Probability and expectancy

Probability of finishing in the money

The risk-neutral probability that the underlying is beyond the strike at expiration. It is N(d2), not N(d1), and therefore not delta.

Field	Value
Formula	P(S_T > K) = N(d2); P(S_T < K) = N(-d2)
Call ITM	N(d2)
Put ITM	N(-d2)
Relation to delta	Delta uses d1 = d2 + sigma*sqrt(T), so delta always exceeds N(d2) for a call
Worked	Reference inputs: S = 100, r = 0.04, q = 0, sigma = 0.20, T = 0.25. At K = 105: d2 = -0.437902, so N(d2) = 0.330729. Call delta at the same strike is 0.367719. The gap is 0.036990, so using delta as the probability overstates it by 3.7 percentage points at this strike

• The gap between delta and N(d2) widens with sigma and with T, because it is driven by sigma*sqrt(T). On a long-dated or high-volatility contract the delta proxy is badly wrong.
• This is a risk-neutral probability. It is the probability under the measure that prices the option, which is not the probability under which the world evolves, and the two differ by the risk premium.
• A quoted "probability ITM" from a broker platform is normally N(d2) computed from that platform implied volatility. Changing the volatility input changes the number.

Probability of touch

The probability that the underlying reaches a level at any time before expiration, rather than merely finishing beyond it. For a driftless underlying it is approximately twice the probability of finishing beyond the level, by the reflection principle.

Field	Value
Formula	P(touch K) approx 2 * P(S_T beyond K) = 2 * N(d2), capped at 1
Basis	Reflection principle for Brownian motion: every path finishing beyond the barrier has touched it, and paths that touch and return contribute the second half
Cap	The approximation must be capped at 1; it breaks down for levels close to spot
Worked	Reference inputs: S = 100, r = 0.04, q = 0, sigma = 0.20, T = 0.25. At K = 105, N(d2) = 0.330729, so P(touch) is approximately 0.661458, twice the 0.330729 probability of finishing above. At K = 100 the formula gives 0.960122, which is near its cap and should be read as "almost certainly"

• Probability of touch is roughly double probability in the money for out-of-the-money levels. Any position that can be closed or assigned before expiration is exposed to touch, not to terminal probability.
• The doubling is exact only with zero drift and a constant volatility. With a drift term the two halves of the reflection are no longer equal.
• For a short option that will be managed rather than held to expiry, touch is the operative probability and terminal probability is the irrelevant one.

Probability of profit

The probability that a position finishes on the profitable side of its breakeven, which is a different level from any strike in the structure. It is computed by evaluating the terminal distribution at the breakeven price, not at a strike.

Field	Value
Formula	POP for a single breakeven B, profitable above = N(d2 evaluated at K = B)
Two-sided structures	POP = N(d2 at lower breakeven) - N(d2 at upper breakeven)
Long premium structures	POP is the complement: the probability of finishing outside the breakevens
Worked	Reference inputs: S = 100, r = 0.04, q = 0, sigma = 0.20, T = 0.25. Bull put spread 90/95 collected for 1.15 has breakeven 93.85. POP = N(d2 at 93.85) = 0.753241. The 90/95/105/110 iron condor collected for 1.60 has breakevens 93.40 and 106.60, so POP = 0.768156 - 0.277886 = 0.490270

• POP and probability of maximum profit are different numbers. The condor above has a 0.490270 probability of any profit and a smaller probability of the full 160.00.
• A high POP with a small maximum profit and a large maximum loss carries no information on its own. Pair it with the breakeven win rate from the payoff ratio before it means anything.
• The condor worked example has a risk-neutral POP of 0.490270 against a breakeven win rate of 0.68. Those two figures being inconsistent is a signal about the premiums used in the example, which are stated inputs rather than model prices, not a signal about the structure.

Expected value of a defined-risk position

The probability-weighted average outcome. For a two-outcome simplification it reduces to one line; for the true continuum it is an integral over the terminal distribution.

Field	Value
Formula	EV = p * MaxProfit - (1 - p) * MaxLoss
Two-outcome form	EV = p*MaxProfit - (1-p)*MaxLoss
General form	EV = integral over S of Payoff(S) * f(S) dS, where f is the terminal density
Breakeven probability	p* = MaxLoss / (MaxProfit + MaxLoss)
Equivalent form	p* = 1 / (1 + b), where b = MaxProfit / MaxLoss
Worked	MaxProfit 1.60, MaxLoss 3.40. b = 0.470588. p* = 3.40 / 5.00 = 0.68 exactly, and 1 / 1.470588 = 0.68 confirms it. At p = 0.70: EV = 0.70 * 1.60 - 0.30 * 3.40 = 1.12 - 1.02 = +0.10 per share, 10.00 per contract. At p = 0.68: EV = 1.088 - 1.088 = 0.00. At p = 0.60: EV = 0.96 - 1.36 = -0.40

• The two-outcome form ignores every partial outcome between the breakevens and the wings, and for a condor or butterfly those are a large share of the distribution. It is an upper-bound sketch, not the expectancy.
• Under the pricing model own measure a fairly priced position has an expected value of zero net of carry. Every positive EV produced by this arithmetic comes from the probability input, which is an assumption supplied from outside the model.
• EV per contract is EV per share times the multiplier. EV per unit of capital is EV divided by the buying-power reduction, and it is the only version comparable across structures.

