Why do we have to use in-the-money paths in LSMC, and how?

Question

In Longstaff's original LSMC paper (Valuing American Options by Simulation: A Simple Least-Squares Approach, 2001 (link)), it is claimed that one should only use in-the-money paths for regression at each time step, in order to improve efficiency. It seems to be accompanied by no proof. The reason the author gives is that only in-the-money paths are relevant for comparing exercise vs continuation values.

Concerning the usage of ITM paths, there are two questions:

1). Would there be a serious problem if I have to use also OTM paths? For example, serious biasedness?

2). For some derivatives, it may be difficult to identify what exactly "in-the-money" even means. For example, consider a convertible bond with an embedded call option, then there is surely more than one way in which the convertible can be early terminated:

• Voluntary conversion: when in the convertible period and conversion value exceeds continuation value.

• Forced conversion: when in the convertible and callable period and call price is smaller than both continuation value and conversion value.

• Call: when in the callable period and call value is smaller than continuation value but greater than conversion value.

In this case I don't think there is a clear definition of "in-the-money" or the "relevance" to exercising. Currently, I'm following the Tsiveriotis and Fernandes framework (Valuing Convertible Bonds with Credit Risk, Jrl Fixed Income, 1998) and splitting the convertible bond cashflows as equity-only (which arises from conversion only) and cash-only (which arises from call only), and I'm considering doing two regressions each step for the two parts of cashflows respectively. So now comes the problem, how to define "in-the-money" for the two respective cases? And for the equity-only part, should we distinguish between voluntary conversion and forced conversion when we regress?

Answer 1

Here's my $1/50. Please be free to raise any suggestions.

1. Don't regress the split cashflows respectively as in TF, just regress the whole continuation value instead.

2. When the bond is in the callable period, we'll have to use all paths; Otherwise, when in the convertible period, only consider the paths where conversion value > straight bond value (or should be straight callable bond value, can't be very sure here); and when not in the convertible period either, no regression is needed at all.

Let's go back to Longstaff and what they mean by "where exercise is relevant". In essence, picking only ITM paths is doing an a priori estimate that filters out some of the paths that you know won't have a chance for early exercise. Specifically, an a priori lower bound of the continuation values, which are not yet known, is that it >= 0, therefore if the intrinsic value (of immediate exercise) is 0 (i.e. OTM), then there's no chance for early exercise. So choosing only the ITM paths for regression is just an a priori filtering of the paths where early exercise is definitely impossible.

Following this thread, if we want to achieve something similar for our callable convertible bond, we will also have to do an a priori filtering of the paths for which early termination isn't possible, i.e.

1. Early conversion isn't possible AND

2. Early redemption (call) isn't possible.

The first filtering is only relevant when in the convertible period, in which case a convenient a priori condition would be $$\text{Conversion Value} \le \text{Straight (Callable) Bond Value}$$ because we know a priori that Straight (Callable) Bond Value <= Continuation Value.

The second one is however much trickier. When in the callable period, we know that the issuer will call if and only if call price < continuation value. Thus to do an a priori estimate for such a condition, we will have to provide a convenient, a priori upper bound for the continuation value. Is any such bound known? I doubt it.

Therefore, the best treatment I can think of is:

1. If a path is convertible & callable, pick it unconditionally.

2. If a path is convertible & (not callable), pick it if and only if conversion value > straight (callable) bond value.

3. If a path is (not convertible) & callable, pick it unconditionally.

4. If a path is (not convertible) & (not callabe), don't pick it at all.

At each step we only pick the relevant paths according to the criteria above for regression. There'll be a point when there'll be no longer any paths to be picked (i.e. before convertible period and before when the earliest callable path becomes callable), and that's going to be when we stop rolling back, and we simply discount all paths at this point to the initial time, to get the final value.