Calculator inputs
Use percentage fields for volatilities and the risk-free rate. Correlation must stay between -0.999 and 0.999.
Example data table
This sample row uses the page defaults and illustrates a typical pricing case for a call spread option.
| Type | S1 | S2 | K | β | σ1 | σ2 | ρ | r | T | Illustrative premium |
|---|---|---|---|---|---|---|---|---|---|---|
| Call | 105 | 95 | 8 | 1.00 | 28% | 24% | 0.45 | 5% | 0.75 | 10.7793 |
Formula used
Underlying payoff: max(S1 − βS2 − K, 0) for a call, or max(βS2 + K − S1, 0) for a put.
Kirk approximation base: A = βS2 + Ke−rT
Weight term: w = βS2 / A
Spread volatility: σspread = √(σ1² − 2ρσ1σ2w + σ2²w²)
d1: [ln(S1 / A) + 0.5σspread²T] / [σspread√T]
d2: d1 − σspread√T
Call price: S1N(d1) − AN(d2). Put price: AN(−d2) − S1N(−d1).
This model is widely used for European spread options because it turns a two-asset problem into a practical one-factor approximation. The page also runs a Monte Carlo estimate to give you a second reference point and a probability of finishing in the money.
How to use this calculator
- Choose whether you want a call or put spread option.
- Enter current prices for asset 1 and asset 2.
- Set the strike, hedge ratio, volatilities, correlation, rate, and time.
- Provide contract count, multiplier, and Monte Carlo paths.
- Press Calculate to show the result directly above the form.
- Review the premium, Greeks, breakeven, and Monte Carlo comparison.
- Use the chart to inspect pricing, payoff, and profit behavior.
- Download CSV or PDF when you need reporting output.
Frequently asked questions
1) What does this calculator price?
It prices European options whose payoff depends on the difference between two assets, adjusted by a hedge ratio and a strike. These are common in energy, commodities, crack spreads, spark spreads, and relative-value trading.
2) Why use Kirk approximation?
Kirk approximation is popular because it is fast, stable, and usually accurate for many practical spread option cases. It converts the two-asset structure into a simpler expression with an effective spread volatility.
3) What does the hedge ratio mean?
The hedge ratio β scales asset 2 inside the spread. A value of 1 compares one unit of each asset. Other values reflect contract size differences, conversion factors, or economic exposure adjustments.
4) Why is correlation important?
Correlation changes how strongly the two assets move together. Higher positive correlation often lowers spread uncertainty, while lower or negative correlation can widen spread risk and increase option value.
5) Why show both Kirk and Monte Carlo prices?
The analytical value is fast and convenient. Monte Carlo gives an independent simulation-based check, along with a standard error and an estimated chance of finishing in the money.
6) Are the Greeks exact?
No. The page estimates Greeks numerically with central differences. They are useful for sensitivity analysis, but they remain approximations that depend on the chosen step sizes and model assumptions.
7) What does breakeven represent here?
Breakeven is the asset 1 expiry price needed to offset the premium paid, assuming asset 2 stays at the input level. It is a practical planning marker rather than a guaranteed outcome.
8) When should I be cautious with this model?
Use caution when markets have strong jumps, complex carry costs, American exercise rights, or unusual contract specifications. In those cases, a fuller simulation or desk-specific model may be more appropriate.