Probability of touch vs expiry ITM

A crossing can reverse before expiry

A stock can rise from $100 to $110, then finish at $103. The $110 barrier was touched, but a $110 call expires out of the money. An upper barrier tracks the maximum price; a lower barrier tracks the minimum. Neither event tells you whether an option position earns a profit after its premium and costs.

Match the inputs to the contract

1. Check the underlying and quote timestamp in TradingView or your contract data. Enter the underlying spot price and the strike or other fixed barrier in the same currency.
2. Use the remaining horizon to the contract expiry. A 30-day input means 30/365 years here, not 30 trading sessions.
3. Enter annualized volatility as a percent: 20 means 20%. Contract IV is a pricing input. Historical volatility is an alternative scenario assumption, not a current contract quote.
4. Enter annual continuously compounded risk-free rate and dividend yield. Zero is an explicit assumption; leave an unknown field blank to suppress the result. This model uses continuous yield, not scheduled cash dividends.

A reproducible 30-day example

Set spot to $100, upper barrier to $110, time to 30 days, volatility to 20%, rate to 5%, and yield to 0%. The log-price drift is 0.05 − 0.20²/2 = 0.03 per year. The model gives 10.3545% probability of touch and 5.2698% probability of ending above $110. Their ratio is about 1.965, rather than a fixed multiplier.

For the same inputs with a $90 lower barrier, the direction of log-price drift reverses in the hitting calculation. The calculator then compares touching $90 with ending below $90, the expiry ITM event for a $90 put. Changing the barrier does not change the underlying drift assumption.

The continuous-barrier formula

Under the pricing measure, log(Sₜ/S₀) = m t + σ Wₜ, where m = r − q − σ²/2. Set a = |ln(H/S₀)| and d = m for an upper barrier or −m for a lower barrier. For T > 0, σ > 0 and H ≠ S₀:

Φ is the standard normal cumulative distribution. The second term counts paths that hit and then return. This follows from the Brownian reflection principle with a drift change; MIT's Brownian motion notes derive the reflection principle. Our lower-barrier formula applies it to the negative log-price process.

Why twice delta is not an identity

When log-price drift d is zero, the two formula terms coincide: touch is twice the terminal crossing probability. That special result concerns a terminal event, not option delta. In Black-Scholes, a call's delta is e^(−qT)Φ(d₁), while its expiry ITM probability is Φ(d₂), with d₁ = d₂ + σ√T. The Options Industry Council defines delta as premium sensitivity. Doubling it is a trading shortcut, not this calculator's method.

What the model leaves out

• Risk-neutral drift uses r − q for the price process. A physical probability needs an estimated real-world price drift and a suitable return model. Entering historical volatility alone does not turn this result into a physical forecast.
• Continuous monitoring counts every crossing. Checking only daily closes can miss intraday crossings, so its touch probability is no greater under the same paths and barrier. Broadie, Glasserman and Kou study this monitoring difference. This tool does not apply a discrete-monitoring correction.
• GBM has continuous paths and constant volatility. Earnings jumps, overnight gaps and changing volatility violate those assumptions. A gap can cross a level without trading at it; contract barrier rules and stop execution determine what that means in practice.
• Longer time cannot reduce a fixed barrier's cumulative hit probability when the other model inputs stay fixed. Volatility changes both dispersion and log drift here, so a universal monotonic volatility claim is unsafe.
• Touch does not price a barrier option, predict assignment, or calculate a multi-leg strategy's probability of profit. Premiums, exercise rules and execution costs require separate analysis.

Frequently asked questions

Educational model estimates, not investment advice. Positive spot and barrier prices and nonnegative time and volatility are required. Negative rates are allowed. Inputs that exceed floating-point range return an error instead of a probability.