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Graduate-Level Modeling · Derivatives Risk

Option Greeks Surface Visualizer

A single Greek at a single strike tells you almost nothing about how your risk actually behaves as the underlying moves and time passes. See the full surface at once, across every strike and every day to expiration.

How To Use This Model

Reading This Tool

Set the spot price, rate, and volatility, then pick which Greek to visualize across the strike/maturity grid.

Rows are strike prices from deep in-the-money to deep out-of-the-money; columns are time to expiration. Watch how Gamma and Theta both concentrate near the money as expiration approaches.

Market Inputs

Strike rows span from 70% to 130% of spot. Time-to-expiration columns span from 1 week to 1 year.

At-The-Money, 30-Day Snapshot

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Delta

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Gamma

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Theta (Per Day)

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Vega (Per 1% Vol)

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Greek Surface: Strike (Rows) × Time To Expiry (Columns)

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What This Surface Actually Shows

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Why Gamma And Theta Spike Near Expiration

As time to expiry shrinks, an at-the-money option's Gamma rises sharply, small underlying moves have an outsized effect on Delta right before expiration, since the option is right on the edge of being exercised or not. Theta similarly accelerates, the option is losing its remaining time value fast in the final days, a well-known pattern options traders call "gamma risk" near expiration.

Reading Vega\u2019s Pattern

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Derivatives Risk Sensitivities

The Core Formulas (Black-Scholes)

Δcall = N(d1), Δput = N(d1)−1
Γ = φ(d1) / (Sσ√T)
Vega = S·φ(d1)·√T (per 1.00 vol; divide by 100 for per 1%)
Θcall = [−Sφ(d1)σ/(2√T) − rKe−rTN(d2)] / 365

φ(·) is the standard normal density, N(·) the standard normal CDF. Gamma and Vega are identical for calls and puts at the same strike and maturity; only Delta and Theta differ by option type.

When To Actually Use This Model

  • Understanding how a delta-hedged options book's risk actually evolves as expiration approaches, not just today.
  • Teaching option Greeks and their term structure in a derivatives or financial engineering course.
  • Planning around known high-gamma-risk periods, like the final week before a large options position expires.
  • Comparing vega exposure across a portfolio's different expiration buckets before an earnings or macro event.

Key Assumptions & Limitations

  • Assumes constant volatility across the whole surface; real implied volatility varies by strike (the volatility smile) and by maturity (term structure).
  • Assumes no dividends and continuous, frictionless trading, the standard Black-Scholes assumptions.
  • These are instantaneous sensitivities, valid for small, local moves, not exact predictors of a large discrete price change.
  • Real-world Greeks for American options, especially puts, can diverge from these European closed-form values.

Foundational Reference

Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654.

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