Black-Scholes Formula

This formula is representing a options trading framework used to evaluate options.

The parameters that we care about when trying to vealuate the price of the stock option are:

• $S_0$ - stock price
• $X$ - exercise price
• $r$ - risk-free rate
• $T$ - time to expiration (maturity)
• $\sigma$ - standard deviation of log returns (volatility)

Let us use $Call_{european}$ for present value of European call option contract and $Put_{european}$ for present value of European put option contract.

$Call_{european} = S_0 \cdot N(d_1) - X \cdot e^{-rT} \cdot X \cdot N(d_2)$

$Put_{european} = - e^{-rT} \cdot X \cdot N(-d_2) - S_0 \cdot N(-d_1)$

where,

$d_1 = \frac{ln(\frac{S_0}{X}) + (r + \frac{\sigma^2}{2}) \cdot T}{\sigma \cdot \sqrt{T}}$

$d_2 = \frac{ln(\frac{S_0}{X}) + (r - \frac{\sigma^2}{2}) \cdot T}{\sigma \cdot \sqrt{T}}$

$N(x)$ - Standard normal distribution - probability that a random variable variable is less or equal to x. It will give a value in range (0,1).

European Call Option

A European call option gives the owner the right to acquire the underlying security at expiry. For an investor to profit from a call option, the stock’s price, at expiry, has to be trading high enough above the strike price to cover the cost of the option premium.

European call option can only be exercised one time and that is on the day of expiration.

$Call_{european} = S_0 \cdot N(d_1) - X \cdot e^{-rT} \cdot X \cdot N(d_2)$ –> $Call_{european} = WhatYouGet - WhatYouPay$

$S_0 \cdot N(d_1)$ -> What you get

$X \cdot e^{-rT} \cdot X \cdot N(d_2)$ –> What you pay

$X \cdot e^{-rT}$ -> exercise price dsicounted back to today (present value of the exercise price)

The higher volatility $\sigma$ then $d_1$ will go up and $d_2$, so the value of $S_0 \cdot N(d_1)$ will go up and the value of $S_0 \cdot N(d_1)$ will go down.

Hence,the value of the European call option will go up as the volatility goes up and vice versa.