Calculator Inputs
Example Data Table
This sample uses S=105, K=100, T=0.75 years, r=5%, σ=22%, q=1%.
| Scenario | Call Price | Put Price | Call Delta | Gamma | Vega |
|---|---|---|---|---|---|
| Reference Example | 12.1969 | 4.3009 | 0.6894 | 0.017390 | 31.6342 |
| Interpretation | The call carries higher value because spot exceeds strike, while positive time and volatility still support both option premiums. | ||||
Formula Used
The Black-Scholes model for European options with continuous dividend yield uses these core equations:
d1 = [ ln(S/K) + (r - q + 0.5σ²)T ] / [ σ√T ]
d2 = d1 - σ√T
Call = Se-qTN(d1) - Ke-rTN(d2)
Put = Ke-rTN(-d2) - Se-qTN(-d1)
Here, S is underlying price, K is strike, T is time in years, r is risk-free rate, q is dividend yield, σ is annual volatility, and N(.) is the cumulative standard normal distribution.
This file also computes Delta, Gamma, Theta, Vega, and Rho to support deeper sensitivity analysis.
How to Use This Calculator
- Enter the current underlying price and the option strike price.
- Set time to expiry in years, then enter annual risk-free rate and volatility.
- Add dividend yield if the underlying distributes cash continuously.
- Choose call or put, then enter contracts and contract size.
- Optionally enter the live market premium for mispricing comparison.
- Press the calculate button to show results above the form.
- Review premium, Greeks, breakeven, ITM probability, and the Plotly curve.
- Download a CSV or PDF report for documentation or sharing.
Frequently Asked Questions
1. What does this calculator estimate?
It estimates European call and put prices using Black-Scholes assumptions. It also returns Greeks, breakeven, total contract value, and a pricing comparison against an observed market premium.
2. Why is volatility so important?
Volatility measures expected uncertainty. Higher volatility usually increases option value because the underlying has a greater chance of moving into a profitable region before expiry.
3. What does Delta tell me?
Delta estimates how much the option price changes for a small move in the underlying. Calls usually have positive delta, while puts usually have negative delta.
4. Why is Theta often negative?
Theta reflects time decay. As expiry approaches, the chance of favorable movement shrinks, so option time value usually declines each day, especially for long positions.
5. Does this work for American options?
No. Standard Black-Scholes is designed for European options, which are exercised only at expiry. American options may need a binomial tree or finite-difference approach.
6. What does the ITM probability represent?
It approximates the risk-neutral probability of finishing in the money at expiry. It is useful for intuition, but it is not a guaranteed real-world probability forecast.
7. Why include dividend yield?
Dividend yield reduces the present value of future underlying ownership. That tends to lower call values and support put values when all other inputs remain unchanged.
8. What does mispricing mean here?
Mispricing is the difference between observed market premium and theoretical premium. A positive value means the market premium is above model value; a negative value means below.