Pricing convertible bonds with call protection

Abstract

In this paper we deal with the issue of pricing numerically by simulation convertible bonds. 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 at certain sub-periods 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 in this paper various possible clauses of call protection. We propose in each case a reference, but heavy, if practical, deterministic pricing scheme, as well as 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. One thus gets a practical and mathematically justified approach to the problem of pricing by simulation convertible bonds 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 systems of 2 scalar coupled partial differential equations that arise in the context of the application at hand in this paper.