Black-Scholes Formula for Valuing European Options. HP 12C Platinum Financial Calculator, 12c

194 Section 13: Investment Analysis

Black-Scholes Formula for Valuing European Options

This program implements the Black-Scholes formula which has been used extensively in option markets worldwide since its publication in the early 1970’s.

The five inputs are simply keyed into the five financial variables and then t displays the call option value, and ~ shows the put option value. The option values produced are accurate to at least the nearest cent for asset and strike prices under $100.

Reference: Tony Hutchins, 2003, Black-Scholes takes over the HP12C, HPCC

(www.hpcc.org) Datafile, V22, N3, pp13-21.

KEYSTROKES

(RPN mode)

DISPLAY

KEYSTROKES

(ALG mode)

DISPLAY fs fs fCLEARÎ

:n

000, fCLEARÎ

001, 45 11 :n

000,

001, 45 11

:¼ b

Þ g>

:M

§

?4

~ gr

:P b

?3

:$

:4 z

002, 45 12 §

003, 25 :¼

004, 16 b

005, 43 22 }

006, 45 15 Þ

007, 20 g>

008, 44 4 §

009, 34 :M

010, 43 21 }

011, 45 14 ?4

012, 25 :n

013, 44 3 gr

014, 45 13 §

015, 45 4 :P

016, 10 b

002, 20

003, 45 12

004, 25

005, 36

006, 16

007, 43 22

008, 20

009, 45 15

010,

011, 44 4

012, 45 11

013, 43 21

014,

016,

36

20

015, 45 14

25

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Printed Date: 2005/8/1 Dimension: 14.8 cm x 21 cm

Section 13: Investment Analysis 195

KEYSTROKES

(RPN mode) g°

DISPLAY

KEYSTROKES

(ALG mode)

017, 43 23 }

DISPLAY

017, 36

~ 018, 34 ?3 018, 44 3 z gF

019, 10 :$

020, 43 40 z

019, 45 13

020, 10

021, 45 4 2

?5 z

+

?6

:3

-

?3

\

§ gr gF

021, 2 :4

022, 44 5 }

023, 10 g°

024, 40 z

025, 44 6 ~

026, 45 3 }

027, 30 :3

028, 44 3 z

029,

030,

36

20

031, 43 21

2

?5

+

032, 43 40 ~

022,

023, 43 23

024,

025,

026,

027, 45 3

028,

029,

36

10

34

36

10

2

030, 44 5

031, 40

2 z

Þ g>

~

3

.

0

033,

034,

037,

038,

039,

040,

2

10

34

3

48

0

-

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2

?6

035, 16 :3

036, 43 22 }

032, 34

033, 30

034, 44 6

035, 45 3

036,

037, 44 3

038, 43 20

039,

36

10

040, 2

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196 Section 13: Investment Analysis

KEYSTROKES

(RPN mode)

0

DISPLAY

KEYSTROKES

(ALG mode)

041, 0 }

DISPLAY

041, 36

6 z

1

+ y

§ gF

042, 6 Þ

043, 10 g>

044, 1 :3

045, 40 g¡

046, 22 gr

047, 20 z

042,

043, 43 22

044, 45 3

045, 43 20

046, 43 21

047,

16

10

048, 3 gF

1

8

7

§

048, 43 40 3

049, 43 40 .

050, 1 0

051, 8 0

052, 7 6

053, 20 +

049,

050,

051,

052,

053,

48

0

0

6

40

2

4

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8

7

+

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.

2 b

054,

055,

056,

2

4

30

1

} y

060, 40 }

061, 20 1

062, 48 8

063, 2 7

064, 25 §

057, 20 ?2

058, 8 §

059, 7 ~

054, 1

055, 36

056, 22

057, 44 2

058, 20

059,

060,

061,

062,

063,

064,

34

36

1

8

7

20

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Section 13: Investment Analysis 197

KEYSTROKES

(RPN mode)

:3

~

DISPLAY

KEYSTROKES

(ALG mode)

