Pricing convertible bonds with call protection

Stéphane Crépey, Abdallah Rahal

Abstract

ABSTRACT

We deal with the issue of pricing convertible bonds numerically, using simulation. A convertible bond can be seen as a coupon-paying and callable American option. Moreover, call times are typically subject to constraints, called call protections, preventing the issuer from calling the bond in certain subperiods of time. The nature of the call protection may be very path-dependent, like a path dependence based on a large number, d, of Boolean random variables, leading to high-dimensional pricing problems. Deterministic pricing schemes are then ruled out by the curse of dimensionality, and simulation methods appear to be the only viable alternative. We consider various possible clauses of call protection. In each case, we propose a reference, a heavy if practical deterministic pricing scheme, and a more efficient (as soon as d exceeds a few units) and practical Monte Carlo simulation/regression pricing scheme. In each case we derive the pricing equation, study the convergence of the Monte Carlo simulation/regression scheme and illustrate our results by reports on numerical experiments.We thereby obtain a practical and mathematically justified approach to the problem of pricing convertible bonds by simulation with highly path-dependent call protection. More generally, this paper is an illustration of the real abilities of simulation/regression numerical schemes for high- to very high-dimensional pricing problems, like the systems of 2dD30 scalar coupled partial differential equations that arise in the context of the application at hand in this paper.