In other articles we built binary tree model with three steps for the call option. Increasing the number of steps, i.e. reducing the time interval between steps, will lead to an absolutely similar result, which can be obtained by using Black-Scholes model, widely known in the financial industry. The model was developed by academics Black, Scholes and Merton in the seventies of the twentieth century and is now widely used both in the academic environment, and in leading financial corporations.

For European options, the Black-Scholes model looks like this:

C = SN(d_{1}) – Xe^{-rt}N(d_{2})

P = Xe^{-rt}N(–d_{2}) – SN(–d_{1})

Where:

C - call option value

P - value of put option

S - spot of the underlying asset

X - strike

N(d) - function of standard normal distribution

ln(x) - natural logarithm with base e

σ - volatility of the underlying asset in annual terms

t - time before expiration, expressed in years

r - risk-free interest rate

The formula for a call option shows that the value of the call (C) is calculated as the difference between the spot price (S) of the underlying asset and the discounted strike value (X); the S and X values are weighted using risk factors N(d1) and N(d2). As in the binary method, the formula is based on the assumption that options can be freely delta-hedged through trading on the underlying asset, also depositing and borrowing at a risk-free interest rate should be possible (Read Delta hedging: analysis and examples). The Black Scholes model implies a normal distribution of returns on the underlying asset.

The N(d2) factor measures the probability that the call option will expire in the money and be exercised. While the N(d1) factor is the option delta, i.e. the hedging factor. Usually, the N(d2) function counts the area to the left of d2 under the normal distribution curve with an average value of 0 and deviation 1. Thus, N(d2) calculates the probability that the variable, which has the standard normal distribution, will be below d2.

**Example**

In other article, we looked for the value of a European option using a binary tree model with data:

Underlying asset price C = $300

Strike price X = $250

Risk-free interest rate r = 10%

Time to expiration t = 0.25 years

Volatility σ = 40%

According to the Black-Scholes model, the cost of a call option is:

C = $300 × 0.8721 - $250 × e^(-0.1 x 0.25 × 0.8255) = $60.36

Where:

d1 = (ln($300/$250)+(0.1× 0.25)+(0.42 × 0.25/2))/(0.4 ×√0.25) = 1.1366

d2 = 1,1366 – (0,4 ×√0,25) = 0,9366

N(d1) = 0.8721

N(d2) = 0.8255

The risk-neutral probability that option will be exercised is 82.55% as the option is initially in-the-money.

**Black-Scholes with dividends**

The model can be adjusted for assets that pay regular dividends. The formula described below assumes that dividends are paid on a continuous basis, i.e. paid to the owner of the underlying asset every second. Let us denote the dividend yield by the letter q:

C = Se^{-qt}N(d_{1}) – Xe^{-rt}N(d_{2})

P = Xe^{-rt}N(–d_{2}) – Se^{-qt}N(–d_{1})

Where:

In the context of an individual share, the assumption of a continuous dividend payment is unrealistic, as issuers typically pay dividends either once a year or every six months. Therefore, to address this issue, we can replace the spot price of the asset with the spot price less the discounted value of the dividend payout expected during the period of the option. Dividend payments should be discounted at a risk-free interest rate.