## Inputs in Black-Scholes Option Pricing Model Formula

• S0 = underlying price
• X = strike price
• σ = volatility
• r = continuously compounded risk-free interest rate
• q = continuously compounded dividend yield
• t = time to expiration

For,

• σ = Volatility = India VIX has been taken.
• r = 10% (As per NSE Website, it is fixed.)
• q = 0.00% (Assumed No Dividend)

Note: In many resources, you can find different symbols for some of these parameters.

For example,

• The strike price is often denoted `K` (here it is `X`).
• Underlying price is often denoted `S` (without the zero)
• Time to expiration is often denoted `T – t` (difference between expiration and now).

In the original Black and Scholes paper (The Pricing of Options and Corporate Liabilities, 1973) the parameters were denoted x (underlying price), c (strike price), v (volatility), r (interest rate), and t* – t (time to expiration).

The dividend yield was only added by Merton in Theory of Rational Option Pricing, 1973.

## Call and Put Option Price Formulas

Call option `C` and put option `P` prices are calculated using the following formulas:  where `N(x)` is the standard normal cumulative distribution function.

The formulas for `d1` and `d2` are:  ## Original Black-Scholes vs. Merton’s Formulas

In the original Black-Scholes model, which doesn’t account for dividends, the equations are the same as above except:

• There is just `S0` in place of `S0 e-qt`
• There is no `q` in the formula for `d1`

Therefore, if the dividend yield is zero, then `e-qt = 1` and the models are identical.

## Black-Scholes Formulas for Option Greeks

### Delta  ### Theta  … where T is the number of days per year (calendar or trading days, depending on what you are using).

### Gamma ### Vega ### Rho  ## Excel/Google Sheet Formulas for Calculation of Black Scholes Model

• Underlying Price: `B1`
• ATM Strike Price: `B2`
• Today’s Date: `B3`
• Expiry Date: `B4`
• Historical Volatility: `B5`
• Risk-Free Rate: `B6`
• Dividend Yield: `B7`
• DTE (Years): `B8`

d1, d2 Calculation

• d1 = `(LN(B1/B2)+(B6-B7+0.5*B5^2)*B8)/(B5*SQRT(B8))`
• Nd1 = `EXP(-(B10^2)/2)/SQRT(2*PI())`
• d2 = `B10-B5*SQRT(B8)`
• Nd2 = `NORMSDIST(B12)`

Calculation Of Greeks

If You see the above formulas, these are derived directly from those formulas –

• Call Theta = `(-((B1*B5*EXP(-B7*B8))/(2*SQRT(B8))*(1/(SQRT(2*PI())))*EXP(-(B10*B10)/2))-(B6*B2*EXP(-B6*B8)*NORMSDIST(B12))+(B7*EXP(-B7*B8)*B1*NORMSDIST(B10)))/365`
• Put Theta = `(-((B1*B5*EXP(-B7*B8))/(2*SQRT(B8))*(1/(SQRT(2*PI())))*EXP(-(B10*B10)/2))+(B6*B2*EXP(-B6*B8)*NORMSDIST(-B12))-(B7*EXP(-B7*B8)*B1*NORMSDIST(-B10)))/365`
• Call Premium = `EXP(-B7*B8)*B1*NORMSDIST(B10)-B2*EXP(-B6*B8)*NORMSDIST(B10-B5*SQRT(B8))`
• Put Premium = `B2*EXP(-B6*B8)*NORMSDIST(-B12)-EXP(-B7*B8)*B1*NORMSDIST(-B10)`
• Call Delta = `EXP(-B7*B8)*NORMSDIST(B10)`
• Put Delta = `EXP(-B7*B8)*(NORMSDIST(B10)-1)`
• Gamma = `(EXP(-B6*B8)/(B1*B5*SQRT(B8)))*(1/(SQRT(2*PI())))*EXP(-(B10*B10)/2)`
• Vega = `(EXP(-B6*B8)/(B1*B5*SQRT(B8)))*(1/(SQRT(2*PI())))*EXP(-(B10*B10)/2)`
• Call Rho = `(1/100)*B2*B8*EXP(-B6*B8)*NORMSDIST(B12)`
• Put Rho = `(-1/100)*B2*B8*EXP(-B6*B8)*NORMSDIST(-B12)`