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.
| Option Type: Call Put | Values | ||||
|---|---|---|---|---|---|
| x | 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 |
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:
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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.
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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
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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.
