Trading Options: Guide on How to Calculate Probability of Profit

The mysteries of POP and learn how it can be your guiding light in the tumultuous world of trading.

Attention, fellow traders! Today, we're delving into the intriguing realm of options trading, and there's one crucial metric that can significantly impact your success: Probability of Profit (POP). So, let's embark on this journey together to unravel the mysteries of POP and learn how it can be your guiding light in the tumultuous world of trading.

Understanding the Essence of Probability of Profit (POP)

Probability of Profit, or POP, is the likelihood of making at least 1 paise on an option trade (in simpler terms, avoiding any penny of loss). At its core, POP signifies the likelihood of making a profit in an options trade. Picture it as your guide through the maze of At-the-money (ATM), In-the-money (ITM), and Out-of-the-money (OTM) scenarios. Calculated using Delta, POP tells you the chances of your trade concluding with at least a modest gain of 1 paise.

If you buy or sell an ATM option that hypothetically has a delta of around 50, the probability of the option expiring in-the-money (ITM) or out-of-the-money (OTM) respectively is 50 per cent, i.e. the trade has a 50 per cent chance of being profitable at expiry.

Similar probability calculations are also done for the OTM options.

For an option buyer, a far OTM call/put option with a delta of 15 has a 15 per cent probability of expiring ITM and being profitable or an 85 per cent (100-15) chance of expiring OTM and worthless at expiry. For an options seller, it is an 85 per cent chance of making a profit.

It’s easy to calculate POP for naked puts or calls but gets a little complicated when calculating it for multi-legged strategies like strangle, straddle, etc.

What is the POP of the option seller who sells 1 lot 21,200 CE, with Delta of 29.73? Now, the person who sells one lot of 21,200 CE, his POP would be 100-29.73= 70.27.

Linear Relationship

Just for clarification, delta and the probability of expiring in the money are not the same thing. What we mean is that delta is usually a close enough approximation to the probability.

One way to think about it is to look at the probabilities and deltas of In the Money, Out of the Money, and At the Money options.

- A deep in-the-money option has a really high chance of expiring in the money, around 100 per cent, and it has about a 100 delta.

- A far out-of-the-money option has a really low chance of expiring in the money, around 0 per cent, and it has about a 0 delta.

- An at-the-money option has about a 50 per cent probability of being in the money because there is a 50-50 chance the stock will go up or down, and it has about a 50 delta.

In these cases, the delta and probabilities are about the same. In fact, if you look at an options chain with delta and probabilities, you can see that they are all about the same. In other words, there is a linear relationship between delta and probability.

Also, note that Delta varies as implied volatility changes. So does our POP.

Here is an example of the increasing complexity of POP calculation. Let’s calculate the POP of an ATM short straddle. Before we dive into more details, we need to discuss important concepts of mathematics.

Bayes Theorem Unveiled:

Bayes’ Theorem is a way of finding a probability when we know certain other probabilities.

The formula is:

Formula: P(A|B) =P(B|A) *P(A)/P(B)

Where:

P(A|B) – the probability of event A occurring, given event B has occurred

P(B|A) – the probability of event B occurring, given event A has occurred

P(A) – the probability of event A

P(B) – the probability of event B

When we know:

How often B happens given that A happens, written P(B|A)

And how likely A is on its own, written P(A)

And how likely B is on its own, written P(B)

Probability of Independent Events

But, Let’s say, there is no connection between two events

P(A|B) = P(A).

P(B|A) = P(B).

The probability of A, given that B has happened, is the same as the probability of A. Likewise, the probability of B, given that A has happened, is the same as the probability of B. This shouldn’t be a surprise, as one event doesn’t affect the other.

So, there are two cases –

When two events, A and B, are independent, the Probability of both occurring is P (A and B) = P(A) * P(B).

When two events, A and B, are independent, the Probability of one of them occurring is P (A or B) = P(A) + P(B)

Example in Action:

Probability of Profit for Short Straddles

Right now, NIFTY’s LTP is 20,862. Let’s say we have sold 20900 PE and 20900 CE of Dec 21 expiry to form a short straddle setup. The call options lower the breakeven of the downside while the put options lower the breakeven of the upside. Hence, POP of this setup is dependent on each other.

Current Setup –

- Sell NIFTY 21st Dec 20900 PE at 108.8

- Sell NIFTY 21st Dec 20900 CE at 154.45

To calculate POP, the correct approach is to find the Breakevens of this setup. In this case –

- Lower Breakeven: 20900 – (108.8 + 154.45) = 20,636.75

- Upper Breakeven: 20900 + (108.8 + 154.45) = 21,163.25

As we notice, chances of breaking Lower Breakeven and Upper Breakeven are independent events! And, in a similar manner, we can calculate the POP of our breakeven breaking.

We can restructure the question as follows –

- What is the POP of 20,650 PE?

- What is the POP of 21,150 CE?

We need to take the nearest strike with respect to the Breakeven points.

- Delta of 20,650 PE is 22.3

- Delta of 21,150 CE is 23.28

So,

- Probability of Profit if we buy 20,650 PE is 22.3% or 0.223

- Probability of Profit if we buy 21,150 CE is 23.28% or 0.232

Now, if one of our breakeven breaks i.e. if one of 20,650 PE and 21,150 CE buys goes into profit, that will be our probability of profit for the short straddle.

So,

Probability of Profit if we buy 20,650 PE is 22.3 per cent or 0.223

Probability of Profit if we buy 21,150 CE is 23.28 per cent or 0.232

Now, if one of our breakeven breaks i.e. if one of 20,650 PE and 21,150 CE buy goes into profit, that will be our probability of profit for the short straddle.

P (A or B) = P(A) + P(B)

P (Buy 20,650 PE or Buy 21,150 CE)

= P (Buy 20,650 PE) + P (Buy 21,150 CE)

= 0.223 + 0.232

= 0.455

This is our probability of loss for the short straddle. So, the Probability of profit for the short straddle will be 1 – 0.455= 0.545 i.e. 54.5 per cent.

Conclusion:

Armed with a deep understanding of POP and the powerful tool that is Bayes' Theorem, you're now equipped to navigate the intricate world of options trading with confidence. Remember, while probabilities provide valuable insights, the strategic application of Bayes' Theorem in conjunction with POP is the key to making informed and profitable decisions. Happy trading!