Numerical methods for American options

Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1

American options The holder of an American option has the right to exercise it at any moment up to maturity. If exercised at t an American call option has the payoff (S t K) +, an American put option has the payoff (K S t ) +. Theorem. If the underlying does not pay dividends, the price of an American call option with maturity T and exercise price K is equal to the price of a European call option with exercise price K expiring at T. Important! It is only true if there are no dividends. Above theorem does not apply for options on the foreign exchange market and on stock indices. Computational Finance p. 2

American instruments Definition. American type instrument with maturity T and payoff function f is a contingent claim that can be exercised at any moment up to T. Its payoff at t equals f(s t, t). Theorem. The price of an American claim at time t can be written as V (S t, t) for some function V : (0, ) [0, T] R. Computational Finance p. 3

Properties V (S t, t) is the price of the instrument at t, so it values future payoffs from the instrument. f(s t, t) is the payoff of the instrument if it is exercised at t. At the terminal time T V (s, T) = f(s, T), s > 0. To preclude arbitrage V (s, t) f(s, t), (s, t) (0, ) [0, T]. Computational Finance p. 4

If the inequality is strict, i.e. V (s, t) > f(s, t), the holder should not exercise the option. Because it is more profitable to sell the option for V (s, t) than to exercise it for f(s, t). If V (s, t) = f(s, t), it is optimal to exercise the option immediately. By holding it, the holder risks loosing money. General rule. The holder should exercise the option as soon as V (S t, t) = f(s t, t). Computational Finance p. 5

American option PDE Up to the exercise of the option, its price forms a value process of the replicating strategy. It is therefore a martingale with respect to the risk-neutral measure. Applying Itô s formula gives dv t = d ( V (S t, t) ) = + ( rs t V (S t, t) s ( σs t V (S t, t) s + 1 2 σ2 S 2 t ) dw t + 2 V (S t, t) s 2 rv (S t, t) + V (S ) t, t) dt. t This process is a martingale if and only if the term by dt is zero. Computational Finance p. 6

Until the optimal exercise we have rs V (s, t) s + 1 2 σ2 s 2 2 V (s, t) s 2 rv (s, t) + And of course, we have V (s, t) > f(s, t). At the optimal exercise moment, we have V (s, t) = f(s, t), V (s, t) t = 0. and rs V (s, t) s + 1 2 σ2 s 2 2 V (s, t) s 2 rv (s, t) + V (s, t) t 0. The inequality in the second formula comes from the fact that V (s, t) may not represent the value process of some strategy after the optimal exercise moment. Computational Finance p. 7

Free-boundary problem We summarize our findings in the following system of equations and inequalities called the free-boundary problem V (s, t) f(s, t) 1 2 σ2 s 2 2 V (s,t) s 2 + rs V (s,t) s rv (s, t) + V (s,t) t 0 1 V (s, t) = f(s, t) or 2 σ2 s 2 2 V (s,t) s 2 + rs V (s,t) s rv (s, t) + V (s,t) t = 0 boundary conditions... The free-boundary is the set of points where V (s, t) = f(s, t). In these points the system is not governed by the equation with partial derivatives. Computational Finance p. 8

Boundary conditions Boundary conditions for the American put option: Terminal condition Left-boundary condition Right-boundary condition V (s, T) = (K s) +. lim V (s, t) = K. s 0 lim V (s, t) = 0. s Computational Finance p. 9

Complete free-boundary problem For s > 0, t [0, T]: V (s, t) (K s) + 1 2 σ2 s 2 2 V (s,t) s 2 + rs V (s,t) s rv (s, t) + V (s,t) t 0 V (s, t) = (K s) + 1 or 2 σ2 s 2 2 V (s,t) s 2 + rs V (s,t) s rv (s, t) + V (s,t) t = 0 V (s, T) = (K s) + lim s 0 V (s, t) = K lim s V (s, t) = 0 Computational Finance p. 10

Simple example We compute the price of an American put option by a simple numerical method. First we make the following change of variables (the same as for the Black-Scholes PDE) x := log s + (r 1 2 σ2 )(T t), τ := σ2 2 (T t), 2τ r(t y(x, τ) := e σ 2 ) V ( 2r x ( e σ 2 1)τ, T 2τ ), σ 2 where x R and τ [ 0, σ2 2 T ]. Computational Finance p. 11

