finding the optimal decision value for two dependent random events.

Question

I have been struggling with this problem regarding options (bermuda) for some time now.

You can exercise this option on two seperate occasions namely at $T_1$ or $T_2$ with a strike price $E$. The price of the stock are simulated from known distributions named $S_1$ and $S_2$. Note: these are not independent random variables; $S_2$ depends in some way on the value of $S_1$.

A simple pricing of this option would be: if $S_1 > E$: $V = \max(E - S_1, 0)$ else $V = \max(E - S_2, 0)$ (note: I left out any irrelevant details such as the price discount rates)

Now suppose we have some a for which if $S_1 > (E + a)$: $V = \max(E - S_1, 0)$ else $V = \max(E - S_2, 0)$

How can I find the optimal a for which the average option price $V$ is optimal/maximalized? Can someone tell me how to approach this problem? - I have tried using monte carlo simulations, which all failed to give useful answer. Even though my intuition says this isnt a particularly had question.

Answer 1

OK, here's one way you can get this. Since you are stating this in a very general fashion, there will not be a general analytical solution -- instead, this will require Monte Carlo and some numerical optimization (gradient search). Also, I think the first terms in your max's are flipped. As it stands, you would never sell on the first time period, since you would be losing money... also, I am assuming this is a call option, correct, since your first conditional is premised on $S_1>E$? So if the price is greater than the strike price at T1, then you stand to make the spread, if not, then you can try again at T2 and then you are out of luck.

First, lets re-formulate this to get an equation for the conditional expected value of V, given $E,a$ and $S_0$: $E(V|S_0,E,a)=P(S_1> E+a|S_0,a)E(S_1-E|S_1>E+a)+P(S_2>E|S_1\leq E+a)E(S_2-E|S_2>E,S_1\leq E+a)$

You can use Monte Carlo to solve this problem by generating the joint distribution of $(S_1,S_2|S_0)$ for a pre-specified value for $S_0$. Then you simulate $S_1=S_0*D_1,(S_2|S_1) = S_1*D_2$. Get a large number of such $(S_1,S_2)$ pairs and save them to a database. If you use enough trials, then you will have faithfully replicated the joint distribution of $(S_1,S_2|S_0)$.

Now, to find the optimal $a$, you need to write a routine (in R or your favorite numerical computation package) that performs the following steps for each value of $a$, with the possible values of $a$ being either pre-specified or based on the rate of increase in the expected value of V between successive valeus of $a$:

1. Find the pairs in the simulated dataset where $S_1 > E+a$ and save in some data array.
2. Count the number of such pairs and divide by the total size of your simulated dataset, this will be an estimate of $P(S_1> E+a|S_0,a)$
3. Take the average value of $S_1-E$ for each pair in this set, this will approximate $E(S_1-E|S_1>E+a)$
4. For all pairs not identified in step 1, find the number of pairs where $S_2>E$ and divide by the toal number of pairs not part of the set in step 1. This will approximate $P(S_2>E|S_1\leq E+a)$
5. Find the average value of $S_2-E$ for pairs from step 4 where $S_2>E$. This will approximate $E(S_2-E|S_2>E)$
6. Substitute the above approximations into the formula for the conditional expected value of V given a.

Now, you just need to search in both directions from your initial $a_0$ to identify local maxima of the values in Step 6 above (or THE maximizer, if you have a unimodal curve). This procedure is computationally intensive, but conceptually simple and you only have to run a Monte Carlo simulation once...but it needs to be a big enough run to properly reproduce the joint distribution of stock values at two different time periods.