How to calculate the probability of 2 options ending in money with different expiration dates?

Question

Lets say I make a trade that consists of buying one put and 2 calls of the same underlying but with different expiration dates and different strikes.

Example trade:

Long call - strike @ $100 -  exp 3/17/2015
    Long put - strike @ $110 - exp: 3/30/2015
cost of trade = $15

- Current price = $90, current date = 1/1/2015

If the trades expired on the same day the trade would net profit if the underlying price at expiration is greater than $105. The probability of the trade being profitable would be equal to the probability of the underlying being above $105 on the expiration date.

The trades unfortunately do not expire on the same day which complicates things. How do I calculate the probability of the trade being profitable when the options expire on different dates?

Is it
= (probability underlying > $105 @ 3/17/2015) * (probability underlying > $105 @ 3/30/2015)

I feel that the above is not correct as it does not account for the relationship between the two probabilities. Maybe a more correct equation would be:

= (probability underlying > $105 @ 3/17/2015) * (probability underlying > $105 starting from 3/17/2015 to 3/30/2015 with a starting price of $105)

Even this equation seems to be incorrect because if the option expires on the earlier date above the $105, then it can expire below $105 on the later expiration date.

Answer 1

First some notation, let

• $S_1$ and $S_2$ and be the stock price at the expiration dates;
• $K_1$ and $K_2$ the strike prices;
• and $C$ be the cost.

then the profit is given by

$$\textrm{profit} = 2 \times (S_1 - K_1)^+ + (K_2 - S_2)^+ - C.$$

At the first expiration date three mutually exclusive cases can be distinguished:

1. $S_1 > K_1 + \frac{C}{2}$, the entire trade is profitable, this happens with probability $P\left(S_1 > K_1 + \frac{C}{2}\right)$;
2. $K_1 \leq S_1 \leq K_1 + \frac{C}{2}$, a part of the investment is earned back;
3. $S_1 < K_1$, the call option expired worthless.

If the calls expire out of the money (the third case), the put has to make up for the investment and the probability of that happening is $P(S_2 < K_2 - C \:\lvert\: S_1 < K_1)$. The second case is the most complex:

$$P\left(K_2 - S_2 > C - 2(S_1 - K_1) \:\lvert\: K_1 \leq S_1 \leq K_1 + \frac{C}{2}\right).$$

We can just at these as they are mutually exclusive so:

$$P(\textrm{profit > 0}) = P\left(S_1 > K_1 + \frac{C}{2}\right) + P\left(K_2 - S_2 > C - 2(S_1 - K_1) \:\lvert\: K_1 \leq S_1 \leq K_1 + \frac{C}{2}\right) + P(S_2 < K_2 - C \:\lvert\: S_1 < K_1).$$

I see no way to simplify without more assumptions this but if you have a model, in which you can calculate conditional probabilities, calculating the result should be straightforward.