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

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!

• If you are able to write the pricing problem as partial differential equations with boundary values, then you can use Feynman Kac to solve for the risk neutral expectation. So finite difference methods can be used if the problem is properly formulated as PDEs. I've seen it used to price barrier, lookback and other types of exotic options. – Slade Mar 18 at 12:36
• @Slade thanks. Therefore in PDE methods, we only need to figure out the various boundaries between various regions. The problem is that, for example, the callability in a convertible is a "soft call", meaning that the call option isn't started until a path dependent condition is met, which implies we cannot determine, at a given point, whether the call is started, even when it would indeed be optimal for the issuer to call if the call condition is indeed met. In other words, the convertible is called only if it's optimal to call and the call is activated already. – Vim Mar 18 at 12:50

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.

• Thanks. So it seems I can model the parisan clause as a one touch clause for stock price well above or well below 130. Do you have any suggestions for how to deal with the close to 130 cases, besides explicitly introducing a big set of auxiliary variables? – Vim Mar 18 at 17:29
• See edit. Also if close to 130 some ideas are to use approximations such as computing separately the probability of 15/30 conditional on one touch and reweight the sof call payoff by this number. – Antoine Conze Mar 19 at 7:58
• Thanks. In my case though, call/put/reset provisions are all parisan e.g. (n/30, x%). Replacing them all with one-touch clauses would imply more optionality to the issuer via call/reset but also more optionality to the holder via put. However, in practice put clauses are more difficult to trigger than reset clauses (which allows the issuer to downward adjust the conversion price, and are invented exactly to preclude early redemption via put and to create higher chances for conversion), so I think you're still right saying that one-touch replacement gives sub replication prices. – Vim Mar 19 at 9:03
• The parisian clause likely really starts to affect the delta hedge once it has started to accumulate, for instance if you've already been above 130 for 14 days in the last 20 days, then it's one touch within the next 10 days. So probably pricing with one touch with reweighting with conditional probability would give a fair enough approximation. – Antoine Conze Mar 19 at 9:57

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.