Option Trade Lab formulas

Every formula Option Trade Lab uses, in the order it calculates them

The option price, the Greeks, the implied volatility search, P&L, payoff at expiry and the "what moved the price" breakdown. A worked example with a real FIG put is near the end.

European optionsno dividendscalendar days ÷ 365

1. Inputs and notation

Every price and Greek comes from five numbers. The lab reads them from your ticket and the scenario sliders.

Symbol	Meaning	Where it comes from in the lab
$S$	Stock price	Stock price slider, or the ticket's price then / now
$K$	Strike price	Ticket: strike
$T$	Time to expiry, in years	Days to expiry ÷ 365 (calendar days)
$σ$	Implied volatility, as a decimal	IV slider ÷ 100, so 68% becomes 0.68
$r$	Risk-free interest rate, as a decimal	Rate slider ÷ 100, so 4% becomes 0.04
$N (x)$	Standard normal cumulative distribution	Section 2
$n (x)$	Standard normal density	Section 2
$s$	Side of the trade	+1 if you bought, −1 if you sold
$Q$	Shares controlled	100 × number of contracts

days = expiry date - as-of date

T = \frac{days}{365}

Days are calendar days and never go below 0.

2. Standard normal functions

Black-Scholes needs the normal distribution. The lab computes it from the error function, using the Abramowitz and Stegun approximation 7.1.26. It's accurate to about 1.5 × 10⁻⁷.

n (x) = \frac{e^{- \frac{x^{2}}{2}}}{\sqrt{2 π}}

N (x) = \frac{1}{2} [1 + \erf (\frac{x}{\sqrt{2}})]

\erf (x) \approx 1 - (a_{1} t + a_{2} t^{2} + a_{3} t^{3} + a_{4} t^{4} + a_{5} t^{5}) \cdot e^{- x^{2}}

t = \frac{1}{1 + 0.3275911 x}

a₁ = 0.254829592, a₂ = −0.284496736, a₃ = 1.421413741, a₄ = −1.453152027, a₅ = 1.061405429. For negative x the lab uses erf(−x) = −erf(x).

3. Black-Scholes option price

d_{1} = \frac{\ln (\frac{S}{K}) + (r + \frac{σ^{2}}{2}) T}{σ \sqrt{T}}

d_{2} = d_{1} - σ \sqrt{T}

Call = S N (d_{1}) - K e^{- r T} N (d_{2})

Put = K e^{- r T} N (- d_{2}) - S N (- d_{1})

$K e^{- r T}$ is the strike discounted back to today. The lab floors the result at $0 to avoid tiny negative rounding errors.

At expiry, when $T$ = 0 (or $σ$ = 0), the formula is replaced by intrinsic value:

Call at expiry = \max (S - K, 0) Put at expiry = \max (K - S, 0)

At expiry, delta is +1 (call) or −1 (put) when the option is in the money and 0 otherwise. Every other Greek is 0.

4. The Greeks, per share

The textbook formulas give theta per year, vega per 1.00 of volatility and rho per 1.00 of rate. The lab rescales them to units you can read directly: theta per calendar day, vega per 1 IV point, rho per 1 percentage point of rate.

Greek	Call	Put	Lab unit
Delta	$N (d_{1})$	$N (d_{1}) - 1$	$ per $1 stock move
Gamma	$\frac{n (d_{1})}{S σ \sqrt{T}}$	same as call	delta per $1 stock move
Theta	$\frac{- \frac{S n (d_{1}) σ}{2 \sqrt{T}} - r K e^{- r T} N (d_{2})}{365}$	$\frac{- \frac{S n (d_{1}) σ}{2 \sqrt{T}} + r K e^{- r T} N (- d_{2})}{365}$	$ per calendar day
Vega	$\frac{S n (d_{1}) \sqrt{T}}{100}$	same as call	$ per 1 IV point
Rho	$\frac{K T e^{- r T} N (d_{2})}{100}$	$- \frac{K T e^{- r T} N (- d_{2})}{100}$	$ per 1% rate

For your position, each Greek is multiplied by the shares you control and by the side of the trade:

Position Greek = s \cdot Q \cdot Greek

Example: a short put (s = −1) has negative delta per share, so the position delta is positive. You gain when the stock rises.

5. Implied volatility (solved, not calculated)

No formula turns a price back into a volatility, so the lab searches for it. It looks for the $σ$ at which the model price equals a known price, using bisection: try the midpoint and keep the half that contains the answer.

find σ such that P (S, K, T, σ, r) = target price

Start with low = 0.5% and high = 500%.
Price the option at mid = (low + high) / 2.
If that price is below the target, set low = mid. Otherwise set high = mid.
Repeat 90 times. The answer is (low + high) / 2.

IV at trade uses your fill price, the stock price then and the days left then. IV now uses the option price you enter, with the current stock price and days left. If the target is at or below the price at 0.5%, or at or above the price at 500%, there's no solution. The lab then reports it (at trade) or falls back to IV at trade (now).

6. Intrinsic value, time value and probabilities

Intrinsic (call) = \max (S - K, 0) Intrinsic (put) = \max (K - S, 0)

Time value = option price - intrinsic value

Time value is floored at $0.

Chance it ends in the money: call = N (d_{2}) put = N (- d_{2})

This is the risk-neutral probability built into the price, not a forecast of what will happen.

±1 SD move by expiry = S \cdot σ \cdot \sqrt{T} range = S \pm move

daily move (%) = \frac{IV%}{\sqrt{365}} move by expiry = σ \sqrt{T}

The lab uses this simple linear range for readability. A log-normal range, $S e^{\pm σ \sqrt{T}}$ , is slightly wider on the upside.

