# How Does the Risk-Neutral Probability Concept Relate to Delta? ⎊ Path

**Published:** 2025-12-04
**Author:** Greeks.live
**Categories:** Path

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## How Does the Risk-Neutral Probability Concept Relate to Delta?

In the Black-Scholes framework, the Delta of a call option is mathematically equal to the cumulative probability distribution function N(d1), which is the risk-neutral probability that the option will expire in-the-money. This probability is derived under the assumption that the expected return on the underlying asset is the risk-free rate.

- [How Is Delta Used as a Probability Estimate for an Option Expiring ITM?](https://learn.greeks.live/path/how-is-delta-used-as-a-probability-estimate-for-an-option-expiring-itm/)

- [What Is the Relationship between the Option’s Delta and Its Probability of Expiring In-the-Money?](https://learn.greeks.live/path/what-is-the-relationship-between-the-options-delta-and-its-probability-of-expiring-in-the-money/)

- [What Does a Delta of -0.80 on a Bitcoin Put Option Indicate?](https://learn.greeks.live/path/what-does-a-delta-of-0-80-on-a-bitcoin-put-option-indicate/)

- [Does an ITM Option Lose Time Value Faster or Slower than an OTM Option, All Else Equal?](https://learn.greeks.live/path/does-an-itm-option-lose-time-value-faster-or-slower-than-an-otm-option-all-else-equal/)

- [What Is the ‘Delta’ of an Option?](https://learn.greeks.live/path/what-is-the-delta-of-an-option/)

- [What Is the Relationship between Delta and the Probability of an Option Expiring In-the-Money?](https://learn.greeks.live/path/what-is-the-relationship-between-delta-and-the-probability-of-an-option-expiring-in-the-money/)

- [How Does the Black-Scholes Model Mathematically Represent This Acceleration?](https://learn.greeks.live/path/how-does-the-black-scholes-model-mathematically-represent-this-acceleration/)

- [Define In-The-Money (ITM) for Both a Call and a Put Option](https://learn.greeks.live/path/define-in-the-money-itm-for-both-a-call-and-a-put-option/)

## Glossar

### [Probability](https://learn.greeks.live/area/probability/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)

Modeling ⎊ Probability is fundamental to quantitative finance, providing the mathematical framework for modeling asset price movements and calculating risk.

### [Probability Distribution Function](https://learn.greeks.live/area/probability-distribution-function/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)

Concept ⎊ The Probability Distribution Function (PDF) is a mathematical concept that describes the relative likelihood for a random variable, such as the future price of a cryptocurrency asset, to take on a given value.

## Discover More

### [How Does the Gamma Greek Relate to the Frequency of Rebalancing a Delta Hedge?](https://learn.greeks.live/path/how-does-the-gamma-greek-relate-to-the-frequency-of-rebalancing-a-delta-hedge/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![How Does the Gamma Greek Relate to the Frequency of Rebalancing a Delta Hedge?](https://learn.greeks.live/wp-content/uploads/2025/10/Hinge-Mechanism-Aesthetics-in-Decentralized-Finance-Protocol-Module-Interoperability-Visualization.webp)

High Gamma means Delta changes quickly, requiring more frequent and costly rebalancing of the Delta hedge.

### [How Does the Concept of 'Counterparty Risk' in Derivatives Contrast with 'Key Compromise Risk' in Crypto?](https://learn.greeks.live/path/how-does-the-concept-of-counterparty-risk-in-derivatives-contrast-with-key-compromise-risk-in-crypto/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![How Does the Concept of 'Counterparty Risk' in Derivatives Contrast with 'Key Compromise Risk' in Crypto?](https://learn.greeks.live/wp-content/uploads/2025/10/Layered-Modular-Architecture-Illustrating-Decentralized-Finance-Derivatives-Stacking-Protocols.webp)

Counterparty risk is the financial default risk of the contract partner; key compromise risk is the security failure leading to irreversible asset theft.

