Black Scholes Calculator

It is a theoretical value of derivatives and other investment instruments by taking time and other risk factors into account.

You will find the formulas and step-by-step process to solve the Black Scholes value along with a few solved example problems from the below sections. Also, visit onlinecalculator.guide where you will get the step-wise procedure to calculate several other math-related concepts.

Black-Scholes Option Pricing Formula

Here are the formulae used by our balck scholes calculator to calculate option pricing is as follows:

C = SP e^-dt N(d₁) - ST e^-rt N(d₂)

P = ST e^-rt N(-d₂) - SP e^-dt N(-d₁)

d₁ = { ln(SP/ST) + [r - d + (σ²/2)] t } / σ √t

d₂ = ( ln(SP/ST) + (r - d - (σ²/2)) t ) / σ √t = d₁ - σ √t

Where:

• C - value of call option,
• P - value of put option,
• N (.) - cumulative standard normal distribution function,
• SP - current stock price
• ST - strike price
• e - constant (2.7182818),
• ln - natural logarithm,
• r - current risk-free interest rate in decimal
• t - time of expiry in years
• σ - annualized volatility of the stock in decimal
• d - dividend yield in decimal

What is the Black-Scholes option pricing model?

We can compare & check the prices of our options using this model. The Black-Scholes model helps in determining the best possible option for pricing for investors and lenders.

How to Determine Black-Scholes Manually?

The Black-Scholes option-pricing model delivers a simple mechanism for valuing calls under certain assumptions. The steps for calculating the Black Scholes value are given below:

• Find out the values for d1 and d2 from the given formulas,

d₁= [ln(S₀/X)+(r_f+σ²/2)T]/[σ√T]

d₁= d₁- σ√T

• Next, by either using a z-table or by using a suitable estimation function from a statistics manual, find out the normal density functions for d₁and d₂

• Finally, take the values of N(d1) and N(d1) and put them in the given formula:

c₀= S₀N(d₁)-X/e^rf^T(N(d₂))

• Hence the estimated value of the call can find out, using which further decisions can be made.

Solved Example of Calculating Black- Scholes value with Steps

Example:

An investor has the opportunity to purchase a one-year call option for $9.00 on a stock that is currently selling for $80. And $90 is the exercise price of the call and the present riskless rate of return is 20% per annum. The variance of annual returns on the underlying stock is 16%. At its current price of $9.00, does this option represent a good investment?

Solution:

Black-scholes calculator with steps is provided here for easy calculations and quick results. Click on the calculate button and get the answer in no time.

First, we note the model inputs in symbolic form:

T=1

r_f=0.20

σ=0.4

S₀=80

X=90

σ^2=0.16

e≈2.71828

Calculating d₁ and d₂,

d₁={ln(80/90)+(0.2+0.5×0.16)×1}/0.4={ln(0.8888)+0.28}/0.4=0.04

d₂=d1−0.4×1=0.04−0.4=−0.36

N(d₁)=N(0.04)=0.515

N(d₂)=N(−0.36)=0.359

Taking the values of N(d1) and N(d₁) and putting in the formula to calculate C0,

C₀=80×(0.515)−(90×0.8187)⋅(0.359)=14.747

Since the 14.747 estimated value of the call exceeds its 9.00 market price, the call should be purchased.

FAQs On Free Online Black-Scholes Calculator with Steps

1. What is the most important contribution of the Black-Scholes formula?

The Black-Scholes formula is one of the most significant concepts in modern financial theory. The formula estimates the theoretical value of derivatives other investment instruments, taking into account the impact of time and other risk factors.

2. How Black-Scholes model can be used to value options?

The Black-Scholes model can be used to determine the fair price or theoretical value for a call.

3. What are the limitations of the Black-Scholes model?

One of the limitations of the Black-Scholes model is that it starts assuming the constant values for the risk-free rate of return and volatility over the option duration while none of those may remain constant in the real world.

4. What happens when the volatility is zero in the Black-Scholes Merton model?

When the volatility is zero it means the price of the underlying at expiry. If the value is higher than the strike price then the option is worth the present value of the difference while when it is less than the strike price then the option is valueless.