Estimate risk-neutral probability that an option expires in-the-money using Black–Scholes.
Option Parameters
Enter option specifications to calculate the probability of expiring in-the-money (ITM)
Formula Used
P(ITM) = N(d₂) for calls | P(ITM) = N(-d₂) for puts
d₂ = [ln(S/K) + (r - σ²/2)T] / (σ√T)
The probability is derived from the Black-Scholes d₂ term, representing the risk-neutral probability that the option expires in-the-money. Note that N(d₁) (delta) is related but slightly different.
**ITM Probability** is the probability that an option will expire in-the-money (ITM)—meaning it will have intrinsic value at expiration. For a call option, this means the underlying price is above the strike; for a put, it means the underlying is below the strike.
Why ITM Probability Matters
ITM probability is fundamental to options trading because it helps traders:
**Assess Trade Quality**: Higher probability trades have lower payoff potential; lower probability trades offer asymmetric upside.
**Select Strikes**: Balance probability of profit against premium paid.
**Position Sizing**: Adjust position sizes based on expected win rates.
A critical distinction: ITM probability from Black-Scholes is **risk-neutral probability**, not real-world probability.
Understanding the Difference
**Risk-Neutral Probability**: Used for pricing derivatives. Assumes all assets grow at the risk-free rate. This is a mathematical construct, not a forecast.
**Real-World Probability**: Incorporates risk premiums and expected returns. Equities have historically earned more than the risk-free rate.
**Implication**: Real-world call ITM probability is typically slightly higher than risk-neutral for upward-drifting assets like stocks.
Why Risk-Neutral Still Matters
Despite not being "real" probability, risk-neutral probability is used because:
It's the probability embedded in option prices.
It enables consistent valuation and hedging.
Differences from real-world probability are typically small for short-dated options.
Delta as a Probability Proxy
**Delta** (N(d₁)) is often used as an approximation for ITM probability. While not exactly equal to N(d₂), they are closely related.
Delta vs N(d₂)
**Delta = N(d₁)**: The hedge ratio; also represents the expected number of shares to replicate the option.
**ITM Prob = N(d₂)**: The actual risk-neutral probability of expiring ITM.
**Relationship**: d₁ = d₂ + σ√T. Delta is always slightly higher than ITM probability for calls (lower for puts in absolute terms).
Practical Usage
For most practical purposes, traders use delta as a probability proxy because:
Delta is readily available on trading platforms.
The difference from N(d₂) is small, especially for shorter-dated options.
A delta of 0.30 means approximately 30% ITM probability.
Trading Applications and Strategy Selection
ITM probability guides strategy selection and strike choice.
High Probability Trades (60%+)
ITM or deep ITM options have high probability but expensive premiums.
Better suited for **selling**: covered calls, cash-secured puts.
Risk is that losses can be large when the trade fails despite high probability.
Moderate Probability Trades (30-60%)
ATM and slightly OTM options balance probability and cost.
Suitable for directional trades with conviction.
Consider spreads to define risk and reduce cost basis.
Low Probability Trades (<30%)
OTM and far OTM options have low probability but cheap premiums.
Use small position sizes or as part of multi-leg strategies.
Conclusion
**ITM Probability** is a fundamental concept for options traders, providing insight into the likelihood of an option having value at expiration. While derived as risk-neutral probability (not a real-world forecast), it helps traders assess trade quality, select appropriate strikes, and balance risk versus reward.
Combined with delta analysis, payoff diagrams, and position sizing, ITM probability forms the quantitative foundation for systematic options trading strategies.
Frequently Asked Questions
Common questions about ITM probability and options expiration
What is ITM probability?
ITM probability is the likelihood that an option will expire in-the-money, meaning it will have intrinsic value at expiration. For calls, this means spot > strike; for puts, spot < strike. It's calculated using the Black-Scholes d₂ term and represents risk-neutral probability.
Is ITM probability the same as delta?
No, but they're closely related. Delta equals N(d₁), while ITM probability equals N(d₂). Delta is always slightly higher than ITM probability for calls. The difference is σ√T. For practical purposes, delta is often used as a probability proxy since the values are close.
Why is this called "risk-neutral" probability?
Risk-neutral probability assumes all assets grow at the risk-free rate, removing risk preferences from the calculation. It's a mathematical construct used for pricing, not a real-world forecast. Real-world probability would incorporate expected returns and risk premiums.
How does volatility affect ITM probability?
Higher volatility increases the ITM probability of OTM options (more potential for large moves) but decreases the ITM probability of already-ITM options. ATM options are least affected. Volatility creates more uncertainty about the final outcome.
Does time remaining affect ITM probability?
Yes. For OTM options, more time increases ITM probability (more time for favorable moves). For ITM options, time has mixed effects. As expiration approaches, probabilities converge toward 0% or 100% depending on whether the option finishes ITM or OTM.
What ITM probability is considered good for buying options?
There's no universal answer—it depends on strategy. Low probability (20-40%) options are cheap and offer asymmetric upside but usually lose. High probability (60%+) options are expensive with limited upside. Many traders target 40-50% probability for balanced risk/reward.
How do option sellers use ITM probability?
Option sellers want high probability of the option expiring worthless (OTM). They target low ITM probability strikes—typically 70-85% probability of OTM expiration. This provides consistent premium income but with occasional large losses when trades fail.
Does the model account for dividends?
Basic Black-Scholes doesn't include dividends. For dividend-paying stocks, the spot price should be adjusted (reduced by present value of expected dividends) for accurate probability calculation. Dividends reduce call ITM probability and increase put ITM probability.
How accurate is ITM probability for real trading?
ITM probability is a model estimate, not a precise prediction. Real markets have fat tails (extreme moves more frequent than model assumes), volatility isn't constant, and risk-neutral differs from real-world. Use it as a guide, not gospel.
Can I use ITM probability for American options?
ITM probability from Black-Scholes applies to European options (exercise only at expiration). American options can be exercised early, which affects timing but not the final ITM/OTM outcome at expiration. For most practical purposes, the probabilities are similar.
Summary
The ITM Probability Calculator estimates the risk-neutral probability of an option expiring in-the-money using the Black-Scholes d₂ term.
Use ITM probability to assess trade quality, select appropriate strikes, and balance probability of profit against potential payoff.
Remember that risk-neutral probability differs from real-world probability and should be used as a guide alongside other analysis.
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Estimate risk-neutral probability that an option expires in-the-money using Black–Scholes.
How to use Probability of Expiring ITM (Options) Calculator
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Frequently asked questions
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