Option buyers choose volatility to reduce premium

The implied volatility plays a crucial role in the Black-Scholes equation to calculate the option prices:

For a European call option:

C = S₀ × N(d₁) − K × e^(−rT) × N(d₂)

For a European put option:

P = K × e^(−rT) × N(−d₂) − S₀ × N(−d₁)

Where:

• (C) = Call option price
• (P) = Put option price
• (S₀) = Current price of the underlying asset
• (K) = Strike price
• (r) = Risk-free interest rate (annualized)
• (T) = Time to maturity (in years)
• (σ) = Volatility of the underlying asset (annualized)
• (N(x)) = Cumulative distribution function (CDF) of the standard normal distribution

Also, the coefficients (d₁) and (d₂):

d₁ = [ln(S₀ / K) + (r + (½) × σ²) × T] / (σ × √T),
d₂ = d₁ − σ × √T

As it is clear from the abovementioned formula, if we decrease the σ (volatility), the d₁ and d₂ increase respectively. However, the key point here is that even if N(x) increases, the strike N(d₂) increases more than the spot N(d₁), and the overall premium decreases.

Also, in the case where N(d₂) becomes more than N(d₁), the contract makes the premium become zero:

        call = strikeNd2 <= spotNd1 ? spotNd1 - strikeNd2 : 0