Black-Scholes Option Model
The Black-Scholes Model was developed by three academics: Fischer Black, Myron Scholes and Robert Merton. It was 28-year old Black who first had the idea in 1969 and in 1973 Fischer and Scholes published the first draft of the now famous paper The Pricing of Options and Corporate Liabilities.
The concepts outlined in the paper were groundbreaking and it came as no surprise in 1997 that Merton and Scholes were awarded the Noble Prize in Economics. Fischer Black passed away in 1995, before he could share the accolade.
The Black-Scholes Model is arguably the most important and widely used concept in finance today. It has formed the basis for several subsequent option valuation models, not least the binomial model.
What Does the Black-Scholes Model do?
The Black-Scholes Model is a formula for calculating the fair value of an option contract, where an option is a derivative whose value is based on some underlying asset.
In its early form the model was put forward as a way to calculate the theoretical value of a European call option on a stock not paying discrete proportional dividends. However it has since been shown that dividends can also be incorporated into the model.
In addition to calculating the theoretical or fair value for both call and put options, the Black-Scholes model also calculates option Greeks. Option Greeks are values such as delta, gamma, theta and vega, which tell option traders how the theoretical price of the option may change given certain changes in the model inputs. Greeks are an invaluable tool in portfolio hedging.
Black-Scholes Equation
Call Option = 
Where:
Given Put Call Parity:
The price of a put option must therefore be:
Black-Scholes Excel
Black-Scholes VBA
Function dOne(UnderlyingPrice, ExercisePrice, Time, Interest, Volatility, Dividend)
dOne = (Log(UnderlyingPrice / ExercisePrice) + (Interest - Dividend + 0.5 * Volatility ^ 2) * Time) / (Volatility * (Sqr(Time)))
End Function
Function NdOne(UnderlyingPrice, ExercisePrice, Time, Interest, Volatility, Dividend)
NdOne = Exp(-(dOne(UnderlyingPrice, ExercisePrice, Time, Interest, Volatility, Dividend) ^ 2) / 2) / (Sqr(2 * 3.14159265358979))
End Function
Function dTwo(UnderlyingPrice, ExercisePrice, Time, Interest, Volatility, Dividend)
dTwo = dOne(UnderlyingPrice, ExercisePrice, Time, Interest, Volatility, Dividend) - Volatility * Sqr(Time)
End Function
Function NdTwo(UnderlyingPrice, ExercisePrice, Time, Interest, Volatility, Dividend)
NdTwo = Application.NormSDist(dTwo(UnderlyingPrice, ExercisePrice, Time, Interest, Volatility, Dividend))
End Function
Function CallOption(UnderlyingPrice, ExercisePrice, Time, Interest, Volatility, Dividend)
CallOption = Exp(-Dividend * Time) * UnderlyingPrice * Application.NormSDist(dOne(UnderlyingPrice, ExercisePrice, Time, Interest, Volatility, Dividend)) - ExercisePrice * Exp(-Interest * Time) * Application.NormSDist(dOne(UnderlyingPrice, ExercisePrice, Time, Interest, Volatility, Dividend) - Volatility * Sqr(Time))
End Function
Function PutOption(UnderlyingPrice, ExercisePrice, Time, Interest, Volatility, Dividend)
PutOption = ExercisePrice * Exp(-Interest * Time) * Application.NormSDist(-dTwo(UnderlyingPrice, ExercisePrice, Time, Interest, Volatility, Dividend)) - Exp(-Dividend * Time) * UnderlyingPrice * Application.NormSDist(-dOne(UnderlyingPrice, ExercisePrice, Time, Interest, Volatility, Dividend))
End Function
You can create your own functions using Visual Basic in Excel and recall those functions as formulas within your chosen workbook. If you want to see the code in action complete with Option Greeks, download my Option Trading Workbook.
The above code was taken from Simon Benninga's book Financial Modeling, 3rd Edition. I highly recommend reading this and Espen Gaarder Haug's The Complete Guide to Option Pricing Formulas
. If you're short on option pricing formulas texts, these two are a must.
Model Inputs
From the formula and code above you will notice that six inputs are required for the Black-Scholes model:
- Underlying Price (price of the stock)
- Exercise Price (strike price)
- Time to Expiration (in years)
- Risk Free Interest Rate (rate of return)
- Dividend Yield
- Volatility
Out of these inputs, the first five are known and can be found easily. Volatility is the only input that is not known and must be estimated.
