1 Explicit Euler Scheme (or Euler Forward Scheme )

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1 Numerical methods for PDE in Finance - M2MO - Paris Diderot American options January 2018 Files: We look for a numerical approximation of the american put option v = v(t, s), t (0, T ) and s Ω := (S min, S max ), solution of the following Partial Differential Equation: with min( t v + Av, v ϕ) = 0, (t, s) (0, T ) Ω, (1a) v(t, S min ) = v l (t), t (0, T ), (1b) v(t, S max ) = v r (t) 0, t (0, T ), (1c) v(0, s) = ϕ(s), s Ω (1d) Av := σ2 2 s2 s,s v rs s v + rv, and σ, r, K are strictly positive constants. A logical choice for v l (t) is v l (t) : ϕ(s min ) = K S min. For the numerical applications we will chose the following financial parameters: K = 100, S max = 200, T = 1, σ = 0.2, r = 0.1, and S min = 0. We will consider two differents payoff functions ϕ 1 ou ϕ 2 for the tests: a classical payoff function (payoff=1): ϕ 1 (x) := (K x) +, with v l (t) := K S min, and a barrier payoff function (payoff=2): { K for K ϕ 2 (x) := x K otherwise }, with v l (t) = 0. 1 Explicit Euler Scheme (or Euler Forward Scheme ) Notations. We adopt the usual notations : mesh s j = S min + jh, j = 1,..., I, h = (S max S min )/(I + 1) (so that s 0 = S min and s I+1 = S max ), and t n = n,

2 0 n N, = T/N. We then look for Uj n, an approximation of v(t n, s j ). We choose to work with the unknown vector of R I : U n := U n 1. U n I Let us consider the explicit Euler Forward scheme (EE) with centered approximation: min ( U n+1 j Uj n U n 0 = v l (t n ), U n I+1 = v r(t n ),. + σ2 U 2 s2 j 1 n + UPj n Uj+1 n j ) h 2 U n+1 j ϕ(s j ) rs j U n j+1 U n j 1 2h + ru n j, = 0, 1 j I, for n = 0,..., N 1. The scheme is initialized with Uj 0 = ϕ(s j ). We denote by A the discretization matrix associated to the operator A, of size I, and q(t) R I, such that (AP + q(t)) j := + σ2 P j 1 + 2P j P j+1 P j+1 P j 1 2 s2 j rs h 2 j + rp j, 1 j I. 2h = (α j β j )P j 1 + (2α j + r)p j (α j + β j )P j+1, 1 j I. with α j = σ2 2 s 2 j h 2 and β j = rs j. We recall that A is the tridiagonal matrix 2h tridiag ( (α j β j ), 2α j + r, (α j + β j )), and q(t) := ( α 1 + β 1 )v l (t) 0. 0 ( α I β I )v r (t). This matrix A and vector q(t) are the same as the one used for European options. Let also g be the vector of R I with components g j := ϕ(s j ). We finaly obtain the following equivalent form of the scheme (EE) in R I : min( U n+1 U n U 0 = g. + AU n + q(t n ), U n+1 g) = 0, n = 0,..., N 1, (2)

3 (where the min must be understood component wise). We finaly check also that the main iteration can also be written or, in vector form, U n+1 i = max(u n i (AU n + q(t n )) i, g i ), 1 i I, U n+1 = max(u n (AU n + q(t n )), g). Download the program tp2.m (or tp2.sci): a matlab file (resp. scilab file) to complete. This is a similar program than the one for European options. For simplification, we will work with full matrices (TYPE=FULL). Modify the variable SCHEME to the value EE-AMER (this will correspond to the iterative scheme for the american option), and program the corresponding Euler Forward scheme case EE-AMER in the main loop. Check that the program does give a stable solution with the parameters I=20 and N=20. Check that there is an unstable behavior with other parameters (such as I=50 and N=20). 2 A first implicit scheme: the splitting scheme For stability reasons, we now turn on the implicit schemes. We propose a first implicit simplified scheme. 1 Although it might be less precise than exactly solving the implicit scheme (see next section), it is much simplier to program. The scheme is as follows: (i) compute U n+1,(1) s.t. U n+1,(1) U n + AU n+1,(1) + q(t n+1 ) = 0, (3) (ii) compute U n+1 s.t. U n+1 = max(u n+1,(1), g). (4) Program this method (SCHEME= EI-AMER-SPLITTING ). What is the advantage of this method with respect to Newton s method? (see for instance the scheme (5)) Propose a variant of Crank-Nicolson type (θ = 1 scheme). However we notice that 2 this Crank-Nicolson type method is not truly second order in time (as well as any method proposed here), because of the nonlinearity of the PDE. 2 3 Implicit Euler Scheme For stability reasons, we now turn on the time-implicit Euler Scheme for the american option, which naturally takes the following form: min( U n+1 U n + AU n+1 + q(t n+1 ), U n+1 g) = 0, n = 0,..., N 1, (5) U 0 = g. 1 For this method, we refer to Barles, Daher and Romano (1994). 2 A true second order method is proposed in Osterlee (2003), Bokanowski and Debrabant (2016).

