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Black-Scholes and the Greeks

A working trader's guide to the model that prices every option Tradient evaluates — and the Greeks it produces.

10 min readgreekspricingmath

Every option price you see in Tradient flows through a Black-Scholes pricing engine. Every Greek (delta, gamma, theta, vega, rho) comes from the same model. You don’t need a finance degree to use Tradient, but you do need to understand what these numbers mean — they’re the steering wheel of every options position.

What Black-Scholes actually does

Black-Scholes takes five inputs and returns a fair price for a European call or put:

  • S — current stock price
  • K — strike price
  • T — time to expiration (in years)
  • r — risk-free interest rate
  • σ — implied volatility (annualized)

The output is a price. Tradient’s pricing engine uses the standard closed-form formula:

  • d1 = (ln(S/K) + (r + 0.5σ²)T) / (σ√T)
  • d2 = d1 − σ√T
  • Call = S·N(d1) − K·e^(−rT)·N(d2)
  • Put = K·e^(−rT)·N(−d2) − S·N(−d1)

N is the standard normal CDF. That’s the entire model. Everything else — Greeks, POP, Monte Carlo — is built on top.

Aside
Tradient prices American-style equity options with the Black-Scholes formula even though the model is technically for European options. For non-dividend-paying stocks the gap is negligible; for dividend payers we apply a small correction. This is the standard industry shortcut.

The Greeks

Greeks are partial derivatives of the option price with respect to each input. They tell you how the option’s value will change as the world moves around it.

Delta — direction

Delta is the change in option price per $1 change in the underlying. Calls have positive delta (0 to 1); puts have negative delta (−1 to 0). An ATM call has delta around 0.50; a deep ITM call approaches 1.0; a far OTM call approaches 0.

Two practical uses:

  • Hedging: a portfolio with net delta of −15 will lose ~$15 if the stock goes up $1. You can neutralize delta by adding offsetting positions.
  • Probability shortcut: the absolute value of delta roughly equals the probability the option finishes ITM. A 0.16 delta short put has ~84% chance of expiring worthless. This is why tastytrade-style strategies sell 16-delta strikes.

Gamma — convexity

Gamma is the rate of change of delta. ATM options have the highest gamma; deep ITM and deep OTM have very little. Gamma is what makes options nonlinear — and what makes a delta hedge fail when the stock moves.

Long options are long gamma (good when the stock moves a lot, in either direction). Short options are short gamma (bad when the stock moves; the delta you thought you had can flip on you overnight).

Theta — time decay

Theta is the change in option price per day, assuming nothing else moves. Long options have negative theta (you lose money to the clock). Short options have positive theta (you collect from the clock).

Theta is not linear in time. It accelerates sharply in the last 30 days, then again in the last week. This is why the income-trader playbook centers on 30-50 DTE entries and early exits — you want the steepest part of the curve without the gamma risk of the final week.

Vega — volatility sensitivity

Vega is the change in option price per 1 percentage point change in implied volatility. Long options are long vega (price up when IV goes up). Short options are short vega (price up when IV goes down).

Vega is the hidden Greek that gets income traders. You sell a 16-delta strangle in calm vol, the market panics, your strikes haven’t been touched but the position is deep red — that’s vega expansion. The position will recover if you wait, but only if you have the margin to.

Rho — interest rate sensitivity

Rho is the change in option price per 1 percentage point change in the risk-free rate. For most retail timeframes (under 90 days) rho is small enough to ignore. We compute it and display it for completeness, not because it should drive your decisions.

Reading the Greeks together

Greeks are most useful when you read them as a vector for the entire position, not leg by leg.

  • Iron condor at entry:delta near 0, theta positive (you’re collecting), vega negative (you’re short vol), gamma negative (the position gets worse if the stock starts moving).
  • Long straddle at entry:delta near 0, theta negative (you’re bleeding), vega positive (you’re long vol), gamma positive (you love movement).
  • Bull call spread: delta positive (smaller than a long call alone), theta slightly negative, vega near zero (the long and short legs cancel).

Tradient’s Focus mode shows the aggregate Greeks for any scan result so you don’t have to add them by hand.

IV is the model's biggest assumption
Black-Scholes assumes σ is constant for the life of the option. Real markets have IV that changes by the hour. The Greeks are a snapshot at one IV value; if vol moves, every Greek shifts. Use them as the steering wheel, not as a guarantee.

Implied volatility — the input we don’t observe

Of the five Black-Scholes inputs, four are observable: S, K, T, and r are facts. σ is not — it’s solved backwardsfrom the market price of the option. That’s why it’s called “implied”: the price implies what vol must be for Black-Scholes to match.

This means IV is whatever traders are willing to pay. When the market panics, IV goes up because people bid up puts. When the market is calm, IV drifts down because nobody wants protection. Tradient’s IV rank measures this drift on a per-name basis.

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