Why is ATM Delta not always 0.50?

[CJFerreira 1]

My understanding has always been that the delta of an ATM option is 0.5. The fact that the number varied slightly I attributed to small variance in market values or even inaccuracies in the calculation.

 

But when I look at LEAP greeks, I see the furthest expirations have a delta that is not even close. For example, at this time an ATM call of 0.61 for calls and -0.39 for puts. The SPY options are much closer - 0.52 for calls and -0.50 for puts (odd - I thought these two numbers always should add up to 0?).

 

I notice also that the calls tend to run a little over 0.5, while puts tend to run a little under.

 

Does anyone have an explanation for this behavior?

 

 

[jfouche 12]

Well, the option is concerned with the value of the underlying at expiration, which depends on dividend yield and cost of borrowing funds. For a stock with a decent yield, the price to take delivery of the stock in 1-2 years will be significantly lower than today's price, so the apparently ATM call is actually an OTM call. I'd be curious where you saw the ATM call having >50 delta... possibly in an unusual situation.

[CJFerreira 1]

Yeah, I thought about dividends making a difference, but this is Netflix (NFLX), and there aren't any.

You can see the ATM calls on NFLX for the Jan 2015 are showing a delta of over 0.60 using TOS as well as the CBOE option calculator.

[Marco 221]

The delta of the 'ATM forward' option will be 0.5.

The forward price for a stock is calculated from current spot price and 'cost of carry' until option expiration.

Cost of carry is determined by interest rates, dividends and borrow cost. (Here a formula if you want to know the exact calculation http://en.wikipedia.org/wiki/Forward_price)

If a stock has a very high div yield and a low IV than for American style options a certain probability of an early exercise can affect the delta too.

As NFLX doesn't have a div, interest rates are pretty low and any borrow cost on the stock would bring the forward (and therefore the delta of the call) down non of that explains the high delta though.... (Scratching head)

With the stock at 284 and 0.66% for interest rates the forward price is about 287$ but if you stick that in your option calculator you still get a delta well over 50. The last thing that I can think of is that an option calculator will take into account that a stock can't go below zero. This is particularly relevant for long dated options and high IV like in this case. If you lower IV in your option calculator you will see the delta for the 287 strike will go to about 55 (closer to 50 but still not quite there)

Basically if you imagine a graph with the possible price of the stock in Jan15 on the x axis and the probability of NLFX to be at that price on the Y axis and a bell curve shaped graph with the pivot point at the ATM forward stock price (287 in this example)

If you have a low IV than the probable stock prices in Jan-15 will all be near that 287 price and say prices of 200 and 374 and below/above have a very low probabily and the graph hits pretty much zero at these levels. If IV is very high and you look at a long time until expiration the probabily of a stock price of zero may hit say 5%. So the left tail of your 'bell curve' (it isn't a normal distribution so speaking of a bell curve is technically not correct) would cross the y axis at 5%. As stock prices can go lower than zero the remaining 5% probability have to go somewhere in that curve - so that will move to the right hand tail hence increasing the probability (and therefore the delta) of calls and decreasing the delta of puts. If you look at http://en.wikipedia.org/wiki/Log-normal_distribution than the blue graph in the top right corner would be an extreme example.

Edited September 1, 2013 by Marco

[jfouche 12]

Yeah, the 0.5 thing relies on some approximations that aren't working when the remaining time and IV are very high. Look at the LEAPS on TSLA, the 50% delta is the 215 strike!

 

I suspect this is for the reasons Marco points out -- the future is not symmetric in those cases.

[CJFerreira 1]

Is this a flaw in the pricing, or in our understanding? With no dividends to account for and interest rates very low, the diversion shouldn't be off by 10 points.

[jfouche 12]

Well, here's an answer based on the Black-Scholes solution for call prices.

 

http://quant.stackexchange.com/questions/4612/why-do-atm-call-options-have-a-delta-of-slightly-bigger-than-0-5-and-not-0-5-exa?rq=1

 

So, the 0.5 thing is not exactly true except when you're close to expiration. 

[samerh 6]

It is because options pricing theory uses the lognormal distribution which is "a continuous probability distribution of a random variable whose logarithm is normally distributed". Note it assumes the logarithm of the price changes are normally distributed (vs the price changes being normally distributed).

 

The reason for this is that Normal distribution is symmetrical: e.g. a $100 stock would have the same probability of rising to $210 as it does to dropping to NEGATIVE $10 (in each case a $110 change). As stocks can't go negative, the normal distribution can't be used, instead a lognormal distribution is used which skews the distribution of final prices so that the final price never drops below zero.

 

You can think of it this way, assuming two continuously compounded 10% increases from $100 gives you $121, a difference of $21.

However 2 10% decreases from $100 gives you $81 - a difference of $19.

=> As the move up gives a bigger number than the move down, Calls have a larger absolute delta than puts

 

Lol - just typed this up and saw that Marco had already answered.

Edited September 3, 2013 by samerh

[CJFerreira 1]

Thank you all. This has been an interesting read!