Numerical Methods For Pricing American And Asian Options

In early financial market there are only four kinds of financial tools: bank deposit, bank draft (bank acceptance), bond and stock. The earliest bank deposit appeared in the 13th centuries. It is known that Ricciardi bank of Luca is the earliest one. During the year from 1272 to 1310, this bank provided a loan of 400,000 pound to the royalty of England. Although the default of this loan caused this company's bankruptcy, it did give an example of the financial risk of early days. Bill of Exchange, which is the creditor's rights with the goal to pay and can be circulated and transferred, arises almost at the same time with bank deposit. However, all kinds of bond, in which the issuer owes the holders a debt and is obliged to repay the principal and interest (the coupon) at a later date (termed maturity), didn't appeared until the 16th century. The first real government bond is Grand Parti of Francis I in 1955. It was issued not just for a handful of banks, but faced all the investors. Stock originated from the British East India company in 1600. The earliest stock is a certificate of pooling capital to finance the building of ships. The first permanent stock company to issue shares of stock was the Dutch East India Company, in 1602. From 1613, this company started to distribute the profit every four voyages, which was commonly regarded as a predecessor of "stockholder" and "distributed dividends".Thus, we can see that the early financial investment is relatively simple. Investors followed some plain philosophy principles such as "lowly buy - highly sell" , "don' t put all the eggs into one basket" , etc. It was not until the latter half of 20th century has the financial market grown up rapidly and mathematical tools took more and more part in it. These changes were mainly due to the two financial revolutions in the Wall street.The first financial revolution started from the paper[Markowitz( 1952)] "Portfolio Selection" , which is Harry Markowitz' s doctoral thesis. The early version of this thesis discussed how to get the portfolio which could maximize the anticipated income through the combination of risk assets (dispersal investment) and at the same time it could keep the risk of single security at an acceptable level. The mathematical tool in his thesis was called Markowitz mean-variance analysis) [Markowitz(1959)]. Before his portfolio theory, investors also discussed risk and profit, but as they couldn' t quantify some important indices, their portfolios were usually very subjective and hardly to make clear why they could get such anticipated profit. Markowitz' s theory solved these problems. Later, many people, such as Willam Sharp, made further research on the problem of profit and risk when markets reached the "balance" (supply and demand are equal) and gave capital asset pricing model - CAPM [Sharpe(1964)]. This model indicates that when markets achieve balance, the factors which determine the asset profit (i.e. pricing assets) are the ,β-measure system risk, whilst non-system risk plays no role in pricing assets, and it is a kind of linear relationship between expecting profit andβ. The standard CAPM gives a complete answer to the problem of the determining mechanism of asset profits when the markets achieve balance. Because of their excellent work, Markowitz and Sharp shared the Nobel prize in economics with Merton Miller in 1990.Before those years we mentioned above, we should point out especially that in 1900, Louis Bachelier published his thesis "Theorie de la Speculation" [Bachelier(1900)], which is a milestone of modern financial study. In his thesis, he used random walk for the first time to describe stock prices and also he mentioned the option pricing problem.At the beginning of the 70's, Black and Scholes obtained a significant breakthrough, which raised the second financial revolution. They deduced a differential equation which should be satisfied for any price of derivative securities based on any non-dividend payment stock.In their breakthrough paper, Black and Scholes solved their differential equation successfully and got the precise formula for European call and put options. But in the real market, the precise formula can' t be obtained for pricing many derivative securities. But we can use numerical methods to solve it.The linear form of backward stochastic differential equations (BSDEs) was first introduced in [Bismut(1978)] in 1978. Later, [Pardoux & Peng(1990)] studied the existence and uniqueness of a kind of nonlinear backward stochastic differential equations under Lipschitz condition. In [Duffie & Epstein(1992b)], they introduced a special case of backward stochastic differential equations