# Black Scholes Calculator You can use this Black-Scholes Calculator to determine the fair market value (price) of a European put or call option based on the Black-Scholes pricing model. It also calculates and plots the Greeks – Delta, Gamma, Theta, Vega, Rho.

Black-Scholes Option Calculator

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Option Type: Call Put Values
Variable Symbol Input Value From To
Spot Price SP
Strike Price ST
Expiry Time (Y) t
Volatility (%) v
Rate (%) r
Div. Yield (%) d

## Option Type: Call Option

y Axis Symbol Result
Value
d1
d2
Delta
Gamma
Theta
Vega
Rho

Related

## The Black-Scholes Option Pricing Formula

You can compare the prices of your options by using the Black-Scholes formula. It's a well-regarded formula that calculates theoretical values of an investment based on current financial metrics such as stock prices, interest rates, expiration time, and more. The Black-Scholes formula helps investors and lenders to determine the best possible option for pricing.

The Black Scholes Calculator uses the following formulas:

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  is the value of the call option,

P  is the value of the put option,

N (.)  is the cumulative standard normal distribution function,

SP  is the current stock price (spot price),

ST  is the strike price (exercise price),

e  is the exponential constant (2.7182818),

ln  is the natural logarithm,

r  is the current risk-free interest rate (as a decimal),

t  is the time to expiration in years,

σ  is the annualized volatility of the stock (as a decimal),

d  is the dividend yield (as a decimal).

## What is Black Scholes Calculator

A Black-Scholes calculator is a tool used to estimate the price of options contracts using the Black-Scholes model. The Black-Scholes model is a mathematical model developed by economists Fisher Black and Myron Scholes in 1973 to calculate the theoretical value of European-style options.

Here's how a Black-Scholes calculator typically works:

1. Inputs: The calculator requires several inputs to estimate the option price:

• Underlying Asset Price: The current market price of the underlying asset (e.g., stock, index).
• Strike Price: The predetermined price at which the option can be exercised.
• Time to Expiration: The length of time until the option contract expires.
• Risk-Free Interest Rate: The risk-free interest rate over the option's term.
• Volatility: The standard deviation of the underlying asset's returns, representing its price fluctuation.
2. Calculation: Using the provided inputs, the Black-Scholes calculator applies the following formula to estimate the option price:

• For a call option: C = S * N(d1) - X * e^(-rt) * N(d2)
• For a put option: P = X * e^(-rt) * N(-d2) - S * N(-d1) where:
• C/P = Call/Put option price
• S = Underlying asset price
• X = Strike price
• t = Time to expiration
• r = Risk-free interest rate
• N() = Cumulative standard normal distribution function
• d1 = (ln(S/X) + (r + σ²/2)t) / (σ * sqrt(t))
• d2 = d1 - σ * sqrt(t)
• σ = Volatility
3. Displaying the Result: The calculator provides the estimated price of the call or put option based on the Black-Scholes model as the output.

It's important to note that the Black-Scholes model assumes certain factors, such as constant volatility, no dividends, and efficient markets. Therefore, the calculated option price may not perfectly match the market price due to real-world complexities and assumptions.

Black-Scholes calculators are widely used in financial markets, options trading, risk management, and quantitative finance for pricing and evaluating options contracts. They provide a useful tool for investors, traders, and analysts to assess the fair value of options and make informed investment decisions.

## Black Scholes Calculator Example

Certainly! The Black-Scholes model is used to calculate the theoretical price of options. Let's consider an example of using the Black-Scholes formula to calculate the value of a call option.

Assume the following parameters:

• Stock price (S): \$100
• Strike price (K): \$105
• Time to expiration (T): 1 year
• Risk-free interest rate (r): 0.05 (5%)
• Volatility (σ): 0.2 (20%)

Using these parameters, we can calculate the value of the call option using the Black-Scholes formula:

d1 = (ln(S/K) + (r + σ^2/2) * T) / (σ * sqrt(T)) = (ln(100/105) + (0.05 + 0.2^2/2) * 1) / (0.2 * sqrt(1)) ≈ -0.0889

d2 = d1 - σ * sqrt(T) = -0.0889 - 0.2 * sqrt(1) ≈ -0.2889

Using a standard normal cumulative distribution function, we can calculate N(d1) and N(d2). Let's assume N(d1) = 0.4641 and N(d2) = 0.3805.

Call option value (C) = S * N(d1) - K * e^(-rT) * N(d2) = 100 * 0.4641 - 105 * e^(-0.05 * 1) * 0.3805 = 46.41 - 105 * 0.9753 * 0.3805 ≈ 46.41 - 39.90 ≈ \$6.51

Therefore, using the Black-Scholes formula, the value of the call option in this example is approximately \$6.51.

Please note that this is a simplified example for illustrative purposes, and the Black-Scholes model has assumptions and limitations. In practice, option pricing can be more complex, and there are variations and refinements to the basic model to consider.