Calculator Inputs
Example Data Table
| Scenario | Type | Spot | α | T1 | T2 | Vol % | Rate % | Yield % | Approx Unit Price |
|---|---|---|---|---|---|---|---|---|---|
| Base Case | Call | 100.00 | 1.05 | 0.50 | 1.50 | 24.00 | 5.00 | 1.00 | 9.8234 |
| Higher Volatility | Call | 100.00 | 1.05 | 0.50 | 1.50 | 32.00 | 5.00 | 1.00 | 12.9147 |
| Protective Put | Put | 100.00 | 0.95 | 0.50 | 1.25 | 20.00 | 4.00 | 0.50 | 5.2871 |
These sample values illustrate how price changes with volatility, option direction, timing, and strike scaling.
Formula Used
A forward start option begins at time T1 and expires at time T2. Its strike is set later as K = α × S(T1), where α is the strike factor.
Forward period: τ = T2 − T1
d1: d1 = [ ln(1 / α) + (r − q + 0.5σ²)τ ] / [ σ√τ ]
d2: d2 = d1 − σ√τ
Call price: C0 = S0 e−qT1 [ e−qτN(d1) − αe−rτN(d2) ]
Put price: P0 = S0 e−qT1 [ αe−rτN(−d2) − e−qτN(−d1) ]
Here, S0 is current spot price, σ is annual volatility, r is risk-free rate, q is dividend yield, and N(·) is the cumulative standard normal distribution.
How to Use This Calculator
- Choose whether you want a call or put forward start option.
- Enter the current spot price of the underlying asset.
- Set the strike factor α. Values above 1 create higher future strikes.
- Enter forward start time T1 and final maturity T2 in years.
- Provide annual volatility, risk-free rate, and dividend yield percentages.
- Enter number of contracts and contract multiplier for total premium sizing.
- Click the calculate button to show premium, Greeks, d-values, and probability.
- Use the export buttons to download the result set as CSV or PDF.
8 FAQs
1. What is a forward start option?
It is an option whose strike is set on a future date, not today. The strike commonly equals a chosen factor times the asset price at the start date.
2. What does the strike factor α mean?
α scales the future strike relative to the asset price at T1. For example, α = 1.05 means the strike becomes 105% of the asset price on the forward start date.
3. Why must T2 be greater than T1?
The option needs time to exist after the strike is set. If maturity is not later than the start date, there is no valid forward pricing period.
4. Why does volatility matter so much?
Higher volatility increases the range of possible future price ratios between T1 and T2. That wider uncertainty generally raises both call and put forward start option values.
5. What is the ITM probability shown here?
It is the model-based probability of finishing in the money under the pricing assumptions. It is useful for comparison, but it is not a guaranteed real-world outcome.
6. Are the Greeks exact or approximate?
This file computes Greeks numerically using small input changes. That makes the calculator practical and transparent, though tiny differences may appear versus closed-form analytic Greek formulas.
7. Can I use this for equity or index options?
Yes, when the Black-Scholes assumptions are a reasonable approximation. You can set dividend yield, rate, volatility, and contract size to reflect the product you are analyzing.
8. What are the main model limitations?
This approach assumes constant volatility, continuous trading, lognormal prices, and stable rates. Real markets may include jumps, skew, smile effects, and changing volatility surfaces.