**This Black-Scholes Calculator is a sophisticated financial tool used to determine the theoretical price of European-style options. **

Named after economists **Fischer Black** and **Myron Scholes**, who developed the underlying model in 1973, this calculator has become an indispensable instrument in the world of finance and derivatives trading.

At its core, this calculator **Black-Scholes-Merton model** (BSM), which revolutionized the field of quantitative finance. This model provides a mathematical framework for estimating the fair value of options, taking into account various factors that influence option pricing.

## Online Black Scholes Calculator

Let’s do some sample calculations. We’ll use the Black-Scholes formula for call options:

C = S *N(d1) – K* e^(-r *t)* N(d2)

For these calculations, we’ll use the following parameters:

- S (Current stock price): $100
- r (Risk-free rate): 5% (0.05)
- σ (Volatility): 20% (0.20)
- t (Time to expiration): 1 year

We’ll vary the strike price (K) to see how it affects the option price.

Strike Price (K) | d1 | d2 | N(d1) | N(d2) | Call Option Price |
---|---|---|---|---|---|

$90 | 0.7693 | 0.5693 | 0.7791 | 0.7154 | $17.39 |

$95 | 0.5409 | 0.3409 | 0.7057 | 0.6334 | $13.65 |

$100 | 0.3300 | 0.1300 | 0.6293 | 0.5517 | $10.45 |

$105 | 0.1349 | -0.0651 | 0.5536 | 0.4740 | $7.83 |

$110 | -0.0464 | -0.2464 | 0.4815 | 0.4027 | $5.75 |

**Black Scholes Calculation Formula**

The **Black-Scholes formula** is the mathematical engine behind the calculator. It’s expressed as follows:

For a **call option**: C = S *N(d1) – K* e^(-r *t)* N(d2)

For a **put option**: P = K *e^(-r* t) *N(-d2) – S* N(-d1)

Where:

**C**= Call option price**P**= Put option price**S**= Current stock price**K**= Strike price**r**= Risk-free interest rate**t**= Time to expiration (in years)**N**= Cumulative normal distribution function

The variables **d1** and **d2** are calculated as:

d1 = [ln(S/K) + (r + σ^2/2) *t] / (σ* √t) d2 = d1 – σ * √t

Where:

**σ**(sigma) = Volatility of the underlying asset

This formula appears complex at first glance, but each component plays a crucial role in determining the option’s value. The **stock price** (S) and **strike price** (K) establish the option’s intrinsic value.

The **risk-free rate** (r) accounts for the time value of money.

**Volatility** (σ) represents the expected fluctuations in the stock price, while **time to expiration** (t) captures the option’s remaining lifespan.

**What is the Black-Scholes time formula?**

The **Black-Scholes time formula** isn’t a separate equation but rather refers to how time is incorporated into the Black-Scholes model. Time plays a critical role in option pricing, and its effect is captured through several components of the formula:

**Exponential decay**: The term e^(-r*t) in the formula represents the *present value factor*. As time to expiration (t) increases, this factor decreases, reflecting the diminishing present value of the strike price.**Volatility scaling**: Time appears in the calculation of d1 and d2 as √t (square root of t). This scaling factor adjusts the impact of volatility based on the option’s remaining lifespan.**Drift component**: In the numerator of d1, the term (r + σ^2/2) * t represents the expected drift in the stock price over time, accounting for both the risk-free rate and volatility.

## What is d1 and d2 in the Black-Scholes model?

**d1 in the Black-Scholes model:**

d1 = [ln(S/K) + (r + σ^2/2) *t] / (σ* √t)

d1 is a crucial component in the Black-Scholes formula. It represents the factor by which the present value of the stock price exceeds the present value of the strike price. More specifically:

- It measures how many standard deviations the stock price is above the strike price.
- It’s used to calculate N(d1), which represents the delta of the option (the rate of change of the option price with respect to the underlying asset price).
- d1 incorporates all the inputs of the model: stock price, strike price, risk-free rate, volatility, and time to expiration.

**d2 in the Black-Scholes model:**

d2 = d1 – σ * √t

d2 is closely related to d1 and has its own significance:

- It’s used to calculate N(d2), which represents the probability that the option will be exercised in a risk-neutral world.
- d2 can be interpreted as a risk-adjusted measure of the likelihood that the option will be in-the-money at expiration.
- The difference between d1 and d2 (σ * √t) accounts for the volatility effect over the option’s lifetime.

In the Black-Scholes formula:

- N(d1) determines the expected benefit from acquiring the stock outright.
- N(d2) determines the present value of paying the exercise price at expiration.

Together, d1 and d2 help capture the complex relationships between all the factors affecting option prices. They’re essential for accurately pricing options and understanding their behavior under different market conditions.

The values of d1 and d2 (and consequently N(d1) and N(d2)) directly influence the calculated option price. As you can see from the table, as the strike price increases (moving from in-the-money to out-of-the-money options), both d1 and d2 decrease, leading to lower call option prices.