Black-Scholes Model

What is a Black-Scholes model?

After we've discovered some of the basic strategies for an option trading, it is time to discover one more important aspect in theory of options trading, which is a Black-Scholes Model.

Call Option Case

Here is an expression of this model, which computes price of a call contract:

Here t - option's start time, S_t - price of an asset at moment t, K - strike price, T - moment of option's expiration, σ - volatility of asset, r - risk-free interest rate, N - cumulative distribution function of a standard gaussian distribution.

Also, d_+ could be computed, using the next expression:

While d_- is computed as follows:

Note also, that as long as in all expression present only T-t, but not T or t separately, it all comes down to the point, where only length of time period matters, so we can assume that t=0 and use only T, which now corresponds to the length of time period.

Here is a code implementation:


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# Importing necessary libraries
import numpy as np
from scipy.stats import norm
# Defining corresponding function `black_scholes_call`
def black_scholes_call(S, K, T, r, σ):
    d_plus = (np.log(S / K) + (r + 0.5 * σ ** 2) * T) / (σ * T ** 0.5)
    d_minus = d_plus - σ * T ** 0.5
    return S * norm.cdf(d_plus) - K * np.exp(-r * T) * norm.cdf(d_minus)
# Defining all necesaary variables: starting price `S` equals 110, strike price `K` equals 100, length of period `T` is 9 month or 0.75 of a year, risk-free interest rate `r` is 0.04 and volatility `σ` is 0.15
S = 110
K = 100
T = 0.75
r = 0.04
σ = 0.15
# Printing estimated price
print(black_scholes_call(S, K, T, r, σ))

Put Option Case

In case of Put contract, with a previously defined notation, the next expression is used:

Or, it can be rewrited in the next way, showing it's connection with corresponding call option's price:

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Section 2. Chapter 4