independently during their study in stochastic differential utility. They found it could be used to describe the consumable preference under uncertain economic environment (i.e. econometrics foundation- utility function theory). Subsequently, El Karoui and Quenez found that the theoretical price of many important derivative securities (e.g. futures and options) could be solved by backward stochastic differential equations, especially Black-Scholes formula, which is a special linear form of BSDEs. See [El Karoui & Quenez(1995)],[El Karoui, Peng & Quenez(1997)], [Duffie & Lions(1992)], [Duffie, Geoffard, & Skiadas(1994)].In [Peng(1991)], Peng obtained a probabilistic interpretation for system of second order quasilinear parabolic partial differential equation, i.e. nonlinear Feynman-Kac formula, so that it connected the solution of BSDE associated with a kind of SDE with PDE. Therefore we can use the ready-made methods of PDE to solve BSDE, and inversely, some PDE problems can be solved by BSDEs' stochastic algorithm. See[Ma, Protter & Yong(1994)], [Ma & Yong(1995)], [Duffie, Ma, & Yong (1995)], [Ma, Protter, San Martin & Torres(2002)] etc. In [Hu & Peng(1995)], Peng also obtained the existence and uniqueness of a kind of completely coupling SDE and BSDE.The European option price can be obtained by solving an initial and border value problem of a highly dimensional parabolic partial differential equation(s) with mixed derivatives. Usually we use numerical methods, such as difference method, finite element method, limited volume method etc. Different from European option, American option can be executed at any time during the period of validity. It can be changed into a linear complementary problem (LCP) depending on time, [Oxsterlee(2003)].There are many numerical methods to solve LCP, such as [Clarke & Parrott(1996)], [Clarke & Parrott(1999)], which gave a method of PFAS for an approximately linear complementary problem. The advantage is its iterative times are independent with the grid number, and get an effective numerical solution, but it is hard to perform it. PFAS was used and improved in [Oxsterlee(2003)] and the vibration of numerical solutions was avoided effectively. In [Zvan, Forsyth & Vetzal(1998)], they used penalization method to solve the American option which can be executed in advance, i.e. a penalty term was introduced in the previous partial differential inequality so that it became into a partial differential equation. This method avoids the vibration of numerical solutions, but the convergence property decreases with the optimization of discretizing. In [Ikonen & Toivanen(2005)] [Ikonen & Toivanen(2004)], they used another method based on operator splitting to solve the American option problem with constraint. They discreted space operators into several simple ones, and separated every time layer into the same amount with operators. In [Ikonen & Toivanen(2005)], they used this method to solve the pricing of American option under stochastic fluctuating ratio, i.e. they decomposed the difference operators and the constraint of executing in advance into a series of one dimensional LCPs and ap- plied Brennan&Schwartz format [Brennan & Schwartz(1977)] to approximate LCPs. The advantage of this method is to transform a complicated highly dimensional problem into several simple one dimensional ones so that decreasing much complex calculating.At the beginning of 90' s, with diversification of market requirements, it is hard to satisfy the special demand of clients only using the standard options (e.g. European option, American option). So some options with more dealing manners and dealing price appear, which is called exotic options [Hull(2000)].Asian option is a kind of exotic option. The income function in the due day depends on the average of some form at some period at least in the valid period of target assets. The arithmetic or geometric average of target assets in the anticipated period is usually used as it' s average price. At present, Asian option is a financial tool widely used in the OTC (over-the counter) market, but even in the assumption that the target follows geometric Brownian motion, only pricing of geometric average Asian option can be expressed explicitly. However, the target of most Asian option trading in the OTC market is arithmetic average. Pricing of this kind of Asian option, we often use numerical method such as binomial method[Klassen(2001)], characteristic difference method [Jiang & Dai(2002)]or approaching with standard geometric-average Asian options[Seydel(2004)]. This paper is organized as follows:In Chapter 1, a brief history review of financial derivatives is given. Since BSDEs play a important role in mathematical finance, some