Question

The Black-Scholes-Merton model’s implied volatility is:

Group of answer choices

the market’s estimate of the stock’s random volatility over an infinitesimal time interval beginning when the option matures

the volatility that equates the BSM model price to the market price, if all other inputs are known

the market’s estimate of the future value of the stock’s random volatility over the option’s life

the market’s estimate of the future value of the stock’s random volatility over an infinitesimal time interval

please provide explanation

Answer 1

The Black-Scholes-Merton model’s implied volatility is the volatility that equates the BSM model price to the market price, if all other inputs are known

Therefore correct answer is option: the volatility that equates the BSM model price to the market price, if all other inputs are known

Implied volatility is based on the changes in option premium and shows expected future volatility of the underline stock. It is different from the historical volatility which is based on the historical price changes of an underlying stock over a certain time period. The implied volatility is the only input of the model that cannot be directly calculated. The Black-Scholes-Merton equation is used to determine the implied volatility. The Black-Scholes-Merton equation is following –

Following is the formula to calculate the value of call option under the Black-Scholes – Merton Model

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

Formula to calculate d1 and d2 are -

d1 = {ln (S/K) +(r+ σ^2 /2)* t}/σ *√t

d2 = d1 – σ *√t Therefore           

Where,

C = call value

S = current stock price

N = cumulative standard normal probability distribution

t = days until expiration

σ = Implied volatility

K = option strike price

r = risk free interest rate