4 (U n is known and we look for a solution U n+1 ). Let us define B := I d + A, and b := U n q(t n+1 ). For each n, one must solve a solution x R I of the following non-linear system min(bx b, x g) = 0, in R I. (6) Then, we will take U n+1 = x as the solution of the scheme (5). We now propose to test several algorithms for solving (6). 3.1 PSOR Algorithm (PSOR = Projected Successive Over Relaxation ). This is an iterative method based on the decomposition B = L + U where L is the lower triangular part of B and U is the strict upper triangular part. 3 Check that the solution x of min(lx b, x g) = 0 can be solved explicitly. Check that the solution x = x k+1 of min(lx (b Ux k ), x g) = 0 can be programmed using the following pseudo-algorithm: x <- x^k for i=1.. n x(i)=( b(i)- sum_{j<>i} (A(i,j) x(j)) ) / A(i,i) x(i)=max(x(i),g(i)) end Complete the iterative method in the file PSOR.m. (a solution is in PSORsol.m.) Branch on this method in the code : SCHEMA= EI-AMER-PSOR. Observe that the method slows down for larger I values (for instance, test with σ = 0.3, N = 10, I + 1 = 100, and observe a more important number of PSOR iterations at each time iteration). Observe that the method can be accelerated by using a relaxation method based on the following decomposition (instead of B = L + U): B = L w + U w where L w = ( 1 w 1)D + L, D = diag(a), U w = B L w. The parameter w can be tested with values in (1, 2) - for instance w = For a given starting vector x 0 R I, we define x k+1 as the solution of the system min(lx k+1 (b Ux k ), x k+1 g) = 0. Assuming that L i,i := B i,i > 0, the system can be solved explicitly, using that L is also lower triangular. By a fixed point argument the method can be shown to be convergent as soon as B is a strictly diagonal dominant matrix with B i,i > 0 i.

5 3.2 Brennan and Schwartz algorithm - or projected UL method There exists a direct method for solving min(bx b, x g) = 0, when the solution 4 x has a particular shape. The idea is to write a decomposition of the form B = UL (L: lower triangular matrix, and U: upper triangular matrix, with U ii = 1, i), and to use the equivalence, in some cases : min(ulx b, x g) = 0 min(lx U 1 b, x g) = 0. (7) Then, the right-hand-side of (7) has a simple explicit solution given by (i) solve c = U 1 b: upwind algorithm. (ii) solve min(lx c, x g) = 0: downwind algorithm. Set SCHEME== EI-AMER-UL in the main working file. Program the B = UL decomposition of a tridiagonal matrix B. In Matlab, one can fill the file uldecomp.m (to be completed). (In Scilab, one can work in the file uldecomp.sci.) Solution file : uldecomp_sol.m. First check that the decomposition B = UL is working on the specific matrix B := I d + A, in the case I = 10. To do so, one can introduce in the main loop a test that is only performed at iteration n=0, as follows: case SCHEME== EI-AMER-UL if n==0 // Here decompose B=UL and test the decomposition B=... [U,L]=uldecomp(B); // Here test that the norm of B-UL is zero : show norm of B-U*L end... //- here american option scheme Program the projected downwind algorithm (complete the function descente_p), in order to find the solution x of min(lx b, x g) = 0. Program the scheme by using the upwind algorithm (which is given) and the projected downwind algorithm. Test the method with N = 20, I + 1 = 50 (and 4 In the case of the american put with one asset, and for a finite element approach, see the reference of Jaillet, Lamberton and Lapeyere (1990). The algorithm has initially been introduced by Brennan and Schwartz.

6 with the classical payoff function). Check that we do solve correctly the equation min(bx b, x g) = 0 at each time iteration. To this end one can print the norm min(bx b, x g) after each new computation of the vector U in the main loop (example with Matlab): Pold=P; P=... % scheme definition err=norm(min(b*u-uold,u-payoff(s))); fprintf( Check: min(b x- b, x-g)= %f\n, err); Next, run the program again with the particular payoff ϕ 2 instead of ϕ 1. Check that in that case min(bx b, x g) 0 (as soon as n = 0). 3.3 Semi-smooth Newton s method The following proposed method will work whatever the form of the data and payoff functions. 5 We now want to apply a Newton type algorithm for solving F (x) = 0 with F (x) := min(bx b, x g). We consider the following algorithm: iterate over k 0 (for a given x 0 starting point of R I, to be choosen) x k+1 = x k F (x k ) 1 F (x k ), until F (x k ) = 0 (or, that x k+1 = x k ). We will take the following definition for F (x k ) (rwo by row derivative) { F (x k Bi,j if (Bx ) i,j := k b) i (x k g) i, otherwise. δ i,j (Note the specific choice F (x k ) i,j = B i,j when (Bx k b) i (x k g) i. The other choice F (x k ) i,j = δ i,j when (Bx k b) i (x k g) i works also but is less efficient : more iterations might be needed.) Fix the parameter SCHEME= EI-AMER-NEWTON. Program the algorithm by completing the function, in Matlab: newton.m (solution file : newton_sol.m). Test the method with N = 20, I = 50 and the classical payoff function ϕ 1. Test with the particular payoff function ϕ 2, and check that the method works now well, in the sense that there is no error when solving (6). Remark: there exist other schemes which are more or less equivalent. For instance, the Primal-Dual method, the policy iteration algorithm, or Howard s algorithm. 5 An analysis of the scheme can been found for instance in the reference Bokanowski, Maroso, Zidani (2009), although this type of algorithms goes back to Howard s algorithm, 1957.

7 Figure 1: American put option. Evaluated at time t = 0 for terminal time T = 1 (σ = 0.3, r = 0.1; N = 20, I = 20).