### [How Does the Birthday Paradox Affect the Probability of a Collision Attack?](https://learn.greeks.live/path/how-does-the-birthday-paradox-affect-the-probability-of-a-collision-attack/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![How Does the Birthday Paradox Affect the Probability of a Collision Attack?](https://learn.greeks.live/wp-content/uploads/2025/10/Precision-Engineered-Hinge-Mechanism-Representing-Latent-Volatility-in-Digital-Asset-Futures-Contracts.webp)

It means a collision can be found in a hash function after approximately the square root of the expected number of attempts.

### [How Can a Delta-Neutral Hedging Strategy Be Constructed Using Options to Mitigate Impermanent Loss?](https://learn.greeks.live/path/how-can-a-delta-neutral-hedging-strategy-be-constructed-using-options-to-mitigate-impermanent-loss/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![How Can a Delta-Neutral Hedging Strategy Be Constructed Using Options to Mitigate Impermanent Loss?](https://learn.greeks.live/wp-content/uploads/2025/10/X-Shaped-Intersecting-Digital-Ledger-Instruments-Reflecting-Implied-Volatility-Surface-Calibration.webp)

A delta-neutral hedge uses options or futures to offset the price risk of a liquidity position, neutralizing impermanent loss.

### [What Does It Mean for a Portfolio to Be 'Delta-Neutral'?](https://learn.greeks.live/path/what-does-it-mean-for-a-portfolio-to-be-delta-neutral/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![What Does It Mean for a Portfolio to Be 'Delta-Neutral'?](https://learn.greeks.live/wp-content/uploads/2025/10/Abstract-Digital-Modeling-Depicting-Volatility-Surface-Mechanics-Through-Spherical-Representation.webp)

The sum of the Deltas of all positions is zero, making the portfolio theoretically insensitive to small directional price changes.

### [Why Is Basis Trading Considered a 'Market-Neutral' Strategy?](https://learn.greeks.live/path/why-is-basis-trading-considered-a-market-neutral-strategy/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![Why Is Basis Trading Considered a 'Market-Neutral' Strategy?](https://learn.greeks.live/wp-content/uploads/2025/10/Crystalline-Volatility-Surface-Intersected-by-Bearish-Cross-Vectors-Digital-Asset-Hedging.webp)

The offsetting long/short positions neutralize directional risk, making the profit dependent only on the convergence of the basis.

### [Explain the Concept of 'Greeks' in Options Trading and Which One Is Most Critical for Delta-Hedging](https://learn.greeks.live/path/explain-the-concept-of-greeks-in-options-trading-and-which-one-is-most-critical-for-delta-hedging/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![Explain the Concept of 'Greeks' in Options Trading and Which One Is Most Critical for Delta-Hedging](https://learn.greeks.live/wp-content/uploads/2025/10/Hypergeometric-Probability-Distribution-Modeling-Decentralized-Autonomous-Organization-Governance-Mechanism-Symmetry.webp)

Greeks measure option sensitivity; Delta is the most critical Greek for delta-hedging to neutralize directional risk.

### [What Is the Probability of Impermanent Loss Fully Reversing in a Volatile Market?](https://learn.greeks.live/path/what-is-the-probability-of-impermanent-loss-fully-reversing-in-a-volatile-market/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![What Is the Probability of Impermanent Loss Fully Reversing in a Volatile Market?](https://learn.greeks.live/wp-content/uploads/2025/10/Transparent-Fin-Actuator-Assembly-Demonstrating-High-Frequency-Trading-Execution-Parameters.webp)

Low, as prolonged volatility makes it unlikely for the exact initial price ratio to be restored before withdrawal.

### [Explain the Concept of "Delta Neutrality" and Why It Is a Constant Moving Target for an Options Market Maker](https://learn.greeks.live/path/explain-the-concept-of-delta-neutrality-and-why-it-is-a-constant-moving-target-for-an-options-market-maker/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![Explain the Concept of "Delta Neutrality" and Why It Is a Constant Moving Target for an Options Market Maker](https://learn.greeks.live/wp-content/uploads/2025/10/Decentralized-Finance-Cross-Intersection-Modeling-Automated-Market-Maker-Liquidity-Provision-Mechanisms.webp)

Delta neutrality means a portfolio's value is insensitive to small price moves, but it requires constant re-hedging.

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**Original URL:** https://learn.greeks.live/learn/how-does-the-risk-neutral-probability-concept-relate-to-delta/