pricing model are given in BSDEs.In Chapter 2, The correspondence between BSDE and a kind of quasi-linear 2-order parabolic PDE , i.e. the "nonlinear Feynman-Kac formula" is introduced . this formula is given in [Peng(1991)], thus the BSDE problem and PDE problem can be transformed into each other.In Chapter 3, Some pricing method for financial derivatives are given, such as binomial method, Monte Carlo method and some PDE method.In Chapter 4 of my doctoral thesis, I studied finite volume numerical simulation method of pricing for American option. For American option under stochastic volatility, a new kind of 9-point finite volume scheme is proposed, in which using a new technique for the 2-order hybrid cross derivatives, and upwind method for the convection item [Liang & Zhao(1997)]where operators A_x, A_y A_xy and A_yx are consisted of coefficiencies of difference equation from four different direction (the x-direction, the xy-direction, the yx-direction and the y-direction) .whereThe correspondingθ-scheme:Meanwhile, the operator splitting scheme can be proposed for this 9-point scheme [Wang & Zhao(2003)].where k = 0,…,l-1,This operator splitting scheme splits according to x, y, xy, and yx four directions, thus the problem turns into four 1-dim problem in different directions.We have maximum principle and error estimate for the scheme proposed:Let Theorem 0.2.1 (Maximum Principle) Let V_ij^k be a net function on G_h satisfies following inequality:where the A_i,j in L_h satisfies assumptions(4.3.1), Then V_ij^k will not achieve its maximum at the inner points, unless V_ij^k is constant.Theorem 0.2.2 (Stability) Let L_hV_ij^k = 0, thenTheorem 0.2.3 (Error Estimate) Under the condition of theorem 0.2.1, the discrete scheme(0.2.5)has the following error estimate:where E^k is a vector with components of e_ij^k.To show the validity of this scheme, some numerical examples are given, and the result tallies with the PSOR method [Ikonen & Toivanen(2005)]. More general multi-factor American option pricing problem will be considered later.In Chapter 5 of my doctoral thesis, I studied an alternating-direction implicit upwind finite volume method for pricing Asian options. [Seydel(2004)]:Proper boundary conditions are given through equation (0.2.8) , letThen the initial and boundary condition for pricing Asian option(0.2.8) is: For convection-dominated problems, using upwind method to avoid non-physical shock, a new kind of alternating-direction implicit finite volume method according to the Asian option can be proposed:the boundary condition for V_ij^n-1/2 and the initial and boundary condition for V_ijⁿ in (0.2.11) are:Maximum principle for this scheme is theoretically proved and the error estimates also derived:Theorem 0.2.4 (Maximum Principle) Let V_ijⁿ satisfies the following condition:Let V_ijⁿ is not constant on 6, then the positive maximum of V_ijⁿ can only be achieved onde.Theorem 0.2.5 (Stability) For f=0 in pricing model, we have the following stability estimate:where L^∞(Ω) is standard Banach space, Theorem 0.2.6 (Error Estimate) Let , Under the condition (5.3.3) and the assumptions of theorem (0.2.4), we have error estimate:It should be pointed out that this alternating-direction implicit finite volume method is also valid for high-dimensional problems.In Chapter 6, a kind of upwind control volume method are proposed, which is designated for the high dimensional problems arises in financial markets. Maximum principle and error estimates also derived:Based on the analysis in section 6.4, we propose three scheme for problem(6.2.1): the Central Control Volume Scheme(6.5.8), the Upwind Control Volume Scheme I (6.5.9) and the Upwind Control Volume Scheme II (6.5.10) . It should be noted that the traditional Central Control Volume Scheme(6.5.8) and Upwind Control Volume Scheme I (6.5.9) don't satisfy maximum principle, but for the Upwind Control Volume Scheme II we proposed, the answer is yes.Scheme(6.5.10)can be expressed as:By analogy to the forenamed procedure, we can get the following maximum principle, stability analysis and the error estimate.Theorem 0.2.7 (Maximum Principle) LetV_ij^kbe a net function on G_h, which satisfies inequality:Then V_ij^k will not achieve its maximum at the inner points, unless V_ij^k is constant. Theorem 0.2.8 (Stability)Theorem 0.2.9 (Error Estimate) Under the condition of theorem 0.2.1, the discrete scheme (6.5.8)has the following error estimate:where E^k is a vector with components of e_ij^k, C is a constant, doesn't depend on the solution u,V, partition h adnâ–³t.Remark 0.2.1 Splitting scheme for (6.5.10) can also be proposed, together with the maximum principle and error estimates, while the details and the numerical simulation are under verifying.