Isogeometric Analysis for the Pricing of Financial Derivatives with Nonlinear Models: Convertible Bonds and Options

TL;DR

The paper demonstrates Isogeometric Analysis (IGA) with weighted cubic NURBS as an efficient and accurate solver for nonlinear option-pricing PDEs, specifically the Leland European call with transaction costs and the AFV convertible-bond model. By transforming both problems to fixed domains and reformulating inequality constraints as penalty terms, the authors obtain a unified variational framework solved with a theta-time scheme and Newton iterations. Numerical results show that weighted, nonuniform knots achieve solutions matching FDM/FEM accuracy with substantially fewer basis functions, and Greeks such as Delta, Gamma, and Theta can be computed directly from differentiable NURBS bases with reduced oscillations. The approach offers significant computational savings and stability, particularly for problems involving nonlinearity, path-dependency, and early exercise, with clear practical implications for pricing and hedging complex financial derivatives.

Abstract

Computational efficiency is essential for enhancing the accuracy and practicality of pricing complex financial derivatives. In this paper, we discuss Isogeometric Analysis (IGA) for valuing financial derivatives, modeled by two nonlinear Black-Scholes PDEs: the Leland model for European call with transaction costs and the AFV model for convertible bonds with default options. We compare the solutions of IGA with finite difference methods (FDM) and finite element methods (FEM). In particular, very accurate solutions can be numerically calculated on far less mesh (knots) than FDM or FEM, by using non-uniform knots and weighted cubic NURBS, which in turn reduces the computational time significantly.

Figures (17)

• Figure 1: Left figure: Cubic B-spline with uniform knot vector as $\Xi = \{0,0,0,0,0.2,0.4,0.6,0.8,1,1,1,1\}$; Right figure: Cubic B-spline with non-uniform knot vector $\Xi = \{0,0,0,0,0.2,0.4,0.6,0.6,0.6,0.8,1,1,1,1\}$.
• Figure 2: Left figure: Cubic NURBS with uniform knot vector and unequal weight vector as $\Xi = \{0,0,0,0,0.2,0.4,0.6,0.8,1,1,1,1\}$, $\omega = \{1,1,1,4,3,5,1,1\}$; Right figure: Cubic NURBS with non-uniform knot vector and unequal weight vector as $\Xi = \{0,0,0,0,0.2,0.4,0.6,0.6,0.6,0.8,1,1,1,1\}$, $\omega = \{1,1,1,1,4,3, 5,1,1,1\}$.
• Figure 3: Solution of the European call option at $t = 0$, computed using cubic NURBS-IGA with constant weights (thus, cubic B-Splines) with $r = 0.05$, $\sigma = 0.2$, $\hat{K} = \$100$. Left figure: Uniform knots with $nE = 2^8$, $n_{\tau} = 6\times 10^4$; Right figure: Non-uniform knots with $nE = 2^8$, $n_{\tau} = 6\times 10^4$.
• Figure 4: Solution of the European call option at $t = 0$, with $r = 0.1$, $\sigma = 0.2$, and $\hat{K} = \$100$, computed using nonuniform, fitted NURBS with $nE = 2^5$ and $n_\tau = 6 \times 10^4$ (left figure). The weights are given in the right figure.
• Figure 5: Solution of the Leland model calculated using unweighted cubic NURBS IGA on uniform knots, with $r = 0.1$, $\sigma = 0.2$, $\hat{K} = \$100$, and $Le \approx 0.8$. Left: $\Delta\tau/\Delta x = 0.02$, $\Delta\tau/\Delta x^2 = 0.8$; Right: $\Delta\tau/\Delta x = 0.05$, $\Delta\tau/\Delta x^2 = 0.1$.