I'm trying to understand Least-Square Monte Carlo approach for pricing american options. I'm familiar with Tsitsiklis and van Roy (2001) approach where we are going backwards with:
- $V_T = h(S_T)$, where $h$ is a payoff from option if exercise
- for each next step (previous time step on the grid) we have $V_{t_{i-1}}=\max\{{(K-S_{t_{i-1}})^+, E[DF \times V_{t_i}}|\mathcal{F_{t_{i-1}}}]\}$ where $DF$ is a discount factor from $t_{i}$ to $t_{i-1}$ and this expected value is called Continuation Value.
Now, after reading the original paper of Longstaff-Schwarz algorithm, my thoughts are that it works as follows
- when we are going backwards, we don't use $V_{t_i}$ as $Y$ in our regression, but we use the sum of discounted cash-flows from a given path which occurs after time $t_{i-1}$. In case of an american option, we simply take the cashflow from the date when we exercise it (or 0 if there is no exercise on that path). Here at each step we need to adjust the cashflows so that if we exercise at time $t_{i-1}$, the cashflows after that time are set to 0.
I believe that up to that point I'm correct. But now I've read Glasserman book about MC simulations in Finance and I found the following there:
From above, it seems that the only difference between LS and TvR is that in LS, at each step we do not set $C$ (continuation value, conditional expectation) as a value of a trade if it exceede exercise value, but we set the discounted value from the previous step. There is nothing mentioned about summming cash-flows from the future. Is it correct? How is it equivalent to the approach described above where we regress the sum of discounted cashflows instead of discounted value from the previous step?
So in short, what is correct in LS algorithm:
- use the sum of discounted realized future cash-flows as $Y$ for Continuation Value estimation and set this $C$ as option value if it exceedes the exercise value
or
- use the value from the previous step as $Y$ for $C$ estimation but don't set this continuation value as option value even if it exceede the exercise value - in that case use the disconted option value from the previous step.
Additional questions:
- LS recommend to use only ITM paths for regression. But what value should be set for OTM paths? Should we simply set the discounted value from the previous step or we should set the continuation value from the regression? I.e. OTM paths shouldn't be used to estimate the regression coefficient s but those coefficient should be used for these OTM paths to determine Continuation Value?
References
F. A. Longstaff and E. S. Schwartz, “Valuing American options by simulation: A simple least-squares approach,” Rev. Financial Studies, vol. 14, no. 1, pp. 113–147, 2001 (link)
J. N. Tsitsiklis and B. Van Roy, "Regression Methods for Pricing Complex American-Style Options", IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 12, NO. 4, JULY 2001 (link)
P. Glasserman: Monte Carlo Methods in Financial Engineering, Springer, 2003
