Binary options reduce a market outcome to a defined payment. A contract might pay if an exchange rate finishes above a stated level at a stated time, with nothing paid if that condition is not met. The payment can be simple while the probability of receiving it remains difficult to estimate.
The mathematics starts with that distinction. Knowing the amount available from a winning contract does not establish whether its price is favourable. A trader needs to compare the potential gain, potential loss and probability of each outcome, then account for charges and uncertainty.
An advertised payout of 80% does not mean an 80% probability of winning. Two possible outcomes do not imply equal odds. Winning more trades than losing can still produce a loss when the average winning payment is smaller than the average losing stake.
The examples below use hypothetical contracts and assume that their stated settlement terms are honoured. They separate the economics of an individual contract from the evidence needed to assess a strategy. Most calculations concern contracts held to expiry; selling or closing earlier creates a different set of realised outcomes.
Start With the Payout Structure
Profit and returned capital are different amounts
Consider a contract requiring a $100 stake and offering an 80% net profit if it wins. A winning settlement returns $180: the original $100 plus $80 profit. A losing settlement returns nothing, producing a $100 loss.
The relevant outcomes are therefore positive $80 and negative $100. Comparing $180 with a $100 loss incorrectly treats the returned stake as income.
The word “payout” can describe either the profit percentage or the total amount returned. Before calculating anything, establish which meaning the contract uses. A total return of 180% of the stake is equivalent to an 80% profit. A total return of only 80% would leave the holder below their starting capital even after the stated winning outcome.
The SEC’s explanation of binary options describes the predetermined cash-or-nothing structure and explains that the holder does not acquire a right to purchase or sell the underlying asset. The contract settles against a proposition rather than delivering the share, currency or commodity itself.
The winning condition defines the probability
A contract paying if EUR/USD finishes above 1.1000 differs from one paying if it touches 1.1000 at any time before expiry. The price can touch that level and finish below it. The same market path would then produce different contract outcomes.
The contract formats discussed in binaryoptions.net guides provide background on such distinctions. For a mathematical model, the governing input is the actual settlement condition, including the expiry time, reference price and treatment of equality.
A high payment may compensate for a low probability of meeting a difficult condition. A smaller payment may accompany an outcome considered more likely. Neither payment size nor apparent simplicity establishes value without a probability estimate.
Break-Even Win Rates and Expected Profit
Let S represent the stake and r the net profit paid on a winning contract as a fraction of that stake. An 80% profit means r = 0.80. Let p represent the probability of winning.
With a full stake lost on failure, no refunds and no additional charges, expected profit is:
E = p × rS − (1 − p) × S.
Dividing by the stake gives expected return per dollar committed:
E ÷ S = p(1 + r) − 1.
For a $100 stake, an 80% profit rate and a 50% win probability, expectancy is $40 minus $50, or negative $10. Equal chances of winning and losing are unfavourable because the losing outcome costs more than the winning outcome earns.
At a 55% win probability, expectancy is $44 minus $45, or negative $1. The trader can be right more often than wrong and still have negative expectancy.
At 60%, the calculation becomes $48 minus $40, or positive $8 per contract. That figure follows from the assumed probability. It does not establish that the trader can achieve it.
Deriving the break-even threshold
Setting expected profit to zero gives p(1 + r) = 1. The break-even win probability is therefore:
p = 1 ÷ (1 + r).
An 80% profit rate requires approximately 55.56% winning contracts before charges. A 70% rate requires approximately 58.82%, while a 90% rate requires approximately 52.63%.
The difference matters when evaluating a signal. A method estimated to win 56% of the time has slightly positive expectancy at an 80% profit rate but negative expectancy at 70%. An unchanged forecast can become unattractive when the payment changes.
The calculation can also be reversed. For an estimated win probability of 60%, the break-even profit rate is 1 ÷ 0.60 − 1, or approximately 66.67%. Payments above that figure imply positive expectancy under the model; payments below it imply negative expectancy.
Expected profit is not a promised account return
Across 100 equal $100 stakes at an 80% profit rate, exactly 60 wins and 40 losses produce $4,800 in winning profits and $4,000 in losses. The net result is $800.
That is 8% of the $10,000 in cumulative stakes. It is not automatically an 8% account return. Account return depends on the starting balance, whether positions overlap and whether stakes change as capital changes.
Nor does a true 60% win probability guarantee 60 wins in the next 100 attempts. Expectancy describes a probability-weighted average. Actual results depend on the outcomes observed.
Fees, Refunds and Changing Payouts
A separate charge changes the threshold. If each contract costs an additional F dollars regardless of outcome, expected profit becomes p × rS − (1 − p) × S − F.
The break-even win probability is then (S + F) ÷ [S(1 + r)]. With a $100 stake, an 80% profit rate and a $1 charge, it rises from 55.56% to approximately 56.11%.
Some fee schedules charge only on winning settlement or on particular transactions. Those charges belong in the outcome where they occur. Averaging them without considering the probability of paying them can distort the result.
Refunds change the losing outcome
Suppose a hypothetical contract returns 10% of the stake after a loss. The loss is then 90% of the stake. With an 80% winning profit and no fees, break-even becomes 0.90 divided by 0.80 + 0.90, or approximately 52.94%.
