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
JL February 7th, 2011 at 4:53pm
Peter,
Keeping all other variables constant, if I increase the Risk Free Rate the value of the Call option increases. This is counter to what should happen, logically if I can earn a better return in a safer investment then the price of a higher risk investment should be lower.
Peter January 23rd, 2011 at 8:01pm
That's right, they're not the same, so it's up to you what method you use.
BSJhala January 21st, 2011 at 9:30am
Hi peter,
But 4/260 and 7/365 are not same.than the results will vary for the two isn't it.
pls suggest me what will show better result.
Peter January 20th, 2011 at 4:18pm
Hi BSJhala, if you want to use trading days then you can no longer reference a 365 day year; you would need to make your interval 4 / 260. Also, in the actual VBA code for Black and Scholes you would need to change the other references to a 365 day year.
ATM/OTM options will have lower market prices than the ITM options hence the price changes as a result of the delta may actually mean a larger "percentage" change in their value. For example, say ITM option has a price of 10 with a delta of 1, while an OTM option has a price of 1 with a delta of 0.25. If the market moves up 1 point, the ITM option will gain only 10% while the OTM option gains 25%. Is this what you are referring to?
The Risk Free Interest rate refers to the "cost of your money" - i.e. what rate do you need to borrow money to invest? Usually, traders just enter the current bank cash rate.
Let me know if anything is unclear.
BSJhala January 20th, 2011 at 9:06am
Dear peter,
I am not clear on your comment on time diff to be used.
Clarify If black scholes model is used and let today date is 20/jan/2011 and date of expiry is 27/jan/2011:
If normal calculation is done time should be 6/365, but trading days are 4 only than it should be 4/365 what should be used.
Also pls tell what should be risk free interest rate .
One more thing pls tell when market is running ,the option value changes frequently that time the variables that is varying should be stock price .
But why the ATM call premium is increasing than the ITM call premium where delta value is close to 1.
What is causing the ATM/OTM calls to changing more than ITM call.
Correct me if I am wrong anywhere
Peter January 19th, 2011 at 4:44pm
If it is the standard Black and Scholes Model then you would use calendar days as the formula will use 365 in the calculations. You can, however, modify the formula yourself and use your own trading day calendar of days.
The likely reason for the difference between your calculated prices and the actual prices is the volatility input that you use. If your volatility input into the model is based on historical prices and you notice that the actual option prices are higher than your calculated prices then this tells you that the market "implied" volatility is higher than the historical; i.e. that the professionals expect volatility to be at higher than historical levels. But, it could also mean that your other parameter inputs are not correct, such as Interest Rates, Dividends etc.
Your best bet at deriving the prices more closely, assuming all the other inputs are correct, is to change the volatility input.
BSJhala January 19th, 2011 at 11:05am
What should be the time(in years).
Should it be simply the date difference between today date and expiration date. Or it should be the trading days difference between today and expiration date.
Why actual prices are different from calculated prices.
How can we derive the prices closely .
Peter December 5th, 2010 at 5:03pm
Thanks for the feedback Tony!
For the expiration...if you want the Friday to be counted in the valuation of the option then you need to enter the Saturday as the expiration date when using Excel. This is because if you enter Friday's date and then this date is subtracted from today's date the last day is not included in the time calculation.
i.e. 27th - 26th = 1 day. Although in trading terms there are actually two days of trading left.
Know what I mean?
Tony December 4th, 2010 at 11:19am
I've working with both your historical volatility and Black Scholes sheets. Thank you for these tools. They are well written, very fast and I sincerely appreciate your level of technical detail.
1. What date should be used for option expiration?
The Friday date or the Saturday date? For example expiration dates are currently 12/17/2010 for Friday and saturday when all is settled is 12/18/2010.
Peter October 13th, 2010 at 12:44am
Yes, you just set the Dividend Yield to the same value as the Interest Rate. This will make the forward price used for the calculation the same as the base price but still use the Interest Rate to discount the premium.
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