Calculating Profit Probability for Options with Different Expirations
Summary
The document explains how to assess whether a position with options expiring on different dates will be profitable. Its example combines calls expiring first with a put expiring later, and defines total profit from the option payoffs at both dates less the initial cost. Multiplying two unconditional probabilities of finishing above a threshold would not capture the dependence between prices on the two dates.
Instead, the proposed method divides outcomes at the first expiration into three disjoint cases: the calls alone cover the trade cost, the calls recover part of it, or the calls expire worthless. It then adds the probability of immediate profitability to conditional probabilities that the later put makes up any remaining shortfall in the latter two cases. The method requires a model for conditional price probabilities; the document provides no specific model or numerical probability. Its stated payoff expression and case analysis are tailored to the described position, so other option quantities or structures require their own payoff decomposition.
Key ideas
- The position’s profit depends on prices at both expiration dates and the initial trade cost.
- Unconditional probabilities at separate dates cannot simply be multiplied without accounting for dependence.
- Partition outcomes by the call payoff at the first expiration, then evaluate later put outcomes conditionally.
- A probability estimate requires a model that supplies the relevant conditional price probabilities.
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Full text
# How to calculate the probability of 2 options ending in money with different expiration dates?
# How to calculate the probability of 2 options ending in money with different expiration dates?
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 by Bob Jansen (score 1)
https://quant.stackexchange.com/a/15995
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:
- $S_1 > K_1 + \frac{C}{2}$, the entire trade is profitable, this happens with probability $P\left(S_1 > K_1 + \frac{C}{2}\right)$;
- $K_1 \leq S_1 \leq K_1 + \frac{C}{2}$, a part of the investment is earned back;
- $S_1 < K_1$, 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.