Convertible bond valuation: 20 out of 30 day soft-call

“Soft-call” in convertible bonds (CBs) usually means that the bond can be recalled by the issuer only if the stock price has previously closed above a specified trigger price for any 20 out of any 30 consecutive trading days. It is not an easy optionality to value and no method has been implemented besides Monte Carlo. The problem is not very well suited to Monte Carlo due to a large number of possible permutations of stock price closes above or below the trigger over a year period (i.e., 2²⁶⁰) with the result that a Monte Carlo valuation requires a trade off between being slow and not smooth. The soft-call feature is typically modeled in the CB industry by presuming that the bond is called as soon as stock touches the trigger price. After discussion of the exact solution of this problem (requiring valuation of component derivatives on, of order, 2²⁶⁰ grids), a simple algorithm is presented to approximately value this feature for the general n out of m case of soft-call. The algorithm requires merely a subtle change to the call feature of the one-touch model and only one running of a grid or tree and hence it is very fast. The method boils down to making the bond “1-touch” callable on some days and not on others, the precise sequence being a function of the 29 day stock price close history. It gives smooth functional output (the theoretical price jumps from day to day) and very compelling qualitative results. The results are accurate to a dime on the dollar of benefits due to provisional call, and this is determined by comparison to the exact solution for easily calculated cases