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# HP 17Bii, HP 17Bii+ and 19Bii Calculator - Black Scholes Equation

Black-Scholes Option Valuation
The equation in this document uses the theoretical model developed by Fisher Black and Myron Scholes to estimate the value of options to buy or sell financial assets.
The BLK.SCHLS equation estimates the value of a call or put option. One of the variables in this equation is the standard deviation of the rates of return on the stock. This number is not commonly available. However, the value may be estimated by using one of the following:
• A source that publishes Beta statistics for a stock may also publish the standard deviation (or variance, which is the standard deviation squared) of the stock.
• A sequence of stock prices at uniform time intervals can be used in the SD.ROR equation, explained later in this document, to calculate the sample standard deviation of the rates of return.
Entering and using the BLK.SCHLS equation
##### note:
You will need approximately 2500 bytes of calculator memory to enter the BLK.SCHLS equation.
1. Enter the following BLK.SCHLS equation into the Solver. Enter every character shown, except for spaces. All spacing is for readability. Do not enter any spaces into the calculator.
2. Display the BLK.SCHLS equation menu.
3. Store the following variables:
• Stock price per share in {PS}.
• Exercise price of option in {PE}.
• Periodic risk free rate of return as a percentage in {RF%}.
• Number of periods until expiration in {T}.
• Standard deviation, as a decimal, of the periodic returns of the stock in {S}.
4. Press CALLV to calculate the call option value per share of stock.
5. Press PUTV to calculate the put option value per share of stock.
##### note:
The time units for RF%, T and S must be the same. For example, if S (standard deviation) is based on monthly rates of returns, then RF%(the risk free rate) and T (time until expiration) must also be expressed in months.
Example of entering and using the BLK.SCHLS equation
A call option on a stock has an exercise price of 45 and a current price of 52. The standard deviation of monthly rates of return for the stock is .2054, and the monthly T-Bill rate is .5%. The option expires in six months. What is the estimated value of the call?
 Keys Display Description 52, then PS PS=52.00 Stores current stock price. 45, then PE PE=45.00 Stores exercise price. .5, then RF% RF%=0.50 Stores the risk free rate of return. 6, then T T=6.00 Stores time period. .2054, S, then MORE S=0.21 Stores standard deviation of stock rates of return. CALLV CALLV=14.24 Calculates value of call option per share of stock. PUTV PUTV=5.91 Calculates value of put option per share of stock.
Standard deviation of rates of return
The variable S in the BLK.SCHLS equation is the standard deviation of the periodic rates of return of the stock. A sequence of stock prices at uniform time intervals can be used in the SD.ROR equation below to calcuate the sample standard deviation of the rates of return.
##### note:
The equation does not include dividends, which is appropriate if the stock pays either a constant dividend or zero dividend over the period of time that the prices are measured.
Entering and using the SD.ROR equation
The steps for entering and using the SD.ROR equation are shown below:
1. Enter the the following SD.ROR equation into the Solver. Enter every character shown, except for spaces. All spacing is for readability. Do not enter any spaces into the calculator.
2. Enter the stock prices into a SUM list and name the list PRICE.
3. Display the SD.ROR equation menu.
4. Press {S} to calculate the standard deviation of the rates of return of the stock. (You may have to press {S} twice, since the first time you press it the machine may store the value in the display. Pressing {S} a second time will begin the solve calculation.)
Example of entering and using the SD.ROR equation
You have tracked the monthly closing stock prices of a corporation for 61 months. Calculate the sample standard deviaiton of returns for this set of prices.
 Month Price 1 42.000 2 39.875 3 39.750 4 37.875 5 37.375 6 37.875 7 41.750 8 43.125 9 40.375 10 42.750 11 46.625 12 47.625 13 50.125 14 55.000 15 49.125 16 54.875 17 62.750 18 61.500 19 63.875 20 62.625 Month Price 21 57.875 22 63.750 23 67.125 24 68.250 25 64.875 26 61.750 27 63.000 28 63.000 29 65.125 30 56.875 31 57.000 32 52.375 33 49.375 34 52.000 35 57.750 36 60.625 37 61.250 38 64.875 39 67.000 40 75.000 Month Price 41 78.250 42 75.125 43 82.375 44 87.500 45 90.500 46 99.500 47 117.000 48 115.000 49 118.125 50 113.875 51 116.375 52 114.375 53 114.625 54 116.125 55 125.875 56 120.125 57 136.125 58 123.625 59 129.125 60 124.250 61 129.250
##### note:
The solution below assumes that you have the numbers above stored in a SUM or STAT list, named PRICE. As a check, the total of the list should be 4,512.88.
Display the SD.ROR equation menu. Then proceed to the following keystrokes to complete the calculations:
 Keys Display Description {S} {S} S=0.07 Calculates sample standard deviation of the numbers in the SUM list named PRICE. (SHIFT) SHOW 6.59623655232E-2 Shows result to full precision.
Further information
##### note:
One or more of the above Web sites will take you outside of the Hewlett-Packard Web site. HP does not control and is not responsible for information outside of its Web site.

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