7. Premium and P&L

Net premium (sold) = fill \cdot Q - commission cash received

Net premium (bought) = fill \cdot Q + commission cash paid

P&L at option price P = s \cdot (P - fill) \cdot Q - commission

Includes the opening commission only. A real close adds a second commission and fills at the bid (selling) or the ask (buying), not at the model price.

Share of premium kept (sold) = \frac{P&L}{net premium} Return (bought) = \frac{P&L}{net premium}

8. Payoff at expiry

The payoff chart uses the P&L formula above with $P$ set to intrinsic value at each stock price. The second, curved line uses the full Black-Scholes price with the days left on your slider.

Breakeven (call) = K + \frac{net premium}{Q} Breakeven (put) = K - \frac{net premium}{Q}

Position	Max profit	Max loss
Long call	Unlimited	Net premium paid
Long put	$K \cdot Q$ − net premium paid (stock at $0)	Net premium paid
Short call	Net premium received	Unlimited (the stock can keep rising)
Short put	Net premium received	$K \cdot Q$ − net premium received (stock at $0)

9. “What moved the price” breakdown

State 0 is the trade-day inputs ( $S_{0}$ , days₀, $σ_{0}$ , $r_{0}$ ). The scenario is ( $S$ , days, $σ$ , $r$ ) from your sliders. The lab prices five states, changing one more input each time:

State	Inputs	Bar in the chart
$P_{0}$	S₀, days₀, σ₀, r₀ (equals your fill price)	At trade
$P_{1}$	S₀, days, σ₀, r₀	$Time = P_{1} - P_{0}$
$P_{2}$	S, days, σ₀, r₀	$Stock = P_{2} - P_{1}$
$P_{3}$	S, days, σ, r₀	$IV = P_{3} - P_{2}$
$P_{4}$	S, days, σ, r	$Rate = P_{4} - P_{3}$ (only shown if the rate changed)

Time + Stock + IV + Rate = P_{4} - P_{0}

Alone effect = Price(state 0 with only that input changed) - P_{0}

Your P&L effect of a step = s \cdot step \cdot Q

The sequence always adds up to the total. The alone effects usually don't, because the inputs interact. Changing the order of the sequence changes how the total is split.

10. Examples on the Greek cards

The examples reprice the full formula instead of using the linear Greek estimate, so they stay correct for large moves. $P (\cdot)$ is the Black-Scholes price, and every other input stays at the slider values.

Card	Example shown	Formula
Delta	Stock up / down $1	$s Q [P (S + 1) - P (S)]$ $s Q [P (S - 1) - P (S)]$
Gamma	Delta after a $1 rise	$Δ (S + 1)$
Theta	Tomorrow, nothing else changes	$s Q [P (days - 1) - P (days)]$
Vega	IV up / down 5 points	$s Q [P (σ + 5) - P (σ)]$ $s Q [P (σ - 5) - P (σ)]$
Rho	Rate up 1 point	$s Q [P (r + 1) - P (r)]$

11. Worked example: a FIG put

Short 1 FIG Oct 30, 2026 $19.50 put, sold Sep 23 at $0.72 with a $0.77 commission. Inputs from Oct 15, 2026: $S$ = $22.89, $K$ = $19.50, 15 days, $σ$ = 68.0% (solved from the $0.1715 mark), $r$ = 4%.

Step	Calculation	Result
$T$	15 / 365	0.04110
$σ \sqrt{T}$	0.68 × √0.04110	0.13785
$d_{1}$	[ln(22.89 / 19.5) + (0.04 + 0.68² / 2) × 0.04110] / 0.13785	1.2436
$d_{2}$	1.2436 − 0.13785	1.1058
$N (- d_{1})$ , $N (- d_{2})$	normal cumulative distribution	0.1068, 0.1344
Put price	19.5 × e^{−0.04 × 0.04110} × 0.1344 − 22.89 × 0.1068	$0.1716 / share
Intrinsic, time value	max(19.5 − 22.89, 0); price − intrinsic	$0.00, $0.1716
Delta	N(d₁) − 1	-0.1068 (position +$10.68 per $1)
Gamma	n(d₁) / (S σ √T)	0.0583
Theta	per calendar day	-0.0191 (position +$1.91 / day)
Vega	per 1 IV point	0.0085 (position −$0.85 / point)
Rho	per 1% rate	-0.0011 (position +$0.11)
Chance ends in the money	N(−d₂)	13.4%
±1 SD range	22.89 ± 22.89 × 0.13785	$19.73 – $26.05
P&L at model price	−1 × (0.1716 − 0.72) × 100 − 0.77	+$54.07 (76% of $71.23 kept)
Breakeven at expiry	19.5 − 71.23 / 100	$18.79

Option Trade Lab shows the same numbers, rounded. IBKR's own Greeks differ slightly because IBKR uses its own volatility and pricing model.

12. What the model leaves out

Dividends. No dividend yield is modelled. Dividends lower call prices and raise put prices.
Early exercise. US equity options are American style. The model is European, so it can underprice deep in-the-money puts, and calls just before an ex-dividend date.
Volatility skew. Real IV differs from strike to strike and expiry to expiry. The lab uses one σ for every stock price on its charts.
Trading costs. Bid/ask spreads and closing commissions aren't in the P&L.
Day count. Calendar days ÷ 365. Some platforms use trading days ÷ 252, which gives different theta and IV.

Formulas as implemented in Option Trade Lab. For learning, not investment advice.