Volatility

Implied Volatility

3 Topics
Option Greeks

Effects on Greeks

Options Pricing Model

Standard Deviation in Options

- 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 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:

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.

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

- 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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