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)

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