Options Pricing: Black-Scholes Model.
The Black-Scholes formula (also called Black-Scholes-Merton) was the first widely used model for option pricing. It's used to calculate the theoretical value of European-style options using current stock prices, expected dividends, the option's strike price, expected interest rates, time to expiration and expected volatility.
The formula, developed by three economists – Fischer Black, Myron Scholes and Robert Merton – is perhaps the world's most well-known options pricing model. It was introduced in their 1973 paper, "The Pricing of Options and Corporate Liabilities," published in the Journal of Political Economy . Black passed away two years before Scholes and Merton were awarded the 1997 Nobel Prize in Economics for their work in finding a new method to determine the value of derivatives (the Nobel Prize is not given posthumously; however, the Nobel committee acknowledged Black's role in the Black-Scholes model).
The Black-Scholes model makes certain assumptions:
The option is European and can only be exercised at expiration. No dividends are paid out during the life of the option. Markets are efficient (i. e., market movements cannot be predicted). There are no transaction costs in buying the option. The risk-free rate and volatility of the underlying are known and constant. The returns on the underlying are normally distributed.
Note: While the original Black-Scholes model didn't consider the effects of dividends paid during the life of the option, the model is frequently adapted to account for dividends by determining the ex-dividend date value of the underlying stock.
Black-Scholes Formula.
The formula, shown in Figure 4, takes the following variables into consideration:
current underlying price options strike price time until expiration, expressed as a percent of a year implied volatility risk-free interest rates.
The model is essentially divided into two parts: the first part, SN(d1) , multiplies the price by the change in the call premium in relation to a change in the underlying price. This part of the formula shows the expected benefit of purchasing the underlying outright. The second part, N(d2)Ke - rt , provides the current value of paying the exercise price upon expiration (remember, the Black-Scholes model applies to European options that can be exercised only on expiration day). The value of the option is calculated by taking the difference between the two parts, as shown in the equation.
The mathematics involved in the formula are complicated and can be intimidating. Fortunately, you don't need to know or even understand the math to use Black-Scholes modeling in your own strategies. As mentioned previously, options traders have access to a variety of online options calculators, and many of today's trading platforms boast robust options analysis tools, including indicators and spreadsheets that perform the calculations and output the options pricing values. An example of an online Black-Scholes calculator is shown in Figure 5. The user inputs all five variables (strike price, stock price, time (days), volatility and risk free interest rate) and clicks "get quote" to display results.
Black-Scholes Calculator.
$25 one-time payment.
How It Works & Screenshots.
Enter parameters in the yellow cells: underlying price, strike price, volatility, interest rate, dividend yield. The user guide provides detailed explanation of each.
You can set time to expiration either as pricing date and expiration date, or as number of days remaining to expiration. The calculator also works with fractions of days to handle intraday pricing.
You will instantly see the resulting option price and Greeks . The Greeks, like delta, gamma, theta or vega, measure sensitivity of option prices to changes in the individual parameters and therefore are very useful for managing option positions. You can find detailed explanation of the Greeks in the user guide.
For even better understanding of your option’s exposure to different factors you can see the effects of changing underlying price, volatility or time to expiration in the charts .
For instance, the screenshot below shows the effect of time to expiration on a call option’s price and delta, demonstrating how this particular option will lose value as it gets closer to expiration.
You can control the charts in the chart settings area on the left.
The charts can display effects of any of the parameters on an option’s price and/or any of the Greeks.
You can also easily adjust the scale to zoom in or out .
The calculator helps you stay in control , understand the risks and behaviour of a position under any possible situation, and make faster and better decisions .
$25 one-time payment.
User Guide.
Besides detailed step-by-step instructions for using the calculator, the guide also explains the assumptions and theoretical background of the Black-Scholes option pricing model, provides all the formulas for option prices and Greeks, and explains the particular Excel implementation.
Black-Scholes Calculator Guide Contents.
Quick Start …3 Main Sheet Overview …6 Calculating Option Prices …7 The Greeks …12 Simulations and Charts …16 Estimating Volatility …21 Option Pricing Model and Assumptions …24 Formulas Used …27 Calculation Area and Functions Used …29 Common Technical Problems …32 References …33 Contact and Terms of Use …34.
