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The overall value of an option is primarily determined by six factors – strike price, spot price of the underlying stock, number of days to expiration, dividend yield, rate of interest, and the volatility of the underlying stock.Because these six inputs behave in different ways to affect the value of an option, it is possible for the price of the underlying stock to increase while the value of the option falls. The Black Scholes formula may fail when other factors are affecting the value of the option more than current stock price.
Inputs to the Calculator
|Current Market Price of the Stock||So|
|Volatility %||s or Sigma|
|Annual Dividend Yield %||k|
|Days to Expiration||t|
|Rate of Interest||r|
The Call outputs are Call Price, Delta, Gamma, Vega, Theta, and Rho.
Similarly, the Put outputs are the Put Price, Delta, Gamma, Vega, Theta, and Rho.
The model is extremely easy to use. Besides current, strike,and time to expiration in days, you will need to know implied volatility, interest rate, dividend yield beforehand. Popular trading tools to the likes of TD Ameritrade or Active Trader Pro will have implied volatility IV for American and European options. You can also find IV from Market Chameleon’s volatility rankings. They have a premium subscription as well.
Annual dividend yield can be easily found from Google or Yahoo finance. Annual rate of interest from Treasury’s daily yield can be plugged into the model. There you go with feeding all that is necessary to find call, put, and option greeks. I have covered how to interpret option greeks in an earlier post.
Back Testing with Chicago Board Options Exchange (CBOE) Calculator
Wouldn’t it be nice to back test all the hard work you put using CBOE online options calculator? The online calculator is free to use for learning purposes. The only catch is that you cannot have this tool calculate in bulk. It is an educational tool for beginners and is intended to assist individuals in learning how options work.
An American call option for AAL with a strike of $10 expires 4 days from now, 04/03/20. The implied volatility is 212%, which is pretty high when all airlines have been grounded due to COVID19 travel restrictions. The annual interest is 0.98% and the annual dividend amount is $0.40 per share paid at the anniversary. $0.40 dividend amount translates to annual dividend yield of 3.33% ($0.40 / $12.25 which is the current price of AAL).
As you can see – Call, Puts, and option greeks that we just built came very close to CBOE’s online calculator. We are almost right in the money. Scroll down to the end of the post, tweet or like and download for free.
Making all sense from option greeks
Call and Put are $2.49 and $0.24 respectively for Apr-03 Strike at 10. The stock is currently trading at 12.25. So far so good with the option prices. How do you interpret those strange greeks?
If you owned AAL call option and 4 days to expiration, a delta of $0.85 means that the option’s price will theoretically move $0.85 for every $1 move in the price of AAL. As you probably have figured out, put options will have negative delta. At-the-money options usually have a Delta near -0.50. The Delta will decrease (and approach -1.00) as the option gets deeper in the money.
You can also think delta as the probability that a given option will expire in the money. For the AAL call option, a delta of 0.85 means the option has 85% chance of being in the money at expiration on Apr-03-20. This doesn’t guarantee you will profit from the trade. That of course, depends on the price at which you bought or sold the option.
You should also think of delta, as the number of shares of the underlying stock, the option behaves like. A Delta of 0.85 also means that given a $1 move in AAL, the option will likely gain or lose about the same amount of money as 85 shares of AAL.
What about Gamma?
Gamma measures the rate of change in an option’s Delta per $1 change in the price of the underlying stock. You can think of delta as velocity and gamma as acceleration. Reminds you of your high school Physics?
Delta is only valid at a certain price and time. In the delta example above, once the stock has moved $1 and the option has subsequently moved $0.85, the delta is no longer 0.85. This $1 movement causes a call option to be deeper in the money, and therefore the delta will move closer to 1.00. Let’s assume the Delta is now 0.94. This change in Delta from 0.85 to 0.94 is 0.09—this is AAL option’s gamma. Because delta can’t exceed 1.00, gamma decreases as an option gets further in the money and Delta approaches 1.00.
Option Values behind the scenes
The six input factors determine the value of an option in a combined way. It is quite possible for the underlying stock to go up, while value of the option goes down.
The longer the time between option purchased date and its expiration date, the more time there is for the current stock price to reach the strike price. Hence greater the value of the option. You can see the value of an option go down even as the price of the underlying stock goes up. This happens when the expiration date is nearing. In such cases, the price increases may be too little and too late for the stock to reach the strike price. Therefore, the option value will continue to decrease as the expiration date approaches as it becomes less likely that the strike price will be reached.
As strike price plays a crucial role in both the value of the option and the potential for profit, it is important for you to choose the strike price carefully. The strike price is the anticipated value of the underlying stock at the time of expiration and the farther this amount is from the stock price at the time the option was written, the lower the premium. Larger gaps between strike price and market price are harder to attain by the expiration date and leads to increased risk. Lower premiums are charged because it is less likely that the gap can be attained before the expiration date.
However, options with the higher strike prices also have the biggest potential for profit. Many investors choose to buy and/or sell options with different strike prices to help limit losses while maximizing profits. Such strategies include the Option Spread, Iron Condor, Butterfly, and the Buy Straddle to name a few.