Is it possible to model path-dependent clauses using finite difference methods?

Question

I'm trying to build a convertible bond pricer. In my case a convertible bond is a complex derivative with call, put and conversion price reset clauses, and all of the clauses are triggered in a path-dependent fashion. For example, the call clause might be:

In 30 consecutive trading days, if the closing stock price is bigger than 130% of conversion price in 15 trading days, then the issuer has the option to redeem the bond.

At the present stage I'm still looking for the right model. Considering the path-dependencies, I see Monte Carlo as somewhat the only way to go. And considering the call/put/reset-ability which are American in nature, LSMC (Least Square Monte Carlo) seems to be the only choice.

However, being non-deterministic and quite fiddly, LSMC should be a last resort. If possible, I would prefer easier methods such as finite difference methods and tree methods, but none of them seems able to deal with the path-dependencies entailed in the call/put/reset clauses.

Is there any way to somehow accommodate such path-dependencies in tree models/finite difference models instead of LSMC? Thanks!

Answer 1

The usual approach to deal with path dependency in finite differences/lattices solvers is to capture the path dependency trough one or more auxiliary variable(s) that make the problem non path dependent in the augmented space, and to discretize along these auxiliary variable(s).

For instance that's easily done for asian options where the path dependency is captured through $M_t = $the average stock price, with dynamics $M_{t+1} = (t M_t + S_{t+1})/(t+1)$

In your case, a parisian clause with 15 out of 30 consecutive days above 130, it is a bit more complicated, and if I am not mistaken you would need 29 boolean auxiliary variables to keep track of whether or not the stock was above 130 in each of the past 29 days, so that's an added dimensionality of $2^{29}$ which makes the problem untractable.

You can however look at the 15 out of 15 consecutive days above 130 case, in which case you only need one auxiliary variable $N_t = $ number of consecutive days the stock price has been above 130. Then the dynamics for $N_t$ is \begin{eqnarray*} N_{t+1} &=& N_t + 1 \text{ if } S_{t+1} \geq 130 \\ N_{t+1} &=& 0 \text{ if } S_{t+1} < 130 \\ \end{eqnarray*}

In any case since with low interest rates $\mathbb{E}_t[S_{t+n}] \approx S_t$ when there is no dividend, viewed from inception pricing with a 15 days out of 30 consecutive days clause or simply with a one touch clause should not make much of difference. It really starts to matter once you get close to 130 (and in particular if you have started to accumulate days above 130).

Hope it helps.

Edit: you can also note that one touch gives a sub-replication price (more optionality to the issuer than the actual 15/30 clause) and 15/15 gives a super-replication price (less optionality to the issuer than the 15/30 clause), so the true price is is between.

Answer 2

If you are asking whether it is possible to price path-dependent American options in tree based models, the short answer is yes. You simply construct your tree/grid and evaluate the rules in each node (analogous to what you would do in your MC simulations). These rules can be arbitrarily complex. Note, however, that you can only evaluate them at a discrete set of times (just like the fact that you can only observe the stochastic processes at a discrete set of times in MC simulations).

I recommend that you begin by having a look at bi- and trinomial lattice methods for pricing path dependent options. You may want to begin by studying how they are implemented for simple path-dependent derivatives such as Asian options, for which there is a lot of literature, before moving on to more complex products.