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Register a new userPricing American options with the Runge-Kutta-Legendre finite difference scheme
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Authors
Fabien Le Floch
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This paper presents the Runge-Kutta-Legendre finite difference scheme, allowing for an additional shift in its polynomial representation. A short presentation of the stability region, comparatively to the Runge-Kutta-Chebyshev scheme follows. We then explore the problem of pricing American options with the Runge-Kutta-Legendre scheme under the one factor Black-Scholes and the two factor Heston stochastic volatility models, as well as the pricing of butterfly spread and digital options under the uncertain volatility model, where a Hamilton-Jacobi-Bellman partial differential equation needs to be solved. We explore the order of convergence in these problems, as well as the option greeks stability, compared to the literature and popular schemes such as Crank-Nicolson, with Rannacher time-stepping.
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We present a multigrid iterative algorithm for solving a system of coupled free boundary problems for pricing American put options with regime-switching. The algorithm is based on our recently developed compact finite difference scheme coupled with Hermite interpolation for solving the coupled partial differential equations consisting of the asset option and the delta, gamma, and speed sensitivities. In the algorithm, we first use the Gauss-Seidel method as a smoother and then implement a multigrid strategy based on modified cycle (M-cycle) for solving our discretized equations. Hermite interpolation with Newton interpolatory divided difference (as the basis) is used in estimating the coupled asset, delta, gamma, and speed options in the set of equations. A numerical experiment is performed with the two- and four- regime examples and compared with other existing methods to validate the optimal strategy. Results show that this algorithm provides a fast and efficient tool for pricing American put options with regime-switching.
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We present new numerical schemes for pricing perpetual Bermudan and American options as well as $alpha$-quantile options. This includes a new direct calculation of the optimal exercise barrier for early-exercise options. Our approach is based on the Spitzer identities for general Levy processes and on the Wiener-Hopf method. Our direct calculation of the price of $alpha$-quantile options combines for the first time the Dassios-Port-Wendel identity and the Spitzer identities for the extrema of processes. Our results show that the new pricing methods provide excellent error convergence with respect to computational time when implemented with a range of Levy processes.
The main objective of this paper is to present an algorithm of pricing perpetual American put options with asset-dependent discounting. The value function of such an instrument can be described as begin{equation*} V^{omega}_{text{A}^{text{Put}}}(s) = sup_{tauinmathcal{T}} mathbb{E}_{s}[e^{-int_0^tau omega(S_w) dw} (K-S_tau)^{+}], end{equation*} where $mathcal{T}$ is a family of stopping times, $omega$ is a discount function and $mathbb{E}$ is an expectation taken with respect to a martingale measure. Moreover, we assume that the asset price process $S_t$ is a geometric Levy process with negative exponential jumps, i.e. $S_t = s e^{zeta t + sigma B_t - sum_{i=1}^{N_t} Y_i}$. The asset-dependent discounting is reflected in the $omega$ function, so this approach is a generalisation of the classic case when $omega$ is constant. It turns out that under certain conditions on the $omega$ function, the value function $V^{omega}_{text{A}^{text{Put}}}(s)$ is convex and can be represented in a closed form; see Al-Hadad and Palmowski (2021). We provide an option pricing algorithm in this scenario and we present exact calculations for the particular choices of $omega$ such that $V^{omega}_{text{A}^{text{Put}}}(s)$ takes a simplified form.
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