I have been asking this similar question before. However, I really want to be concrete and get and concrete explanation.
I have been reading the paper by Moreni and try to implement the same methodology based on least-square monte carlo for both European and American put options. They payoff function for both these puts is given by $f(t,S_t)=\max(K-S(t),0)$. The only difference is that for American case, we have payoffs in different times from simulations depending on the optimal stopping time.
As usual we have the underlying stock price as geometric brownian motion:
$dS_t = r S_t dt + \sigma S_t dW_t$.
Moreni introduces the change of drift in the underlying:
$dS_t = (r+\theta\sigma )S_t dt + \sigma S_t dW_t$.
When simulating from the drifted process, we have to multiply the PAYOFF functions with the likelihood ratio defined as $L_\theta=\exp\{-\theta W_t+\frac{1}{2}\theta^2 t\}$
so we have $f(t,S_t)=\max(K-S(t),0)\cdot\exp\{-\theta W_t+\frac{1}{2}\theta^2 t\}$
instead of just $f(t,S_t)=\max(K-S(t),0)$
The idea is of course to use negative values for $\theta$ so we get more "in-the money paths".
But when I try to use these modified payoffs with the Longstaff and schwartz algorithm, I don't get any relevant results. The price gets extremly biased whichever value i use for $\theta$.
My question is, should the estimated continuation values in least-square method also be modified with the likelihood ratio? In that case how? Or as mentioned in the article, ONLY the payoff functions be multiplied by the likelihood ratio??
Thank you for your patience
PS* Added by request the steps that I use:
- Generate $N$-scenarios of the drifted with $\theta$ geometric brownian motion
- Save all the standard normal values $Z_t$ from the GBM for use in the likelihood ratio.
- Create the likelihood ratio using the $Z_t$'s and for some value $\theta$
- Use the likelihood ratio and set $f(t,S_t)=\max(K-S(t),0)\cdot\exp\{-\theta W_t-\frac{1}{2}\theta^2 t\}$ for all paths
- Now run the Least-square algorithm from $t=T$ down to $t=0$ to calculate all the continuation values and optimal stopping times.
- Take average of all the scenarios.