Black and Scholes Option Model
Fisher Black, Myron Scholes and Robert Merton were the three academics that developed the much acclaimed option pricing model now called Black and Scholes.
The Black and Scholes option pricing model was the result of the educational paper called Real Options, written by these academics in 1973. The model was so highly praised that they were awarded the prestigious Noble Prize in 1997.
The Black and Scholes model has spawned a family of other pricing variations since then, although the Black and Scholes still remains the foundation model for pricing options today.
Black Scholes Inputs
S = Stock Price
X = Exercise Price
r = Risk Free Interest Rate
T = Time to Expiration (Years)
N(x) = The Cumulative Normal Distribution Function
= Standard Deviation
Black Scholes Formula
The Black Scholes equation states that a:
Call Option = 
Where:
Given Put Call Parity:
The price of a put option must therefore be:
Most of the inputs for the Black Scholes Model are a given, i.e. we know with certainty what the exercise price is, how many days until the option expires, where the underlying instrument is currently trading and we also know what interest rates currently are. However, we do not know what volatility the underlying will experience between the time of trade and the expiry date. That is why volatility is so important when it comes to option pricing - the volatility of the underlying asset really determines how likely the option contract will be in, at or out-of-the-money by the expiration date.
The Greeks
As well as calculating the fair value for an option contract, option pricing models also calculate what-if values, referred to as The Greeks - Delta, Gamma, Vega, and Theta. The importance of these calculations are covered under The Greeks link above.
Limitations
An important assumption in the Black-Scholes option pricing model is a statistical concept known as the Normal Distribution. The Black-Scholes model uses the volatility input to calculate the probability of the underlying price expiring in the money before the option's maturity date. The higher the volatility the more likely it is that the underlying price will be above the exercise price before the maturity date and hence the option will then have a higher premium. If the volatility of the underlying instrument is expected to be low then there is less chance that the option will be profitable and will have a lower expected option value.
Black Scholes Calculator
Download the Option Trading Workbook to see the Black Scholes model in action.
Comments (8)
Peter
April 8th, 2009 at 7:05am
Hi Helen,
You can see my code in the spreadsheet:
http://www.optiontradingtips.com/pricing/free-spreadsheet.html
I've not seen a "reversed" Black-Scholes formula yet. If you find one...please let me know and I'll add it to the pricing spreadsheet.
Helen
April 7th, 2009 at 2:53pm
What will be the best way to calculate the implied volatility on options. Doing the backward of the Black-scholes model?
Admin
March 22nd, 2009 at 6:36am
For American style options you would use the Binomial option pricing model. My spreadsheet currently doesn't price American options...only European options. I plan to add a Binomial model soon.
JT
March 18th, 2009 at 8:08am
One more question. From reading your site, which is fantastic by the way, it seems that this "pricing" strategy is mainly used for Euro style options. What source of pricing model would you use for American style options?
Admin
March 18th, 2009 at 4:43am
Hi JT,
Yes, "theoretically" it would be a good price to buy.
JT
March 17th, 2009 at 12:53pm
Stupid question. Is the theoretical price that is calculated using this method, the "max" price you should purchase this option at? Say the option price was 1.30 for a call with a strike of 2.50 and the theoretical price is 1.80. Would that make it a "good" buy?
Admin
February 1st, 2009 at 3:45am
Hi Hadi,
Yep, I agree. I've corrected the paragraph as noted.
Thanks!
Hadi AK
January 31st, 2009 at 12:53am
" The volatility of an option really determines how likely that contract will be in, at or out-of-the-money by the expiration date. "
4th Paragraph above the Google Ads, last line.
The volatility referred by those academics was the volatility of the underlying stock not the volatility of the option itself,
The price of an option is derived fully from the underlying stock and its provisions ( Strike Price , Maturity , Underlying Price, Int Rate and Volatility OF THE UNDERLYING STOCK )
Nice Webpage i use it frequently,
Rgds
H.A.K.
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