Pricing American bond options using a cubic spline collocation method

Abdelmajid El hajaji,Abdelhafid Serghini,El bekkey Mermri,Khalid Hilal

doi:10.5269/bspm.v32i2.21354

Abstract

In this paper, American options on a discount bond are priced under the Cox-Ingrosll-Ross (CIR) model. The linear complementarity problem of the option value is solved numerically by a penalty method. The problem is transformed into a nonlinear partial differential equation (PDE) by adding a power penalty term. The solution of the penalized problem converges to the one of the original problem. To numerically solve this nonlinear PDE, we use the horizontal method of lines to discretize the temporal variable and the spatial variable by means of trapezoidal method and a cubic spline collocation method, respectively. We show that this full discretization scheme is second order convergent, and hence the convergence of the numerical solution to the viscosity solution of the continuous problem is guaranteed. Numerical results are presented and compared with other collocation methods given in the literature.

Highlights

We adopt the CIR model developed by Cox, Ingersoll and Ross in 1985 [8], is one of the most widely used term structure model and has several favorable features
2θ σ2, θ = κθ, μ = κ + ζ, We concentrate on the numerical solution of the CIR model which can be formulated as parabolic partial differential complementarity (PDC) problem with suitable boundary and terminal condition
In this paper we develop a novel numerical method for solving a PDC problem by using the cubic spline collocation method and the generalized Newton method

Summary

Penalty problem

Let u(r, t) be the value of an American put option on a zero-coupon bond with striking price K, where the holder can receive a given payoff Λ(r, t) at the expiry date T. F is a nonlinear continuous function on uλ and satisfies the following Lipschitz condition:. The following theorem proves the order of convergence of the solution um λ to uλ(r, t). For an analytic function F we may bound the term O((∆t)3) by c(∆t) for some c > 0, and this upper bound is valid uniformly throughout [0, T ] It follows from the Lipschitz condition of Lemma 2.3 and the triangle inequality that. When m = 0 we need to prove that e0 ≤ 0 and that e0 = 0 This is certainly true, since at t0 = 0 the numerical solution matches the initial condition and the error is zero. It is easy to see that Jλ is a nonlinear continuous function on uλ and satisfies the following Lipschitz condition:. In the sequel of this paper, we will focus on the solution of problem (3.6)

Spatial discretization and cubic spline collocation method

Convergence of the method

Numerical examples

Conclusion