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Theoretical question but which one is the best mathematically?

1) Buy one ITM call option with strike (price-5) and one OTM call option with strike (price+5)

2) Buy two ATM call option

Which one is better? Why?

  • What do you mean by "better"? The payoffs will be very different so there's not a definitive answer. – D Stanley Feb 1 at 0:25
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There's no one size fits all answer to your question because option price is heavily dependent on price, the time remaining and implied volatility and they move independent of each other.

For example, if these are for a near term weekly expiration, the ITM + OTM will cost much more than the two ATM if the implied volatility is low. Therefore, there's a very different amount of premium at risk.

If this is a medium term expiration or later, the ITM + OTM have a somewhat similar cost as two ATM regardless of the implied volatility. Therefore, similar price risk.

There are three ways that you can get a feel for which pair performs better.

1) Use a pricing model with time slice graphs and enter the position as long the ITM, long the OTM, and short the two ATM. The risk graph will show you at which price each does better. In this case (L -2*S +L), when P&L rises, the longs are the better choice. When P&L declines, the -2*S are stronger. It doesn't matter which pair is long or short because it's just the juxtaposition to see strength across a range of prices.

2) Again using time slice graphs, look at the net delta of the (L -2*S +L) position. When it's rising across price, the long position is stronger. When falling, the short position is stronger.

3) If you don't have access to a program that does this, just add the delta of the ITM and OTM and compare it to twice the delta of the ATM. The higher total delta is the better choice today for the upside. But this method only depicts today's possible performance. Without the time slices, you can't see the range of option performance over underlying price, say from $95 to $105.

Clear as mud?

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SPY trading at 270

Options expiration, 02/27/2019

Strike 263 at 7.91

Strike 270 at 4.20

Strike 277 at 1.18

The in-the-money option is the least trading risk but the most capital risk. The out-of-the-money option is the most trading risk but the least capital risk.

Buy the ITM at 7.91 and the OTM at 1.18 for a total of 9.09 ?

Or buy two ATM for a total of 8.40 ?

The ATM breaks even at 274.20 . The ITM has a profit of 3.29 at 274.20. The OTM has a loss of 1.18 at 274.20. The ITM and the OTM together have a profit of 2.11 at 274.20.

Or look at this one:

SPY trading at 270, Options expiration Dec 2021, Strike 260 at 38.17 .

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    There's nothing to be learned from calculating P&L at a few random prices. One could spend days, weeks, whatever trying to amass enough data points to craft some semblance of conclusion that explains in what circumstances one pair performs better than the other (and vice versa). These numbers certainly didn't. – Bob Baerker Feb 1 at 4:50
  • I thought the question was a very unique subject and deserved a test. And I used a large liquid test-bed with the S&P 500. Also, I used a nearly one month time frame. Scientific testing is not unusual at all. – S Spring Feb 1 at 5:30

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