#16: Options Greeks: Delta as "Probability of Expiring ITM"
 
In my previous post, I wrote about Delta as "stock equivalence" and how it can be used for hedging.

In this post, I will be writing on a critically useful interpretation of Delta: it's use as a probability.

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Suppose you flipped a coin. 

You know that the probability of it landing on heads is 50%, or 0.50. If you rolled a die, the probability of getting "six" is 1/6, or 0.167.

Now let's say you took a directional bet on the market. 

Say you bought 100 shares of SPY. What's the probability of you being right?

The answer to that is usually wrapped around 52%. Historically, the markets drifted upwards and there is a slightly higher chance of markets going up than coming down.

But most people wouldn't know about that 52% probability. You would have to dig around to find it, and even then the number is disputed. 

Does knowing the probability before you make a trade influence your decision to trade in the first place?

HELL YES!

Most people would not bother making a 50-50 trade, especially if the payoffs are equal on either side. 

50% chance of winning $1 versus 50% chance of losing $1. Well... what's the fucking point of that??

What about a trade with 90% chance of being right?  90% chance of winning $1 versus 10% chance of losing $9.

Sure the payoff is shit, and the cost of being wrong will be painful, but at least you know you stand a very high chance of being right.

Wouldn't having this probability in your hands be a useful metric before you take a trade?

Now suppose I told you that there is a way to know the probability of a trade going in your favour. 

Wouldn't it be awesome to know the probability of a trade before you take it??

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It's odd that Delta can tell us probability. I'm going to be honest and state that I don't know the math derivation that linked delta's number to it's probability of expiring in-the-money (ITM).

I've read many books on Options, and they just pretty much state it as fact. 

The other sites that discuss the Math backing up this claim is either too difficult to understand, or actually dispute the claim altogether.

Nonetheless, it has become a simple rule of thumb for most option traders to use delta as the probability of expiring ITM.

Let's first take a look at an options chain.

I have circled the deltas in the options chain above. Those circled in green are on the call options, and those in blue are for the put options. 

Let's look at the upper most blue circle. The delta shown there is -0.08. Since the negative sign tells us the directionality, we can ignore that for now and focus only on the number 0.08.

A delta of 0.08 means that this put option (strike of 245) has an approximately 8% chance of expiring ITM. It is what we consider to be "FAR out of the money (OTM)".

So if you BUY this option, there is a bloody high chance (92%) that you are wasting your money. The flipside to that is that the option only costs $60.

Low chance of success but cheap cost to buy = Lottery ticket!

Going down the blue circles, we see the following options:

- Strike 255 has delta of 0.17 (OTM) -- costs $116 to buy with 17% chance of expiring ITM.

- Strike 266 has delta of 0.51 (at-the-money or ATM)-- costs $301 to buy with 51% chance of expiring ITM.

So you see that the higher the chance of expiring ITM, the more costly the option. This is logical, of course. 

The pattern is repeated on the call side in green circles:

- Strike 266 has delta of 0.50 (ATM) -- costs $413 to buy with 50% chance of expiring ITM.

- Strike 273 has delta of 0.18 (OTM)-- costs $73 to buy with 18% chance of expiring ITM.

- Strike 276 has delta of 0.08 (Far OTM)-- costs $26 to buy with 8% chance of expiring ITM. 

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You might be thinking, "if the payoffs are inversely correlated to the probabilities, then where's the edge for the trader?"

True. If the probabilities are accurate, and the option prices are priced to perfection, then it doesn't matter whether we bet on 50-50 or 90-10, the trade-offs negate any benefits or losses. 

However there are TWO very critical deviations from the hard numbers that give certain traders the edge:

1) The probabilities are NOT accurate.

2) Probability of expiring ITM does NOT equal probability of being profitable

These two are so critical that it is worth elaborating on them in detail.

1) The probabilities are NOT accurate

How is the number for "delta" generated in the first place? It is derived from a mathematical formula based off of options prices, which is a very important point.

Delta is a result of options prices, therefore if prices are not rational, then delta probabilities are inaccurate!

Why would options prices not be rational? Because of FEAR and GREED.

Options are mainly used as hedging instruments (insurance) and it's prices tend to be inflated more than necessary most of the time. This creates the phenomenon of delta probabilities being OVERSTATED. 

Which means to say when an option has a delta of 30%, the actual statistical data shows that the option only expires ITM about 16%. This is based off TastyTrade's research below.

So there is an edge for the options SELLER because the probabilities of the buyer being right is even lower than what the delta is showing.

2) Probability of expiring ITM does NOT equal probability of being profitable 

For options buyers, expiring ITM doesn't mean making a profit. It could be a loss!

If an option buyer pays $413 to buy that call option with strike of 266, and the option eventually expires with the underlying at 267, the option has gone ITM by $1, but the option buyer would have lost $313 as a result.

(1 options contract = 100 shares. ITM by $1 = gained $100 of intrinsic value. Since he paid $413, gaining $100 leaves him with $313 loss. For him to breakeven, he would have needed the underlying to rise to 266 + 4.13 = 250.13)

So for an options BUYER, probability of expiring ITM is ALWAYS HIGHER than the probability of being profitable. 

If you bought an option with delta 30, you will always have LESS than 30% chance of being profitable. 

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When we combine both points 1 & 2, then it becomes quite clear that the odds are stacked in favour of the options SELLER

Not only are delta probabilities overstated (point 1) due to fear and greed in the seller's favour, there is the additional premium collected that forces the underlying to move further before the option seller suffers a loss (point 2).

So for goodness' sake DO NOT BE AN OPTIONS BUYER!  

Flip the coin over and be a seller. Join the dark side!

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That's it for this three-part series explaining Delta. 

In the next post I shall be writing about the second of the options Greeks, Gamma. It might be the second on the list, but it's definitely the most dangerous of the greeks to the options seller!

Look out for that post coming up in a week's time. 

Till then, happy trading!