Transformation of the problem y(x, τ) e ( 2τ r(t σ 2 ) K e x ( 2r σ 2 1)τ ) +, y τ (x, τ) 2 y x 2 (x, τ) 0, y(x, τ) = e ( 2τ r(t σ 2 ) K e ) + 2r x ( σ 2 1)τ or y t (x, τ) 2 y x 2 (x, τ) = 0, y(x, 0) = e rt (K e x ) +, 2τ r(t lim x y(x, τ) = Ke σ 2 ), lim x y(x, τ) = 0. Computational Finance p. 12

Numerical method Grid x (, ), τ [0, 1 2 σ2 T] discretization of time τ: δτ = 1 2 σ2 T N discretization of space x τ ν := ν δτ for ν = 0, 1,...,N x min, x max δx = x max x min M x i := x min + i δx for i = 0, 1,...,M w i,ν denotes the approximation of y(x i, τ ν ) Computational Finance p. 13

Finite differences We use explicit method (this is important!). The consequence is a small time step for the algorithm to be stable. Computational Finance p. 14

In the node (x i, τ ν+1 ) the expression u 1 = λw i 1,ν + (1 2λ)w i,ν + λw i 1,ν, where λ = δτ (δx) 2, approximates the value of y(x i, τ ν+1 ) in the case of no exercise. If this value is smaller than the payoff from the exercise u 2 = e r(t 2τ ν+1 σ 2 ( ) + ) K e x i ( 2r σ 2 1)τ ν+1 then it is optimal to exercise immediately and w i,ν+1 = u 2. This is written concisely as ( w i,ν+1 = max λw i+1,ν + (1 2λ)w i,ν + λw i 1,ν, e r(t 2τ ν+1 σ 2 ( ) ) + ) K e x i ( 2r σ 2 1)τ ν+1. Computational Finance p. 15

Algorithm for an American put option Input: x min, x max, M, N, K, T and the parameters of the model δτ = σ2 T 2N, δx = x max x min M Calculate τ ν, ν = 0,1,..., N, and x i, i = 0, 1,..., M For i = 0, 1,..., M w i,0 = e rt (K e x i) + For ν = 0, 1,..., N 1 w 0,ν+1 = Ke r(t 2τ ν+1 σ 2 ) w M,ν+1 = 0 For i = 1,2,..., M 1 u 1 = λw i+1,ν + (1 2λ)w i,ν + λw i 1,ν u 2 = e r(t 2τ ν+1 σ 2 w i,ν+1 = max(u 1, u 2 ) ) K e x i ( 2r σ 2 1)τ ν+1 + Output: w i,ν for i = 0,1,..., M, ν = 0, 1,..., N Computational Finance p. 16

General American instrument For a general American instrument we return to original variables only making the change of time t τ = T t. A general American instrument is characterized by a pay-off function g(s, τ). The free-boundary problem for this instrument is given by rs ( V (s,τ) τ V (s,τ) s V (s, τ) g(s, τ) 0, rs V (s,τ) τ V (s,τ) s V (s, 0) = g(s, 0), lim s + V (s, τ) = lim s + g(s, τ). lim s 0 V (s, τ) = lim s 0 g(s, τ). 1 2 σ2 s 2 2 V (s,τ) s 2 + rv (s, τ)) (V (s, τ) g(s, τ) ) = 0, 1 2 σ2 s 2 2 V (s,τ) s 2 + rv (s, τ) 0, This problem is also called the linear complementarity problem for the American instrument defined by a pay-off function g(s, τ). Computational Finance p. 17

Penalty method There are many numerical methods which solve the linear complementarity problem (LCP). We present here the method called the penalty method. This method is simple and efficient (this is particularly visible for more complicated instruments like barrier options). On the other hand the method is only first order (slow convergence). The basic idea of the penalty method is simple. We replace the linear complementarity problem by the nonlinear PDE V (s, τ) τ = rs V (s, τ) s + 1 2 σ2 s 2 2 V (s, τ) s 2 rv (s, τ)+ρ max ( g(s, τ) V (s, τ) ) where, in the limit as the positive penalty parameter ρ, the solution satisfies V g 0. Computational Finance p. 18

Finite difference approximation We use the same grid as for the Black-Scholes equation with Vi n approximation to V (s i, τ n ) and gi n an approximation to g(s i, τ n ). denoting an The nonlinear PDE for the penalty method becomes in the discrete version V n+1 i V n i = (1 θ) ( τ ( + θ τ + P n+1 i j=i±1 j=i±1 (γ ij + β ij ) ( ) V n+1 j V n+1 ) i r τv n+1 i (γ ij + β ij ) ( ) Vj n Vi n ) r τv n i ( g n+1 i V n+1 ) i, where the choice of θ gives the implicit (θ = 0) and the Crank-Nicolson (θ = 1 2 ) scheme. Computational Finance p. 19