Kelly sizing applied to a defined-risk options position

The fraction of capital that maximises the expected logarithm of wealth for a repeated bet with known probability and known payoff ratio. For a defined-risk position the payoff ratio is fixed at entry, so only the probability is an assumption.

Field	Value
Formula	f* = (p*b - (1 - p)) / b, where b = MaxProfit / MaxLoss
Equivalent form	f* = p - (1 - p) / b
Interpretation of f*	Fraction of capital placed at risk, meaning the fraction exposed to MaxLoss, not the notional
Contracts implied	floor(f* * Capital / (MaxLoss * multiplier))
Negative f*	Occurs whenever p is below the breakeven win rate, and indicates no size satisfies the criterion
Worked	MaxProfit 1.60, MaxLoss 3.40, so b = 0.470588. At p = 0.70: f* = (0.70 * 0.470588 - 0.30) / 0.470588 = 0.029412 / 0.470588 = 0.0625, i.e. 6.25 percent of capital at risk. On 50,000.00 of capital that is 3,125.00 at risk, which at 340.00 of maximum loss per contract is 9 contracts. At p = 0.68 the formula returns 0.0000 and at p = 0.60 it returns -0.250000

• Kelly assumes the probability is known, the bet is repeatable, and outcomes are independent. Options positions on correlated underlyings violate the independence assumption, and the probability is never known.
• The criterion is extremely sensitive to the probability input near the breakeven win rate. In the worked case a two-point error in p moves f* from 6.25 percent to zero.
• A fractional Kelly - half of f*, for instance 3.125 percent in the worked case - reduces growth rate modestly and reduces drawdown substantially. That trade-off is arithmetic, not judgement.

Source: Kelly 1956

Also described at: Wikipedia · Wikidata

Win rate versus payoff ratio

The two figures are jointly constrained: for a fixed expected value, a change in the payoff ratio implies an exact change in the required win rate. Neither number is informative alone.

Field	Value
Formula	p_required = (EV_target + MaxLoss) / (MaxProfit + MaxLoss)
For EV = 0	p* = MaxLoss / (MaxProfit + MaxLoss) = 1 / (1 + b)
Sensitivity	dp*/db = -1 / (1 + b)^2
Worked	MaxProfit 1.60, MaxLoss 3.40, total 5.00. For EV = 0: p* = 3.40 / 5.00 = 0.6800. For EV = +0.10: p = (0.10 + 3.40) / 5.00 = 0.7000. For EV = +0.25: p = 3.65 / 5.00 = 0.7300. Sensitivity at b = 0.470588: dp*/db = -1 / 1.470588^2 = -0.4624, so a 0.10 increase in the payoff ratio lowers the required win rate by about 4.6 percentage points

• A high win rate and a low payoff ratio can describe exactly the same expectancy as the reverse. Any comparison quoting one without the other is incomplete by construction.
• The sensitivity term is largest at small b, so structures with a small maximum profit relative to maximum loss are the ones whose required win rate moves most for a small change in the payoff ratio.

Expected move compared to breakeven distance

The comparison that decides whether a structure needs more movement than the premium is charging for. Both sides are computable from stated inputs and neither requires a market view.

Field	Value
Formula	One-standard-deviation move = S * sigma * sqrt(T); Breakeven distance = |Breakeven - S|
One-sigma move	S * sigma * sqrt(T)
Expected absolute move	S * sigma * sqrt(2*T/pi), approximately 0.7979 * the one-sigma figure
Ratio to compare	Breakeven distance / one-sigma move, expressed in standard deviations
Worked	Reference inputs: S = 100, r = 0.04, q = 0, sigma = 0.20, T = 0.25. One-sigma move = 100 * 0.20 * 0.5 = 10.0000. Expected absolute move = 100 * 0.20 * 0.398942 = 7.9788. A long 100 straddle for 6.00 has breakevens at 106.00 and 94.00, a distance of 6.00, which is 0.60 standard deviations. A long 95/105 strangle for 3.00 has breakevens at 108.00 and 92.00, a distance of 8.00, which is 0.80 standard deviations

• Breakeven distance measured in standard deviations is the only form comparable across underlyings and tenors. Measured in points or in percent it is not.
• The expected absolute move is 0.7979 of the one-sigma move for a normal distribution, so a breakeven inside one standard deviation is not the same as a breakeven inside the expected move.
• These figures follow from the volatility input. Changing sigma changes both sides of the comparison, which is why the comparison is a statement about the input, not a prediction.