Variational Model And Numerical Methods For American Put Option Pricing Problem

Option is a kind of financial derivatives which gives holders the right tobuy or sell certain underlying assets by certain price. American put option letholders sell certain underlying assets by the contracted price E at any timeduring the valid period T of the contract. In this paper, we choose a stockwithout dividend as underlying asset and denote the stock price as S t at timet .The value of the American put option on the above stock is represented byV (t , S ) which satisfies the following partial differential equations:where r is the riskless rate of return available in the market and σ is thevolatility of the stock price S t . In fact, above differential model is a parabolic type differential inequalityproblem with moving boundary. The difficulty of solving such differentialinequality problem lies in the moving boundary is unknown and to bedetermined. So far, all the studies in literatures (see [1],[3],[4]) are firstlytransforming the Black-Scholes Model to a standard heat diffusion typeinequality by using a variable transformation x = logS ,hereafter theirtheoretical and numerical investigations are based on the correspondingvariational inequality model. We see that under variable transformationx = logS, the original bounded domain (0, R )(about space variable S ) istransformed to an unbounded domain ( ?∞ ,log R).From the viewpoint ofnumerical computation, such way is not the best, Since to set up a numericalmethod one have to make a truncation and use a approximate boundarycondition on the left side of ( ?∞ ,log R).In this paper, we find a new waywhich could avoid truncation and which makes the theoretical proof mucheasy for the stability and convergence analysis of the numerical solutions.In the first chapter of this paper, we discuss the mathematical model ofAmerican put option pricing problem. We firstly introduce a Sobolev SpaceH S1 ( I ) with weight which consists of all the functions on I = (0, R) boundedin norm2 1 212 2 22V S = { S ??V S + V } = {∫ 0 R( S ??VS+V )d S} .Based on above weighted Sobolev Space H S1 ( I ), We reformulated theproblem as an equal variational inequality problem which is : findV (? , t ) :[0, T ]→ K satisfying( V , W V ) a (V , W V) 0t?? ? + ? ≥,? W ∈ K (1.7)where K is a closed convex set of H S1 ( I ).In the second chapter of this paper, we use finite element method toconstruct a semi-discrete approximation of variational problem (1.7), then westudied the stability and convergence of the semi-discrete approximatesolution under the weighted Sobolev norm. We proved the following lemmaand theorem:Lemma 2.1 The solutions of semi-discrete problem (2.1)Vh (t ) :[0, T ]→ Kh satisfy the following stability estimate2 2 2220 0( ) ( ) 423t th h S I SIV t + γ ∫ V τ dτ ≤ Mγ∫ g dτ +g, ?t ∈ [0, T].Theorem 2.1 Assume V (t ) is the solution of problem (1.7) andVh (t ) is the solution of (2.1) and Vh (0)= gI, then the followingerror estimate holds2 212V (t ) ? Vh (t ) ≤ Ch{ g 2 + ∫0 t Vt (τ ) 2dτ + ∫0 tVt (τ ) + LV (τ ) ( g 2 +V (τ ) 2) dτ},?t ,0≤ t ≤ T,where ? 2 denotes the norm in Sobolev Space H 2( I ).In the third chapter, We set up a fully discrete approximation ofvariational inequality problem (1.7) by using backward Euler-finite elementmethods and discuss the convergence and error estimate of the approximatesolution under the weighted Sobolev norm. We got the following result:Theorem 3.1 Assume V (t ) is the solution of problem (1.7) andkVh is the solution of (3.2),in addition assume V ∈ L∞ (0, T;H 2( I)),Vt , Vt t∈ L2 (0, T;L2 ( I)), h ?t ≤ C0,then we have212 21212(0, ) 12 0 0( ) ( ( ( ) ) )( ){ max ( ) ( ) ( ( ) ) }.k kkk lk h ll h St tH E j kj t ttV t V t V t VC h t g V t V τ dτ V τ dτ=≤ ≤? + ? ?≤ +? + + +∑∫ ∫...