounded domain problem. The method not only makes the solving area smaller in order to accelerate the computation, also can reduce the numerical reflection to guarantee the calculation accuracy. So far, we obtain the bounded regular domain for model(1). Then, we observe that the optimal exercise boundary B(t)and option price V(S, t) are both unknown, and there is dependence between them, how to solve the coupled systems becomes the key to success for the algorithm. Inspired by the previous work, we apply the finite volume method to discrete the simplified problem, and use Newton method to obtain the option price and the optimal exercise boundary,the algorithm is presented in Chapter 2 for details.(c) We display numerical experiments to test the correctness and performance of the proposed method.3. Pricing problem of lookback optionsPart three consists of Chapter 3 and Chapter 4. In this part, we study the lookback options. Compared with the standard American options, lookback options is more complicated, whose payoff function not only depends on the on-the-spot underlying asset price,but also on the maximum or minimum value of the underlying asset price over the history of the whole(or part) validity period closely. Here, we only consider the American put options for example. With the same as standard options pricing model, American lookback option also satisfies two forms: Free boundary problem and linear complementary problem. The main work of this part are concerned on two different pricing models, and proposed fast and efficient numerical methods respectively.? Chapter 3 is devoted to the free boundary problem for American lookback options.At first, we introduce the American lookback put option price V(S, G, t) satisfying the free boundary problem as follows: where the unknown free boundary B(t) stands for the optimal exercise boundary. Letσ, r, q and T be the volatility, interest rate, dividend rate and date of expiration, respectively. Denote Gt:= max0≤τ ≤t Sτis a path-dependent variable, when using equation to describe the pricing model, we will regarded as independent variables, denoted as G for convenience.By analyzing the pricing model(2), we summarize two challenges during solving this problem:(I) The concerned domain is two-dimensional unbounded irregular domain(the region surrounded by S = B(t) and S = G);(II) The optimal exercise boundary B(t) is unknown, which is also depended on option price V(S, G, t). For these difficulties,we adopt following techniques: First, we apply the numeraire transformation to reduce the problem into a one-dimensional free boundary problem. Moreover, using the Landau transformation, we convert the varying domain into [0,1]. So that, the original model is transformed into parabolic nonlinear problem on [0,1]. Then, we deal with the nonlinear parabolic problem, and present a numerical algorithm to obtain both B(t) and V(S, G, t).Similar to the technique for valuation of standard American options, we discretize the simplified model by a finite volume method, and then use Newton method to solve the discreted system, the algorithm is presented in Chapter 3 for details. Finally, numerical simulations are presented to verify the practical and effectiveness of the proposed method.? In Chapter 4, we study the linear complementary problem for American lookback options, and present a more efficient method to solve the problem. The American lookback put option price V(S, G, t) satisfies the linear complementary problem as follows:whereThe main work of this chapter will be deployed from the following aspects:(a) The numeraire transformation is applied to reduce the problem to a one-dimensional unbounded problem.(b) For the unbounded domain, we use the property of optimal exercise boundary, and present precise boundary conditions. We also reformulate the one-dimensional bounded linear complementary problem into a bounded variational inequality. In order to guarantee the symmetry of the discretization matrix, we remove the one-order derivative term from the problem.(c) We discretize the bounded variational inequality by a finite element method, and use the projection and contraction method to solve the discretized system. Moreover, the symmetric positive definiteness of the full-discrete matrix is established, the algorithm is presented in Chapter 4 for details.(d) The numerical simulations are verified the main advantage of the projection and contraction method, whose computing speed is much faster under the same given accuracy.4. Pricing problem of multi-asset optionsIn Chapter 5, we concern on the American multi-asset options. Multi-asset options is a special exotic options, which is a variety of underlying asset options portfolio, and different from single asset options hedging effect. Compared with standard American options,the pricing model of American multi-asset option is a high-dimensional free boundary problem or linear complementary problem, which is difficult to solve. In this chapter, we will take American put options as an example, and present an efficient numerical method to solve it. For the convenience of the expression, we only consider American two-asset options, which can be applied to general cases similarly. The price V(S1, S2, t) satisfies:?whereThe pricing model(4) is a two-dimensional unbounded parabolic linear complementary problem, we will meet the following difficulties when numerically solve the problem:(I) The computing domain for the problem is two-dimensional unbounded one, which is hard to apply numerical method to solve;(II) Choosing a reasonable method to obtain the option price V(S1, S2, t).For the former, we adopt the perfectly matched layer(PML) technique to handle it,which is an efficient method to truncate the unbounded domain problem. The main idea is that in order to reduce the numerical reflection, we add a composed of non-reflective material absorption layer to the truncation of solving domain, so that we can achieve the purpose of reducing errors. For the latter, we use the method of adding a penalty term to transform the linear complementary problem into a nonlinear parabolic problem.Furthermore, we propose a semi-implicit finite element method(the nonlinear term by using the explicit form, the other by applying the implicit scheme) to solve the problem,the algorithm is presented in Chapter 5 for details. In order to illustrate the correctness of the proposed method, this paper also presents some important theoretical analysis and numerical simulations. The convergence of semi-implicit finite element method is presented, and it is proved that the solution is nonnegative. Compared with the far field estimate method, the PML technique is more efficient.In conclusion, this dissertation mainly studies numerical methods for the American options pricing problem under Black-Scholes model. Theoretically, we testify the conver- gence of algorithms and the nonnegativity of solutions; Numerically, we verify that the proposed methods are correctness, efficient and practical.