More assistance is available via support.
$25 one-time payment.
Frequently Asked Questions.
Is it a one-time payment or monthly/recurring?
One-time payment, yours forever.
Does it work in my version of Excel?
The calculator works in all versions of Excel from Excel 97 to Excel 2016. It was developed in Excel 2010 and tested in other versions. For older versions you may need to use a different version of the calculator, which is also included.
Does it work in OpenOffice / LibreOffice / Apple Numbers / other spreadsheet software?
It may work in some, but unfortunately we can’t guarantee it and can’t provide support for software other than Microsoft Excel.
Are the formulas freely available?
Yes. The calculator only uses the basic built-in Excel formulas or their combinations. Everything is freely available, nothing hidden or password protected. You are free to change any formulas and customize the calculator.
You can pay by credit/debit card or PayPal. All payments are processed by PayPal, which provides world-class security and buyer protection. You don’t need a PayPal account to check out when paying with a card.
I have other questions / need more information.
$25 one-time payment.
Related Calculators – Often Bought Together.
Implied Volatility Calculator – Does the inverse of the Black-Scholes Calculator: Calculates IV from option prices and helps you understand the essential volatility input.
Option Strategy Payoff Calculator – Good to learn about individual option strategies and their payoffs at expiration. Draws payoff diagrams; calculates max profit, max loss, risk-reward ratios and break-even points.
By remaining on this website and/or using Macroption content, you confirm that you have read and agree with the Terms of Use Agreement just as if you have signed it. The Agreement also includes Privacy Policy and Cookie Policy. If you don't agree with any part of this Agreement, please leave the website now. All information is for educational purposes only and may be inaccurate, incomplete, outdated or plain wrong. Macroption is not liable for any damages resulting from using the content. No financial, investment or trading advice is given at any time.
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Black-Scholes Excel Formulas and How to Create a Simple Option Pricing Spreadsheet.
This page is a guide to creating your own option pricing Excel spreadsheet, in line with the Black-Scholes model (extended for dividends by Merton). Here you can get a ready-made Black-Scholes Excel calculator with charts and additional features such as parameter calculations and simulations.
Black-Scholes in Excel: The Big Picture.
If you are not familiar with the Black-Scholes model, its parameters, and (at least the logic of) the formulas, you may first want to see this page.
Below I will show you how to apply the Black-Scholes formulas in Excel and how to put them all together in a simple option pricing spreadsheet. There are 4 steps:
Design cells where you will enter parameters. Calculate d1 and d2. Calculate call and put option prices. Calculate option Greeks.
Black-Scholes Parameters in Excel.
First you need to design 6 cells for the 6 Black-Scholes parameters. When pricing a particular option, you will have to enter all the parameters in these cells in the correct format. The parameters and formats are:
S 0 = underlying price (USD per share)
X = strike price (USD per share)
r = continuously compounded risk-free interest rate (% p. a.)
q = continuously compounded dividend yield (% p. a.)
t = time to expiration (% of year)
Underlying price is the price at which the underlying security is trading on the market at the moment you are doing the option pricing. Enter it in dollars (or euros/yen/pound etc.) per share.
Strike price , also called exercise price, is the price at which you will buy (if call) or sell (if put) the underlying security if you choose to exercise the option. If you need more explanation, see: Strike vs. Market Price vs. Underlying Price. Enter it also in dollars per share.
Volatility is the most difficult parameter to estimate (all the other parameters are more or less given). It is your job to decide how high volatility you expect and what number to enter – neither the Black-Scholes model, nor this page will tell you how high volatility to expect with your particular option. Being able to estimate (= predict) volatility with more success than other people is the hard part and key factor determining success or failure in option trading. The important thing here is to enter it in the correct format, which is % p. a. (percent annualized).
Risk-free interest rate should be entered in % p. a., continuously compounded. The interest rate’s tenor (time to maturity) should match the time to expiration of the option you are pricing. You can interpolate the yield curve to get the interest rate for your exact time to expiration. Interest rate does not affect the resulting option price very much in the low interest environment, which we’ve had in the recent years, but it can become very important when rates are higher.
Dividend yield should also be entered in % p. a., continuously compounded. If the underlying stock doesn’t pay any dividend, enter zero. If you are pricing an option on securities other than stocks, you may enter the second country interest rate (for FX options) or convenience yield (for commodities) here.
