How to use the option probability calculator: probability of touch and expected move
- Enter the current stock price and days to expiration.
- Enter implied volatility, ideally the IV of options at the same expiration.
- Enter the risk free rate.
- Set the lower and upper prices you care about, such as short strikes or price targets.
- Read the chance of finishing above, below and between the two prices, and the chance of touching each one.
- Use the 1 and 2 standard deviation ranges and the distribution chart to see the expected move.
What the option probability calculator shows
The calculator answers two different questions. The first is where the stock may finish at expiration: the probability it ends above the upper price, below the lower price, or between them. The second is whether the stock reaches a price at any point before expiration, called the probability of touch. Touching a level is always at least as likely as finishing beyond it, because a stock can touch a price and then move back.
It also works as an expected move calculator. It shows the one standard deviation move implied by the volatility you enter, the 1 and 2 standard deviation price ranges, and a chart of the lognormal distribution of prices at expiration with your two levels marked.
Probability formulas (lognormal model)
The calculator assumes the stock's log return to expiration is normally distributed, as in Black Scholes. T is days ÷ 365, σ is implied volatility, r is the risk free rate, q is the dividend yield (zero unless set) and N(x) is the standard normal cumulative distribution function. U is the upper price and L the lower price. The drift μ is the risk neutral growth rate, r minus q, less the volatility adjustment σ² ÷ 2, over the time to expiration:
μ = (r minus q minus σ² ÷ 2) × T sd = σ × √T P(finish above U) = 1 minus N([ln(U ÷ S) minus μ] ÷ sd) P(finish below L) = N([ln(L ÷ S) minus μ] ÷ sd) P(between) = 1 minus P(above U) minus P(below L) Expected move (1 sd) = ± S × σ × √T 1 sd range = S × e^(μ minus sd) to S × e^(μ + sd) 2 sd range = S × e^(μ minus 2 × sd) to S × e^(μ + 2 × sd)
Probability of touch formula
As a probability of touch calculator, it uses the barrier hitting formula for a stock that follows geometric Brownian motion with annual drift ν = r minus q minus σ² ÷ 2. If a level is already at or beyond the current price, its probability of touch is 100%. Otherwise the calculator uses the two formulas below, one for an upper level U above S and one for a lower level L below S.
When the drift is small, the two terms in each formula are nearly equal. That is where the familiar rule of thumb comes from: the probability of touching a level is roughly twice the probability of finishing beyond it. The formulas, with sd = σ × √T, are:
ν = r minus q minus σ² ÷ 2 P(touch U) = N([ln(S ÷ U) + ν × T] ÷ sd) + (U ÷ S)^(2 × ν ÷ σ²) × N([ln(S ÷ U) minus ν × T] ÷ sd) P(touch L) = N([ln(L ÷ S) minus ν × T] ÷ sd) + (L ÷ S)^(2 × ν ÷ σ²) × N([ln(L ÷ S) + ν × T] ÷ sd)
Worked example: $100 stock, 45 days, 30% IV
The calculator loads with a $100 stock, 45 days to expiration, implied volatility of 30%, a 4.1% risk free rate, a lower price of $90 and an upper price of $110. Then T = 0.1233, sd = 0.30 × √0.1233 = 0.1053 and μ = (0.041 minus 0.045) × 0.1233 = negative 0.00049. The results are:
- Finish above $110: 18.2%, because [ln(1.10) minus μ] ÷ sd = (0.0953 + 0.0005) ÷ 0.1053 = 0.909, and 1 minus N(0.909) = 0.182
- Finish below $90: 16.0%, because [ln(0.90) minus μ] ÷ sd = (negative 0.1054 + 0.0005) ÷ 0.1053 = negative 0.996, and N(negative 0.996) = 0.160
- Finish between $90 and $110: 65.9%
- Touch $110 before expiration: 36.4%
- Touch $90 before expiration: 31.9%
- Expected move: ±$10.53, with a 1 standard deviation range of $89.96 to $111.05 and a 2 standard deviation range of $80.96 to $123.39
Risk neutral probabilities are not forecasts
These are risk neutral probabilities: the odds implied by option prices under the model's assumptions, not predictions of what the stock will do. They use the risk free rate as the stock's expected growth instead of a forecast return, and they inherit the lognormal assumption, which understates the chance of sudden jumps such as earnings gaps. Implied volatility can also include a premium for risk, so these numbers can differ from how often moves actually happen.
In the example, touching each level is about twice as likely as finishing beyond it: 36.4% versus 18.2% at $110, and 31.9% versus 16.0% at $90. That gap matters for short option positions, because a level that is touched can trigger an adjustment or a stop even if the option later expires out of the money. Use the probabilities to compare strikes and expirations on a consistent basis, not as a guarantee or as advice to trade.
Frequently asked questions
How do you calculate the probability of an option expiring in the money?
You calculate it with the lognormal model: the risk neutral probability that a call finishes in the money is N(d2), and for a put it is N(minus d2). This calculator applies the same math to any price level, using 1 minus N([ln(U ÷ S) minus μ] ÷ (σ × √T)) for finishing above a level U. The result depends heavily on the implied volatility you enter.
What is probability of touch?
Probability of touch is the chance that the stock trades at a given price at any time before expiration, not just at expiration. It is always at least as high as the probability of finishing beyond that price, and with little drift it is roughly twice as high. The calculator computes it with the barrier hitting formula for geometric Brownian motion.
How do you calculate the expected move of a stock?
The expected move is the stock price times implied volatility times the square root of the time to expiration in years: S × σ × √T. With a $100 stock, 30% IV and 45 days, that is 100 × 0.30 × √(45 ÷ 365) = ±$10.53. It is a one standard deviation range, which the model expects to contain the price about 68% of the time.
Is delta the probability of expiring in the money?
Delta is only an approximation of the probability of expiring in the money. Call delta is based on N(d1), while the risk neutral probability of finishing in the money is N(d2), which is lower. The gap grows with higher volatility and longer expirations.
Are option probabilities accurate predictions?
No, option probabilities are risk neutral estimates implied by prices, not predictions. They assume lognormal returns at constant volatility and use the risk free rate as the stock's growth rate. Real stocks can jump and their volatility changes, so actual outcomes can differ widely.
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Educational tool, not financial advice. Calculations are theoretical and depend on your inputs; real fills, fees, early assignment and dividends can change results. Options involve risk and are not suitable for every investor.