The Convergence Of Penalty Method For American Option Pricing

This paper is a comprehensive survey of the recent results obtained from the study of the convergence of penalty method for American option pricing. Monotonic penalty, power penalty and C_k,m penalty methods are proposed to solve the linear complementarity problem arising from the valuation of American option,which produces a nonlinear degenerated parabolic PDE with Black-Scholes operator. Based on the variational theroy, the solvability and convergence properties of these penalty menthods are established in weighted Sobolev space.a power penalty method is developed for VI model of the two-asset American option pricing .This paper consists of five parts.In the second part, we mainly introduce the free-boundary problem and the linear complementarity problem for American option pricing. An equivalent standard form satisfying homogeneous Dirichlet boundary conditions is established as follows is self-adjoint form withpayoff function becomesFree boundary problem is usually very difficult to solve, because we need to determine the unknown boundary during the solving process. An alternative method is to transform it into a linear complementarity problem involving the Black-Scholes differential operator and a constraint on the value of the option to redo the problem with variational Inequality. Pricing function can be obtained directly from the description, and the boundary can be identified from the answer.Variational Inequality will become very promising since it is not obviously dependent on the boundary.In the third part, we maily introduce a monotonic penalty method based on the variational theroy , the solvability and convergence properties of this penalty menthod is established in a weighted Sobolev space.In order to handle the degeneracy of the Black-Scholes operator, the coerciveness and continuity of the bilinear operator Z(u,v) is given on H_0,w¹(I),which is a weighted Sobolev space.where C and M are positive constants. Theorem 1:Variational inequality as follows has a unique solutionwhere K = {v∈H_0,w¹ (I):v≤u^*} C is a convex and closed subset of H_0,w¹(I).According to Theorem 1, the penalized problem for the monotonic penalty method can be given based on the variational theroy .Consider the following PDEwhereλ> 0 is penalty parameter and p(·) is a continuous, monotonic penalty function subject toBased on the fundamental lemma of variation, the variational form corresponding to (4) isLemma 1:Let u_λis the solution to (5),a priori estimate for {u_λ} as followswhere C is a positive constant independent of u_λ.This implies that {u_λ} is uniformly bounded in the sapce L²(0,T;H_0,w¹(I))∩L^∞(0,T;L²(I)).Therefore, there exists a subsequence of {u_λ},still denote it by {u_λ}, such thatWe can deduce that (?) = u and the strong convergence of {u_λ}.On this basis, we have the following convergence result Theorem 2: let u and u_λbe the solution to (4) and (5), ThenIn the forth part, we mainly introduce a nonlinear penalty method-powerpenalty method, and the convergence of this method is obtained. The form of power penalty term as followsConsider the nonlinear PDE in power penalty method as followswhereλ> 1 is the penalty parameter.The nonlinear termλ[u_λ(t)-u^*(t)]₊^1/k is used to penalize the positive part of u_λ(t)-u^*(t).This can be loosely explained as follows: when the constraint (2) is satisfied, [u_λ(t)-u^*(t)]₊=0,thus,(7) reduces to the Black-Scholes equation. When u_λ(t)-u^*(t)>0,(7) gives [u_λ(t)-u^*(t)]₊=λ^-k(-Lu_λ)^k.Therefore, ifλis sufficiently large and Lu_λis bounded,[u_λ(t)-u^*(t)]₊≈0 so that (2) is satisfied within a tolerance depending onλ.The convergence analysis of power penalty menthod consists of two parts. Error bounds for [u_λ-u^*]₊ is given firstly.Lemma 2:If u_λ∈L^p(Ω),then there exists a positive constant C, independent of u_λandλ,such thatWhere k > 0,p=1 +1/k.Let q=1+ k,1/p+1/q = 1 is obtained. Before giving the main convergence theorem of power penalty method, we make the following assumption for the solution u(t) to (4) as followsTheorem 3: Suppose the assumption above, there exists a positive constant C, such thatWhen k=1,it reduces to the linear penalty menthod, (7) is of orderΟ(λ^-1/2).k > 1,it is lower order penalty method, when k is large, we need only a very smallλto achieve a given accuray. This improves significiently the existing theoretical result of the square root rate of convergence mentioned above.C_k,m penalty method is given to make full use of the advantages of the higher order and lower order penalty methods. The penalty term as followsThe combination of these two power penalty methods possesses a good convergence rate with a desirable accurate.Theorem 4:u_λis the solution to the original problem and u is the solution to the penalized problem, there exists a positive constant C,such thatIf p=1 + (?),Theorem 5 is equal to The mechanism of this combined penalty method is that if u is not a 'good' initial guess, then a higher order penalty term(m > 1) palys a dominate role in the behavior of u, controlling it to converge to a near zero as quickly as possible. Then, the problem will behave as a well-defined initial valua problem, hence asymptotically the lower order penalty term(0 < k < 1) will play the dominant role.In the fifth section, we introduce a power penalty method for two-asset American option valuation,the unique solution and convergence properities of a two-dimensional nonlinear parabolic PDE contraining a power penalty term is given based on the variational theory.The convergence results as followsTheorem 5:u_λis the solution to the original problem and u is the solution to the penalized problem, If u_λ∈L^P(θ) and (?)∈L^k+1(θ),k> 0,then there exists a positive C, such thatWhenλâ†'∞,we show that the solution to penalized problem converges to that of original problem at the rateΟ(λ^-k/2).