Using Option Delta and Expected Moves to Estimate Trade Probabilities

# Using delta as probability of an option expiring in the money


# Using delta as probability of an option expiring in the money












I understand that delta can be seen as a probability proxy for an option expiring in the money, as well as deltas for call options ranging from 0 to 1 and deltas for put options ranging from 0 to -1.

How/can you use delta to calculate a probability of profit proxy for spreads or positions with multiple legs? For example, a position that consists of both long and short calls and puts, as well as a common stock position. How would one go about calculating the overall probability of profit proxy of that portfolio using delta?

Consider the following portfolio scenario:

```
Current price of AAPL: 150.00

AAPL Sep 17 2021 157.5 Put (Delta: -0.889)
AAPL Sep 17 2021 149 Put (Delta: -0.427)
AAPL Sep 17 2021 148 Call (Delta: 0.651)
AAPL Sep 17 2021 146 Call (Delta: 0.778)
AAPL Common Stock; 5 shares purchased at 150.00
```

Furthermore, if using delta is not a suggested way to go about calculating the overall probability of profit proxy of a portfolio, how can we use N(d1) from the Black-Scholes formula to do the same?

## Answer by Dr. Michael J. Stefano (score 1)

https://quant.stackexchange.com/a/85331

I dont think the answers you got are very helpful. see if this helps.

example: call bull spread. buy ITM at 70 delta and sell ATM at 50 delta. for example, bought a ITM 40 strike call at 10 and sold the ATM 50 strike call at 5. net debit = 5. so break even = 45, and max gain at 50 strike = 5. max gain roi = 100, acheived at the 50 strike which has delta of .50.

we can say that the spread has a 50% probability of realizing the max gain of 100% roi at expiration because the short strike is ATM where the delta is .50

Therefore, we can calculate what is called an expected value. that would be the max gain multiplied by the probability of it occurring. So in this case that would be the 5, or $500 max gain*50% ( based on using the ATM delta as our probability of profit) so we would have a 50% prob of gaining $250

Now we would get the max loss at the .70 delta where the long call strike is.

we could estimate the probability of that occurence using the delta of .70 to mean that price would remain above the 45 strike 70% of the time and end up below the .70 delta 30% of the time.

this would give us an expected value for max loss as follows:

max loss of the entire net debit cost of the spread ($500)*.30 = (150).

now we would also have smaller gains at expiry in the region between the 2 strikes, making our overall expected value over time to be a positive number.

the casino builds in risk return profiles with negative expected value, so they simply make money on the volume.

This risk return ratio in the example above is not common for call debit spreads though it is more likely in GLD and SLV where the IV skew is to the upside(in calls) due to them acting like volatility assets. it is also available in SPY with put debit spreads due to the significant IV skew to the put side for equities in general but noticeably more in SPY. then you would have positive EV in your spreads.

another easier and commonly accepted way to figure out the probability of your trade profit or staying in a range, as in a calendar spread, is that some platforms will overlay a bell curve with standard deviations marked on your spread/position and you can see the probabilities that way. you just need to know the basic probs for 1,2,3 s.d respectively. this is an easy google search

one other commonly accepted way is to use the The Expected Move, which is also referred to as the Implied Move, which reflects the price range that a security is expected to move from current price. The Expected Move is calculated based on 85% of the value of the at-the-money straddle, for whatever option contract you are using depending on the time frame you are interested in.

so if the straddle premiums add to 5, then 85% of that is 4.25 if the stock price was 100, then the expected move is up to 104.25 and down to 95.75. this would rendere as a perentage, and so in this example it would be 4.25% either way. Again, this would be specific to the expiration of the option contract used, AND, it i important to know that this will be considered an acceptable approximation of a one standard deviation move in either direction, which is a range of about 68%. otherwise there is much tedious data and a comple formula that none of us could or would likely do. that leaves 16% below and 16% above that range, which means that the stock would have a 84% probability of finishing above the lower range of that 1 s.d. calc. in this example that would be at 95.75

so lets say i wanted to sell an OTM put, or OTM put credit spread, I could calc this and decide that this is my comfortable risk profile and then know where to set my short spread strike, say at the 96 strike for the given expiration, which you chose when you did the 85% calc.

however, the idea of expected value is important, because it is highly unlikely that your put credit spread will have a positive EV. why? the IV skew smirks to the downside in equities, making the long put relatively more expensive than the short put. so while you may now have a spread at the lower end of the expected move, giving you a 84% probability that the stock will stay above the short strike, the roi on the spread may only be 14% or likely even lower.

if the roi on the spread is 10% which is say $1, then the loss is therefore 9. if the probability of gain is 90%, then i gain $1 90 times out of every 100 occurrences and lose $9 the other 10 times out of 100. net EV = zero, and actually less because you had costs and gave up the risk free rate.

this is the allure of the OTM put credit spread which really have no positive EV.

on a calendar spread, use the 85% method using the short dated ATM option straddle calc, or look at the deltas of the options at the break even points of the spread (using the short option) to get an estimate of the prob of going above or below the breaks evens and the diff will be the prob of remaining within the upside and downside breakevens.

i hope this helped make more sense out of how to get an idea of probabilities and also remember to consider EV