Stock Call Option Calculator

Black-Scholes Model for Call Options:

\[ C = S_0N(d_1) - Ke^{-rt}N(d_2) \] \[ d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)t}{\sigma\sqrt{t}} \] \[ d_2 = d_1 - \sigma\sqrt{t} \]

Current Stock Price (S₀):

USD

Strike Price (K):

USD

Time to Maturity (t):

years

Risk-Free Rate (r):

decimal (e.g. 0.05 for 5%)

Volatility (σ):

decimal (e.g. 0.20 for 20%)

Call Value (USD):

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1. What is the Black-Scholes Model?

The Black-Scholes model is a mathematical model for pricing an options contract. It estimates the theoretical value of derivatives based on other investment instruments, taking into account the impact of time and other risk factors.

2. How Does the Calculator Work?

The calculator uses the Black-Scholes formula for call options:

\[ C = S_0N(d_1) - Ke^{-rt}N(d_2) \] \[ d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)t}{\sigma\sqrt{t}} \] \[ d_2 = d_1 - \sigma\sqrt{t} \]

Where:

\( C \) — Call option price
\( S_0 \) — Current stock price
\( K \) — Strike price
\( r \) — Risk-free interest rate
\( t \) — Time to maturity
\( \sigma \) — Volatility of the stock
\( N \) — Standard normal cumulative distribution function

Explanation: The formula calculates the theoretical value of a call option based on the current stock price, strike price, time remaining until expiration, risk-free rate, and volatility.

3. Importance of Option Pricing

Details: Accurate option pricing is crucial for traders and investors to determine fair value, assess risk, and make informed trading decisions in the options market.

4. Using the Calculator

Tips: Enter the current stock price, strike price, time to maturity in years, risk-free rate (as decimal), and volatility (as decimal). All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What assumptions does the Black-Scholes model make?
A: It assumes constant volatility, no dividends, efficient markets, no transaction costs, and that returns are lognormally distributed.

Q2: How accurate is the Black-Scholes model?
A: While widely used, it's most accurate for European options (no early exercise) and may not account for all market realities like changing volatility.

Q3: What is implied volatility?
A: The volatility value that makes the model price equal to the market price, often used as a measure of expected future volatility.

Q4: Can this be used for put options?
A: No, this calculator is for call options only. Put options require a different formula.

Q5: What's a typical risk-free rate to use?
A: Often the yield on short-term Treasury bills is used as a proxy for the risk-free rate.