Options Strategy Builder with Greeks
P&L at Expiration
Greeks
| Leg | Premium | Delta | Gamma | Theta | Vega |
|---|---|---|---|---|---|
| Long CALL 100 | $3.63 | 0.5362 | 0.0462 | -0.0638 | 0.1139 |
| Combined | $3.63 | 0.5362 | 0.0462 | -0.0638 | 0.1139 |
This options strategy builder prices each leg of a position with the Black-Scholes model and reports the full set of risk sensitivities known as the Greeks, then combines them across all legs. For every leg it computes the theoretical premium plus delta, gamma, theta, vega, and rho, scaled by the leg quantity so long and short positions net out correctly. It also draws a profit-and-loss diagram across a range of underlying prices, locating breakeven points and the maximum profit and loss at expiration.
Formula
d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T); Call price = S·N(d₁) − K·e^(−rT)·N(d₂)
- S
- Current underlying (spot) price
- K
- Strike price of the leg
- T
- Time to expiry in years = days ÷ 365
- r
- Risk-free rate as a decimal
- σ
- Implied volatility as a decimal
- N(·)
- Cumulative standard normal distribution; delta = N(d₁) for a call, N(d₁) − 1 for a put
How it works
- Set the shared inputs: the current underlying price, the annual risk-free rate (as a percent), and a default implied volatility (as a percent). These feed the Black-Scholes pricing for every leg.
- Add one or more legs. Each leg has a type (call or put), a strike price, days to expiry, a signed quantity (positive for long, negative for short), and an optional per-leg implied volatility override. Time to expiry is converted to years as days ÷ 365.
- The tool prices each leg and computes its Greeks, multiplies them by the quantity, and sums them into combined delta, gamma, theta, and vega. It then plots P&L at expiration across 60 prices from half to one-and-a-half times the underlying to find breakevens, max profit, and max loss.
Worked example
A single long call: underlying at $100, strike $100, 30 days to expiry, quantity +1, 5% risk-free rate, 30% implied volatility.
- Time to expiry T = 30 ÷ 365 ≈ 0.0822 years; d₁ and d₂ are computed from S=100, K=100, r=0.05, σ=0.30.
- Black-Scholes prices the call premium at about $3.63 per share.
- Greeks: delta ≈ 0.5362 (gains ~$0.54 per $1 move), gamma ≈ 0.0462, theta ≈ −$0.064/day, vega ≈ $0.114 per 1% vol, rho ≈ 0.041.
The call costs about $3.63, so the net premium (debit) is $3.63 and the breakeven at expiration is about $103.63 (strike plus premium). Delta near 0.54 reflects an at-the-money call with slightly positive moneyness from the risk-free drift.
Frequently asked questions
- What do the Greeks mean?
- Delta is the change in option value per $1 move in the underlying; gamma is how fast delta itself changes; theta is the daily time decay; vega is sensitivity to a 1% change in implied volatility; and rho is sensitivity to interest rates. Together they describe how a position reacts to market shifts.
- How is the option premium calculated?
- Each leg is priced with the Black-Scholes model using the underlying price, strike, time to expiry, risk-free rate, and implied volatility. The model assumes lognormal price movement and no dividends, producing a theoretical fair value for European-style options.
- How are multi-leg strategies combined?
- Each leg's Greeks are multiplied by its signed quantity so short legs subtract and long legs add, then summed into combined delta, gamma, theta, and vega. The net premium nets debits paid against credits received across all legs.
- Does this model dividends or American-style early exercise?
- No. It uses the standard Black-Scholes framework, which assumes no dividends and European-style exercise at expiration only. Real American options on dividend-paying stocks can differ, so treat the prices and Greeks as theoretical estimates.