# stock price vs. option price vs. profit gain

I know if the stock is cheaper and its call option is also cheap. However, is it better to buy an option of a cheap stock than buy an option of an expensive stock? Assume both stocks do not pay dividend and implied volatility of both stocks is the same.

For example, the current price of stock A is 100 dollars and the current price of stock B is 200 dollars. Assume they all grow 10% in a month: stock A grows to 110 dollars and stock B grows to 220 dollars. Assume the strike prices of the options are 20% higher than the current price. The strike price of stock A is 120 dollars and the strike price of stock B is 240 dollars. The expiration date is the same, one month later.

Is it better to buy the option of stock A? This is my feeling and I just want to confirm with someone. However, if stock B grows faster (percentage wise), how do I decide which option to buy? Thanks

• "Better" in what way? – glibdud Oct 1 '20 at 1:10
• make more profit – Jerry Zhang Oct 1 '20 at 1:12

The strike price of stock A is 120 dollars and the strike price of stock B is 240 dollars. The expiration date is the same, one month later. Is it better to buy the option of stock A? This is my feeling and I just want to confirm with someone. However, if stock B grows faster, how do I decide which option to buy? Thanks

There are two missing missing puzzle pieces in your equation which contribute to an accurate answer.

1. Both stocks pay no dividend and if they do, the dividend of stock B is twice that of stock A

2. The implied volatility of the options of both stocks is identical and will remain identical throughout your ownership.

If all conditions are met, then it makes no difference which options you buy because option price is linear in regard to price. IOW, the cost of a \$240 call on a \$200 stock will be exactly twice the cost of a \$120 call on a \$100 stock. And if they move up the same percent and all conditions remain equal then both options will profit by the same percentage.

The last sentence in your paragraph is problematic. If stock B grows faster (which I interpret to mean that its price rises more percentagewise) then it would be the obvious one to buy.

Which option to buy is a can of worms. You can figure this out with an option pricing formula but it's a bit of grunt work. The path of least resistance is some decent option modeling software that displays graphs.

A really nuanced way to compare two positions in the same stock is to input one option position as long and the other as short. If the positions are equivalent, you'll get a horizontal graph across time and price. If they are not equivalent you'll get a sloping graph and with a bit of further comparison, say expiration values, you'll be able to tell which option leg is better.

Read this answer that I posted earlier today about a similar question. It offers some ideas onhow to determine which option to buy.

• Thanks. I updated my question to reflect your points. But options never pay dividend, right? So, dividend is definitely not a concern here, right? – Jerry Zhang Oct 1 '20 at 1:55
• Do you know any software which provides option modeling and comparison? I use e-trade and interactive broker. Does the trading platform provide any tools to compare options? – Jerry Zhang Oct 1 '20 at 1:59
• Also, I feel the implied volatility of both stocks does not matter here, right? I feel if the price of the stock finally grows the same percentage, the implied volatility should not matter. – Jerry Zhang Oct 1 '20 at 2:02
• Dividends affect option prices, increasing put premium and decreasing call premium. Therefore, a dividend in one stock or a dividend ratio other than 2:1 would break the linear relationship of premium cost across price and time that I discussed. I use IBKR as my primary broker but I have no clue what software they offer I use some proprietary software that's about 25 years old so I don't have any software suggestions for you. – Bob Baerker Oct 1 '20 at 5:18
• Implied volatility definitely matters because it increases or decreases option premium. If it was not equal, this too would break the linear relationship. Imagine two identical \$100 stocks (your parameters). Stock A has an implied volatility of .25 so the one month \$110 call costs 40 cents. Stock B has an IV of .50 so its one month \$110 call costs \$2.50 . If both stocks are \$120 at expiration, call A makes \$9.50 and call B makes \$7.50. An equal stock price move but an unequal result. – Bob Baerker Oct 1 '20 at 5:19