065, 45 3 :2

066, 34 -

DISPLAY

065, 45 2

066, 30

?3

O

~ go g(077

1

?-3

Þ

067, 44 3 2

068, 35 4

069, 34 §

070, 43 34 :2

071,43,33,077 +

067, 2

068, 4

069,

071,

20

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40

072, 8

073, 7

?§3

~

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O

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074, 16 §

075,44 20 3 ~

076, 34 §

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078, 43 35 2

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080, 45 6 }

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2

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35

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~ g(028

085, 35 ~

086, 44 5 go

087, 34 g(093

088,43, 33, 028 1

085, 34

086, 43 34

087,43, 33, 093

088, 1

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198 Section 13: Investment Analysis

KEYSTROKES

(RPN mode)

~

DISPLAY

KEYSTROKES

(ALG mode)

089, 34 ?-3

DISPLAY

089,44 30 3

:3

:$

?-4

§

:6

-

?+4

:4

~

090, 45 3 Þ

091, 45 13 ?§3

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095, 30

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098, 34 g(106

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35

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109,

111,

20

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30

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Section 13: Investment Analysis 199

KEYSTROKES

(RPN mode)

DISPLAY

KEYSTROKES

(ALG mode)

}

DISPLAY

113, 36

?+4 114,44 40 4

:4

~

115, 45 4

116, 34

?5 117, 44 5 g(000 118,43,33, 000 fs

REGISTERS n: Term to expiry i: Interest rate (%) PV: Stock price

FV: Strike price R

0:

Unused R

1

: Unused

R

3

: N(d

1

) R

4

: Put value

R

7

-R

.9

R

5

: Call value

PMT: Volatility (%)

R

2

: Unused

R

6

: QxN(d

2

)

Note:

The n, i and PMT values must all be based on the same time unit (for example: n is measured in years or months and i and PMT are rates per year or per month). i is a continuous percentage rate. PMT is the standard deviation of the continuous percentage stock return (as observed over the

time unit). For sensible output, all inputs should be positive. The PMT=0 case can be simulated by using a PMT arbitrarily close to 0.

Program Instructions

1. Key in the program.

2. Enter the five inputs into the five financial registers. These values are preserved by the program. a. Key in the unexpired term of the option and press n. b. Key in the risk-free interest rate as a percentage and press ¼. c. Key in the current (or spot) stock price and press $. d. Key in the volatility assumption as a percentage and press P. e. Key in the strike price and press M.

3. Press t. The Call value is displayed. Press ~ to see the Put value.

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200 Section 13: Investment Analysis

Example 1: An option has 6 months to run and a strike price of $45. Find

Call and Put values assuming a spot price of $52, return volatility of 20.54% per month and a risk-free interest rate of 0.5% per month. Show how to change the time scale of the inputs between monthly and annual values.

Keystrokes

(RPN mode) f]

6n

.5¼

52$

20.54P

45M t

~

Keystrokes

(ALG mode) f[

6n

.5¼

52$

20.54P

45M t

~

:gAn :gAn

:gC¼

:P

12gr§P t

:ngA :ngA

:¼gC :¼gC

:P

12grzP

:gC¼

:P§

12grP t

:Pz

12grP

Display

6.00

0.50

52.00

20.54

45.00

6.00

0.50

14.22

5.89

0.50

6.00

71.15

14.22

20.54

Time to expiry (months).

Interest rate (% per month).

Stock price.

Volatility (% per month).

Strike price.

Call value.

Put value.

Years to expiry.

Yearly interest rate %.

Yearly volatility %.

Call value (unchanged).

Months to expiry.

Monthly interest rate %.

Monthly volatility %.

The next example is Example 12.7 from Options, Futures, and Other Derivatives

(5th Edition) by John C. Hull (Prentice Hall, 2002).

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Section 13: Investment Analysis 201

Example 2: The stock price six months from the expiration of an option is $42, the exercise price of the option is $40, the risk-free interest rate is 10% per annum, and the volatility is 20% per annum. Find Call and Put values.

Keystrokes

(RPN mode) f]

.5n

10¼

42$

20P

40M t

~

Keystrokes

(ALG mode) f[

.5n

10¼

42$

20P

40M t

~

Display

0.50

10.00

42.00

20.00

40.00

4.76

0.81

Time to expiry (years).

Interest rate (% per year).

Stock price.

Volatility (% per year).

Strike price.

Call value.

Put value.

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