**Introduction**

The Black Scholes-Merton model, often known as the Black-Scholes model, is a model to estimate the call and put options. Even though this model is designed for European options, the model is used on options worldwide. The formula to calculate the call option is:

C = S * N(d1) – K * e^(-r * T) * N(d2)

where:

C = call price

S = stock price

K = exercise price

r = risk free rate (percent)

v = implied volatility (standard deviation of log returns) (percent)

(also symbolized as σ)

T = the time until the stock matures, in the measure in years (if you are given the number days, divide this number by 365 (or 365.25, 366, 360))

N(d1) and N(d2): lower tail probability of the Normal Distribution of points d1 and d2, respectively. (NormCDF( -∞,d1) and NormCDF(-∞,d2))

d1 and d2:

d1 = ( ln(S/K) + ( r^2 + v^2/2 ) * T ) / ( v * √T )

d2 = d1 – v * √T

Normal Distrubtion:

NormCDF(a, b) = 1 / √(2 * π) * ∫( e^(-t^2/2) dt, a, b)

Also note:

Expected Stock Value = S * N(d1)

Expected Strike Value = K * e^(-r * T) * N(d2)

Call Price = Expected Stock Value – Expected Strike Value

**HP Prime: Black-Scholes Model**

There is no need to enter a program to calculate Call Price and Put Price. The Black-Scholes Model is in the Finance app. Select Black-Scholes on the Symb page. Enter and solve for call and put price on the Num screen.

**Casio fx-CG50 Program BLACKSCH**

This program calculates the expected stock value, expected strike value, and the call price.

“BLACK SCHOLES”

“STOCK PRICE: “? → S

“STRIKE PRICE: “? → K

“TIME-EXPIR. (DAYS): “? → T

T ÷ 365.25 → T

“RISK FREE %: “? → R

R ÷ 100 → R

“VOLITALITY σ%: “? → V

V ÷ 100 → V

(ln (S ÷ V) + (R + V^2 ÷ 2 ) * T) ÷ ( V * √T ) → D

D – V * √T → E

S * NormCD(-10,D) → W

K * e^(-R * T) * NormCD(-10, E) → X

W – X → C

“EXEPECTED STOCK VALUE”

RndFix(W,2) ⊿

“EXPECTED STRIKE VALUE”

RndFix(X,2) ⊿

“CALL PRICE”

RndFix(C,2)

**Example**

Input:

Stock Price: $55.00

Strike Price: $41.00

Days to Expiration: 90 days (90/365.25 ≈ 0.25 year)

Risk Free Rate: 2.5%

Volatility Rate: 25%

Output:

Expected Stock Value: $54.64

Expected Strike Value: $40.37

Call Option: $14.27

Source:

Black-Scholes-Merton Brilliant.org. Retrieved April 24, 2019, from https://brilliant.org/wiki/black-scholes-merton/

Eddie

All original content copyright, © 2011-2019. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.