Black-Scholes Volatility
Volatility is the most important factor in pricing options. It refers to how predictable or unpredictable a stock is. The more an asset price swings around from day to day, the more volatile the asset is said to be. From a statistical point of view volatility is based on an underlying stock having a standard normal cumulative distribution.
To estimate volatility, traders either:
- Calculate historical volatility by downloading the price series for the underlying asset and finding the standard deviation for the time series. See my Historical Volatility Calculator.
- Use a forecasting method such as GARCH.
Implied Volatility
By using the Black-Scholes equation in reverse, traders can calculate what's known as implied volatility. That is, by entering in the market price of the option and all other known parameters, the implied volatility tells a trader what level of volatility to expect from the asset given the current share price and current option price.
Assumptions of the Black-Scholes Model
1) No Dividends
The original Black-Scholes model did not take into account dividends. Since most companies do pay discrete dividends to shareholders this exclusion is unhelpful. Dividends can be easily incorporated into the existing Black-Scholes model by adjusting the underlying price input. You can do this in two ways:
- Deduct the current value of all expected discrete dividends from the current stock price before entering into the model or
- Deduct the estimated dividend yield from the risk-free interest rate during the calculations.
You will notice that my method of accounting for dividends uses the latter method.
2) European Options
A European option means the option cannot be exercised before the expiration date of the option contract. American style options allow for the option to be exercised at any time before the expiration date. This flexibility makes American options more valuable as they allow traders to exercise a call option on a stock in order to be eligible for a dividend payment. American options are generally priced using another pricing model called the Binomial Option Model.
3) Efficient Markets
The Black-Scholes model assumes there is no directional bias present in the price of the security and that any information available to the market is already priced into the security.
4) Frictionless Markets
Friction refers to the presence of transaction costs such as brokerage and clearing fees. The Black-Scholes model was originally developed without consideration for brokerage and other transaction costs.
5) Constant Interest Rates
The Black-Scholes model assumes that interest rates are constant and known for the duration of the options life. In reality interest rates are subject to change at anytime.
6) Asset Returns are Lognormally Distributed
Incorporating volatility into option pricing relies on the distribution of the asset’s returns. Typically, the probability of an asset being higher or lower from one day to the next is unknown and therefore has a 50/50 probability. Distributions that follow an even price path are said to be normally distributed and will have a bell-curve shape symmetrical around the current price.
It is generally accepted, however, that stocks – and many other assets in fact – have an upward drift. This is partly due to the expectation that most equities will increase in value over the long term and also because a stock price has a price floor of zero. The upward bias in the returns of asset prices results in a distribution that is lognormal. A lognormally distributed curve is non-symmetrical and has a positive skew to the upside.
Geometric Brownian Motion
The price path of a security is said to follow a geometric Brownian motion (GBM). GBMs are most commonly used in finance for modelling price series data. According to Wikipedia a geometric Brownian motion is a “continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion". For a full explanation and examples of GBM, check out Vose Software.
61 Comments
Paul S July 11th, 2011 at 10:40am
Peter,
Understanding that entering the current price of an option along with all other inputs would give us Implied Volatility, but not being a math whiz, what is the construction of the formula for Implied Volatility?
Thanks
-Paul
Peter March 23rd, 2011 at 7:56pm
Mmm...let me go back to my books and see what I can discover.
Bob Dolan March 23rd, 2011 at 6:39pm
Peter wrote:
"Do you know if there is an available option model for a binary distribution...?"
Actually, the binary distribution is fully described in this web site.
The example given was a stock that had a 0.5 probability of 95 and at 0.5 probability of 105.
But your mileage may differ for a specific security.
The real question is:
How do you establish the binary points and probabilities thereof for any given security?
The answer is research.
How you link 'research' to an Excel model is an open question.
I mean, that's the fun of it.
Bob
Bob Dolan March 23rd, 2011 at 5:59pm
Peter wrote:
"Do you know if there is an available option model for a binary distribution you mentioned?"
Well, shucks, if that option model exists, it certainly isn't easily available through a Google search.
I figure that I/we have to write it.
Hey: 'Once more into the fray'.