Finite difference approximation cont. Coefficients from the previous slide are as follows: P n+1 ρ, for V n+1 i < g n+1 i, i = 0, otherwise, γ ij = β ij = σ 2 s 2 i s j s i (s i+1 s i 1 ), rs i (j i) s i+1 s i 1, for σ 2 s i + r(j i) s j s i > 0, ( ) + 2rsi (j i) s i+1 s i 1, otherwise, where j = i ± 1 and ρ is a penalty factor (a large positive number). Computational Finance p. 20

Finite difference approximation cont. The numerical algorithm can be written in the concise form ( I + (1 θ) τm + P(V n+1 ) ) V n+1 = (I θ τm)v n + P(V n+1 )g n+1, where V n is a vector with entries Vi n and g n a vector with entries gi n, ( [MV n ] i = (γ ij + β ij ) ( ) Vj n Vi n ) rv n i j=i±1 and P(V n ) is a diagonal matrix with entries [P(V n ρ, for Vi n < gi n )] i,i =, 0, otherwise. Computational Finance p. 21

Finite difference approximation cont. Matrix M has the property of strict diagonal dominance. It has positive diagonal and non-positive off-diagonals with diagonal entries strictly dominating sum of absolute values of off-diagonal entries. This property of M is essential for the convergence of the method and is visible from the structure of the upper left corner of the matrix M = 0 B @ r + γ 12 + β 12 γ 12 β 12 0... γ 21 β 21 r + γ 21 + β 21 + γ 23 + β 23 γ 23 β 23............... 1 C A, Note that in the vector on the right hand side (I θ τm)v n the first and the last elements have to be modified to take into account the boundary conditions. Computational Finance p. 22

Convergence Theorem. Let us assume that γ ij + β ij 0, ( 2 θ τ j=i±1 τ s < const. τ, s 0, where s = min i (s i+1 s i ). ) (γ ij + β ij ) + r τ 0, Computational Finance p. 23

Convergence cont. Then the numerical scheme for the LCP from the previous slides solves V (s, τ) τ rs V (s, τ) s V n+1 i g n+1 i C ρ, C > 0, ( V (s, τ) V (s, τ) rs τ ( V n+1 s i g n+1 i C ρ 1 2 σ2 s 2 2 V (s, τ) s 2 + rv (s, τ) 0, 1 2 σ2 s 2 2 V (s, τ) ), where C is independent of ρ, τ and S. ) s 2 + rv (s, τ) = 0 Computational Finance p. 24

Iterative solution Since we get a nonlinear equation for V n+1 it has to be solved by iterations. We shall use here the simple iteration method. Let (V n+1 ) (k) be the k-th estimate for V n+1. For notational convenience, we write V (k) (V n+1 ) (k) and P (k) P ( (V n+1 ) (k)). If V (0) = V n, then we have the following algorithm of Penalty American Constraint Iteration. Computational Finance p. 25

Algorithm Input: V n, tolerance tol. V (0) = V n For k = 0,... until convergence solvè I + (1 θ) τm + P (k) ) V (k+1) = (I θ τm)v n + P (k) g n+1. If V (k+1) i max i V (k) i max(1, V (k+1) i or P (k+1) = P (k) quit. V n+1 = V (k+1) ) < tol Output: V n+1 Computational Finance p. 26

Convergence of iterations Theorem. Let γ ij + β ij 0, then the nonlinear iteration converges to the unique solution to the numerical algorithm of the penalized problem for any initial iterate V (0) ; the iterates converge monotonically, i.e., V (k+1) V (k) for k 1; the iteration has finite termination; i.e. for an iterate sufficiently close to the solution of the penalized problem, convergence is obtained in one step Computational Finance p. 27

Size of ρ In theory, if we are taking the limit as s, τ 0, then we should have ( ) 1 ρ = O min ( ( s) 2, ( τ) 2). This means that any error in the penalized formulation tends to zero at the same rate as the discretization error. However, in practice it seems easier to specify the value of ρ in terms of the required accuracy. Then we should take ρ 1 tol. Computational Finance p. 28

Speed up iterates convergence Although the simple iterates converge to the solution of the nonlinear problem its speed of convergence is rather slow. To make the convergence more rapid we can use the Newton iterates. This requires to write the nonlinear equation in the form F(x) = 0 and solve the iterative procedure where F (x) is the Jacobian of F. x k+1 = x k F (x k ) 1 F(x k ), In the penalty method algorithm the only nonlinear term which requires differentiation in order to obtain F is Pi n. Unfortunately, this term is discontinuous. A good convergence can be obtained when we define the derivative of the penalty term as P n+1 i (g n+1 i V n+1 i ) V n+1 i = ρ, for V n+1 i < g n+1 0, otherwise. i, Computational Finance p. 29

