A dynamic programming approach for pricing options embedded in bonds

Abstract

We propose a dynamic programming (DP) approach for pricing options embedded in bonds, the focus being on call and put options with advance notice. An efficient procedure is developed for the cases where the interest-rate process follows the Vasicek, Cox-IngersollRoss (CIR), or generalized Vasicek models. Our DP methodology uses the exact joint distribution of the interest rate and integrated interest rate at a future date, conditional on the current value of the interest rate. We provide numerical illustrations, for the Vasicek and CIR models, comparing our DP method with finite-difference methods. Our procedure compares quite favorably in terms of both efficiency and accuracy. An important advantage of the our DP approach is that it can be applied to more general models calibrated to capture the term structure of interest rates (e.g., the generalized Vasicek model).

American option pricing with randomized quasi-Monte Carlo simulations

“…We took n = 2 12 and it was already sufficient to reach a nearly-optimal policy, in the sense that the average from the second stage evaluations did not differ significantly (statistically) from the exact value (for the given basis functions). This exact value is estimated to be 0.77871 (obtained from 5000 replications of RQMC with PCA, with n = 16) and is identical to the value reported in Ben Ameur, Breton, Karoui, and L'Ecuyer (2007).…”

“…We consider an example of a callable bond taken from Ben Ameur, Breton, Karoui, and L'Ecuyer (2007). The bond is issued at time t 0 = 0, pays a coupon value c at each of the coupon dates t c 1 < · · · < t c d , and returns a principal value of 1 at the maturity date t c d .…”

“…Conditional on R j−1 , the pair (R j , Λ j ) has a known distribution given in Ben Ameur, Breton, Karoui, and L'Ecuyer (2007), which can be simulated easily by applying a simple transformation to a bivariate normal vector, and an explicit expression is available for E exp (−Λ j ) |R j−1 . We give the DP formulation of this optimal stopping problem from the viewpoint of the issuer, who wants to minimize his cost.…”

Evaluating callable and putable bonds: An eigenfunction expansion approach

“…We consider the callable bond example that has been extensively used in the literature starting from Büttler and Waldvogel (1996) and including d 'Halluin et al (2001), Ben-Ameur et al (2007), andde Frutos (2008), as the test case to compare computational performance of a number of computational approaches to the callable bond valuation. The callable bond was issued by Swiss Confederation in 1987 with maturity in 2012.…”

“…The holding value function is approximated as a finite summation involving the Laguerre polynomials, and the problem is converted to a stiff system of ordinary differential equations (ODEs) for the time-dependent coefficients of the Laguerre expansion. Ben-Ameur et al (2007) (BBKL) propose an alternative dynamic programming approach not based on PDEs in which a piecewise linear approximation is used for the value function and the exact transition probability is used to compute the expectation of the discounted piecewise linear approximation of the value function in the cases of CIR and Vasicek short rate models where the exact analytical expressions are available. All of these papers provide numerical experiments illustrating computational performance of their respective methods on the same example of a Swiss callable bond.…”