Wanna help?
Bob
Peter March 23rd, 2011 at 5:01pm
Thanks for the great comments Bob!
Your approach to finding IV by reversing Black and Scholes sounds almost the same as what I used in my B&S Spreadsheet;
High = 5
Low = 0
Do While (High - Low) > 0.0001
If CallOption(UnderlyingPrice, ExercisePrice, Time, Interest, (High + Low) / 2, Dividend) > Target Then
High = (High + Low) / 2
Else: Low = (High + Low) / 2
End If
Loop
ImpliedCallVolatility = (High + Low) / 2
Do you know if there is an available option model for a binary distribution you mentioned? Perhaps I could make a spreadsheet our of it for the site?
Bob Dolan March 23rd, 2011 at 3:46pm
JL wrote:
"Stock prices rarely follow theoretical models however, so I suppose that is why the authors did not attempt to include any projections."
Well, sure. But also, the authors believed the 'random walk' model of stock pricing. Their skepticism of anyone's ability to forecast prices made it easy for them to embrace a model with no 'oooch' factors.
In 'The Big Short' Michael Lewis describes an analyst who adheres to 'event driven' investing. The concept is simple:
* Black-Scholes assumes a log-normal distribution of stock prices over time.
* But, sometimes, prices are determined by discrete events [law-suits, regulatory approval, patent approvals, oil discoveries]. In these cases, a binary or bipolar distribution of future stock prices is a better model.
* When future stock prices are better represented by a binary distribution, there may be probability arbitrage to be had if an option is priced assuming a long-normal distribution.
* The longer the time frame, the more likely that GBM progressions do not apply. SOMETHING will happen. If the possibility of that something can be foreseen, probability arbitrage is possible.
So, how do you quantify that?
And here I am on your web site.
Bob
Bob Dolan March 23rd, 2011 at 3:23pm
Back to the "reversed" Black-Scholes algorithm [and sorry to find your site a year late] ...
Manually, I use a binary search to get an approximation of the IV needed to produce a given option price. It's actually a two-step process:
Step One: Guess at the IV [say, 30] and adjust the guess until you have the IV bracketed.
Step Two: Iterate a binary search -- each time making the 'guess' half-way between the brackets.
Even doing this manually, I can come up with a close approximation in a reasonable time. Iterating the search in Excel, and comparing the result to some level of 'tolerance', would seem to be a fairly easy work-around.
From a UI standpoint, I think I would specify the 'tolerance' in significant digits [e.g., 0.1, 0.01, or 0.001].
In any event, this would seem to lend itself to some sort of VBA macro.
Bob
Peter February 8th, 2011 at 4:25pm
Hi JL,
Black Scholes doesn't attempt to directionally forecast the stocks price, but it does attempt to forecast the stocks price path with the volatility input.
Also, dividends are indeed incorporated into the Black and Scholes model and form part of the Theoretical Forward price. The reason that call option prices don't decrease with a change in interest rates is because the increase in the Theoretical Forward due to the stock's cost of carry (Stock Price x (1 + Interest Rate)) will always be greater than the present value of future dividends.
JL February 8th, 2011 at 9:06am
Peter,
Thank you for the fast response. Your work has been very helpful in trying to understand option pricing. If I understand your explination correctly, a call option increases in price because the assumed current price of the stock will remain the same and the "Theoretical Forward Price" increases therby increasing the value of the call option. I suppose my main issue is with the Black-Scholes model itself because it makes no attempt to forecast a stocks price, which theoretically should be the present value of all the future dividends. So if interest rates are rising, the prices of stocks should be declining due to the higher discount rate used in the present value calculation, and therby decreasing the current value of the call options sold on those stocks. Stock prices rarely follow theoreticall models however, so I suppose that is why the authors did not attempt to include any projections.
Peter February 7th, 2011 at 6:16pm
Hi JL,
The risk free rate is a measure of the value of money i.e. what your return would be if, other than buying the stock, you were to invest in this risk free rate. Therefore the Black Scholes Model first calculates what the Theoretical Forward price would be at the expiration date. The Theoretical Forward price shows at what price the stock must be trading at by the expiration date to prove a more worthy investment than investing in the risk free rate of return.
As the Theoretical Forward price increase with interest (risk free) rates the value of call options increases and the value of put options decreases.
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