American call option Solving linear complementarity problem for an American call option we have to impose a boundary condition on artificial boundary S max. Let us recall that for an Europen call option the suggested boundary condition is (S max Ke r(t t)) + or for a stock paying dividend of constant rate d ( S max e d(t t) Ke r(t t)) +. Due to the early exercise possibility for an American call option the boundary condition on artificial boundary S max should be ( S max K) +. Computational Finance p. 30

American call option cont. For large S max and r > 0 we have ( ) + ( S max K < S max Ke r(t t)) +. Hence for large S, the price of an American call option is smaller than the price of an European call option, which contradicts the theorem that these prices are equal. The problem comes from the replacement of an original boundary condition for S by a condition at artificial boundary S max. To avoid the problem we should impose the following boundary condition at S max ( ) +, ( max( S max K S max e d(t t) Ke r(t t)) ) +. Computational Finance p. 31

American barrier options Contrary to European barrier options for American barrier options we do not have in-out parity. In general, the sum of prices of knock-in and knock-out options is not equal to the price of the corresponding American vanilla option. Let τ (D) X = inf{t T : S t X} τ (U) X = inf{t T : S t X} be the first time the price of the underlying asset falls below (rise above) the barrier X. Then for options with payoff g(s t, t) we have [ ] [ ] sup E e rτ 1 g(s τ1, τ 1 )1 {τ1 + sup E e rτ 2 g(s <τ ( ) τ 1 T X } τ2, τ 2 )1 {τ2 >τ ( ) 0,T τ 2 T X } 0,T [ ] sup E e rτ 3 g(s τ3, τ 3 ) τ 3 T 0,T and the equality holds only when all stopping times τ 1, τ 2 and τ 3 are equal. Computational Finance p. 32

Down-and-in option. American knock-in options The option condition is: when the asset price S t falls below X the holder of the option receives an American vanilla option, with maturity date T and strike price K. To price that option we consider only the interval [X, + ). In this interval there is no early exercise possibility and we gave an European option which fulfils the Black-Scholes equation V (S, t) t + rs with terminal condition and boundary conditions V (S, t) S + 1 2 σ2 S 2 2 V (S, t) S 2 rv (S, t) = 0. V (S, T) = 0 for S > X. V (S, t) 0 as S, V (X, t) = C(X, t), where C(S, t) is the price of the corresponding American vanila option. Computational Finance p. 33

American knock-in options cont. Up-and-in option. The option condition is: when the asset price S t rises above X the holder of the option receives an American vanilla option, with maturity date T and strike price K. We price the option in the interval [0, X]. In this interval there is no early exercise possibility and we gave an European option which fulfils the Black-Scholes equation with terminal condition and boundary conditions V (S, T) = 0 for S < X. V (S, t) 0 as S 0, V (X, t) = C(X, t), where C(S, t) is the price of the corresponding American vanila option. Computational Finance p. 34

American knock-out options These options are options with early exercise provision. Terminal conditions are as for American vanila options modified by limitations coming from boundary conditions. Boundary conditions: Down-and-out option V (S, t) = h(x, t), for S = X, V (S, t) = boundary value for vanila option, for S > X. Up-and-out option V (S, t) = boundary value for vanila option, for S < X, V (S, t) = h(x, t), for S = X. Here h(x, t) is a payoff function on barrier X (see the next slide). Computational Finance p. 35

American knock-out options cont. From both financial and mathematical point of view, payoff function h(x, t) is equal to zero. But before the stock price crosses the trigger X, the American option is active and its price must be no less than the option s payoff function g(s t, t) ((S t K) + for call option and (K S t ) + for put option). Hence, up to the barrier the price of the option must be bounded from below by the payoff function g(s t, t). Assuming h(x, t) = 0 will create a jump on the barrier since before the stock price crosses the trigger X the price V (S, t) g(s, t). If the option price is continuous up to the barrier then we shall get V (X, t) g(x, t). That suggests to take as the payoff function on the barrier h(x, t) = g(x, t). This choice does not change the resulting price V (S, t) but has obvious numerical adventage: we do not solve a problem with jumps on a boundary. As a result we obtain a numerical algorithm which is more stable (jumps on a boundary can create oscillations in numerical solutions). Computational Finance p. 36