Time to expiration should be entered as % of year between the moment of pricing (now) and expiration of the option. For example, if the option expires in 24 calendar days, you will enter 24/365=6.58%. Alternatively, you may want to measure time in trading days rather than calendar days. If the option expires in 18 trading days and there are 252 trading days per year, you will enter time to expiration as 18/252=7.14%. Furthermore, you can also be more precise and measure time to expiration to hours or even minutes. In any case you must always express the time to expiration as % of year in order for the calculations to return correct results.
I will illustrate the calculations on the example below. The parameters are in cells A44 (underlying price), B44 (strike price), C44 (volatility), D44 (interest rate), E44 (dividend yield), and G44 (time to expiration as % of year).
Note: It is row 44, because I am using the Black-Scholes Calculator for screenshots. You can of course start in row 1 or arrange your calculations in a column.
Black-Scholes d1 and d2 Excel Formulas.
When you have the cells with parameters ready, the next step is to calculate d1 and d2, because these terms then enter all the calculations of call and put option prices and Greeks. The formulas for d1 and d2 are:
All the operations in these formulas are relatively simple mathematics. The only things that may be unfamiliar to some less savvy Excel users are the natural logarithm ( LN Excel function) and square root ( SQRT Excel function).
The hardest on the d1 formula is making sure you put the brackets in the right places. This is why you may want to calculate individual parts of the formula in separate cells, as I do in the example below:
First I calculate the natural logarithm of the ratio of underlying price and strike price in cell H44:
Then I calculate the rest of the numerator of the d1 formula in cell I44:
Then I calculate the denominator of the d1 formula in cell J44. It is useful to calculate it separately like this, because this term will also enter the formula for d2:
Now I have all the three parts of the d1 formula and I can combine them in cell K44 to get d1:
Finally, I calculate d2 in cell L44:
Black-Scholes Option Price Excel Formulas.
The Black-Scholes formulas for call option (C) and put option (P) prices are:
The two formulas are very similar. There are 4 terms in each formula. I will again calculate them in separate cells first and then combine them in the final call and put formulas.
N(d1), N(d2), N(-d2), N(-d1)
Potentially unfamiliar parts of the formulas are the N(d1), N(d2), N(-d2), and N(-d1) terms. N(x) denotes the standard normal cumulative distribution function – for example, N(d1) is the standard normal cumulative distribution function for the d1 that you have calculated in the previous step.
In Excel you can easily calculate the standard normal cumulative distribution functions using the NORM. DIST function, which has 4 parameters:
NORM. DIST(x, mean, standard_dev, cumulative)
x = link to the cell where you have calculated d1 or d2 (with minus sign for - d1 and - d2) mean = enter 0, because it is standard normal distribution standard_dev = enter 1, because it is standard normal distribution cumulative = enter TRUE, because it is cumulative.
For example, I calculate N(d1) in cell M44:
Note: There is also the NORM. S.DIST function in Excel, which is the same as NORM. DIST with fixed mean = 0 and standard_dev = 1 (therefore you enter only two parameters: x and cumulative). You can use either; I’m just more used to NORM. DIST, which provides greater flexibility.
The Terms with Exponential Functions.
The exponents (e-qt and e-rt terms) are calculated using the EXP Excel function with - qt or - rt as parameter.
I calculate e-rt in cell Q44:
Then I use it to calculate X e-rt in cell R44:
Analogically, I calculate e-qt in cell S44:
Then I use it to calculate S0 e-qt in cell T44:
Now I have all the individual terms and I can calculate the final call and put option price.
Black-Scholes Call Option Price in Excel.
I combine the 4 terms in the call formula to get call option price in cell U44:
Black-Scholes Put Option Price in Excel.
I combine the 4 terms in the put formula to get put option price in cell U44:
Black-Scholes Greeks Excel Formulas.
Here you can continue to the second part, which explains the formulas for delta, gamma, theta, vega, and rho in Excel:
Or you can see how all the Excel calculations work together in the Black-Scholes Calculator. Explanation of the calculator’s other features (parameter calculations and simulations of option prices and Greeks) are available in the calculator’s user guide.
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