A refunded tie is different. If ties return the stake with no charge, they have zero profit. The usual break-even rate still applies among decisive wins and losses, provided payouts are unchanged. Counting refunded ties as wins would inflate reported accuracy.
Why average payouts can mislead
Consider 200 equal $100 trades. In the first 100, the winning profit rate is 90%, but only 45 trades win. That group loses $1,450. In the second 100, the profit rate is 50%, and 75 trades win. That group gains $1,250.
The combined result is negative $200. Yet the pooled win rate is 60% and the simple average profit rate is 70%. Substituting those averages into the basic formula incorrectly suggests a positive 2% return on stakes.
The error comes from treating payout and success probability as unrelated. Here, the higher winning rate occurred with the lower payment. Calculating each trade’s cash result, or grouping genuinely comparable contracts, preserves that relationship.
Contract Prices and Implied Probabilities
Some binary contracts are bought at a market price rather than presented as a stake with a quoted profit percentage. The same expected-value logic applies, but the inputs are expressed differently.
Suppose a hypothetical contract costs $42 and pays $100 if its condition is met, or zero otherwise. The buyer can gain $58 or lose $42. Its expected profit before fees is p × $100 − $42.
Break-even occurs at a 42% win probability. If the trader estimates a 48% probability, the model gives $48 minus $42, or $6 expected profit. That is approximately 14.29% of the $42 purchase price, not a guaranteed return.
The CFTC’s explanation of event contracts describes this relationship between purchase price, fixed settlement payment and market expectations. Contract denominations and legal classifications differ, so a $100 settlement value here is an illustration rather than a universal specification.
Market prices are estimates, not measured truth
Dividing price by the winning settlement amount produces a useful implied break-even probability when fees, discounting and settlement risk are ignored. It does not reveal the actual probability with certainty.
A trader’s decision concerns the executable price. If the displayed midpoint is $42 but the lowest available purchase price is $44, the buyer’s threshold is 44% before fees. The midpoint is not an available purchase merely because it appears on a screen.
Charges tied to settlement also affect the calculation. If a $42 contract pays $100 on success but deducts a $1 winning settlement fee, expected profit becomes p × $99 − $42. Break-even is approximately 42.42%.
A high probability of winning can coexist with poor value. Paying $95 for a contract that returns $100 requires a probability above 95% before costs. A belief that the event is “very likely” is not precise enough to establish whether the price is favourable.
Buying both sides does not remove the payout disadvantage
Suppose two contracts cover exactly opposite outcomes, each costs $100 and each pays an 80% net profit if it wins. Buying both costs $200. If exactly one wins, the total returned is $180, producing a $20 loss regardless of direction.
The positions offset uncertainty about which outcome wins, but they do not offset the difference between the amount paid and the amount returned. Any additional charges make the result worse.
For market-priced contracts with mutually exclusive and exhaustive outcomes, the same comparison uses their combined purchase price. If a yes contract costs $42 and a no contract costs $61, and exactly one pays $100, buying both locks in a $3 loss before fees.
This reasoning depends on both contracts having identical settlement definitions and collectively covering every possible outcome. Different expiry times, reference prices or cancellation rules can leave gaps in the supposed hedge. Two contracts with opposite labels do not necessarily provide complementary cash flows.
Expiry and Settlement Are Mathematical Inputs
For a simple above-or-below contract, the relevant probability is that the specified reference price satisfies the condition at settlement. It is not the probability that the market moves in the predicted direction at some point before then.
Suppose a contract pays only if a reference price is strictly above 100.00 at expiry. A settlement at 100.01 wins and one at 99.99 loses. Unless the terms provide otherwise, a settlement exactly at 100.00 fails the strict “above” condition.
The size of the underlying move does not change the terminal payment once the condition has been met. A settlement at 100.01 and one at 105.00 can produce the same profit. This prevents a trader from assuming that occasional large market moves will compensate for numerous losing contracts.
Match the test to the actual contract
A backtest must use the same reference price, timestamps and settlement rule as the contract being evaluated. An average of several observations can differ from the final chart tick. A quoted bid can differ from an ask or midpoint.
Timing errors are especially consequential near expiry. A model that decides to trade using information from a completed candle cannot assume it entered earlier within that candle. It must use an entry that would have been available after the information arrived.
Shorter expiries also create more opportunities to trade within a session, but additional trades do not repair negative expectancy. Under identical assumptions, repeating an unfavourable contract increases expected monetary loss in proportion to total stakes. Overlapping contracts may also expose the account repeatedly to the same movement, making their outcomes dependent.
How Much Evidence Does a Win Rate Provide?
An observed win rate is an estimate. Its precision depends on the amount and quality of the evidence, rather than on how many decimal places a spreadsheet displays.
Suppose a method wins 60 of 100 contracts. Under a model of independent outcomes with a constant win probability, a 95% Wilson confidence interval runs from approximately 50.2% to 69.1%. This calculation uses the method described in NIST’s guidance on confidence intervals for proportions.
The 55.56% break-even threshold for an 80% profit rate falls inside that interval. The sample therefore leaves substantial uncertainty about which side of the profitability threshold the underlying probability lies.
Even 240 wins from 400 contracts gives an interval of approximately 55.1% to 64.7% under the same model. More observations narrow the interval, but they do not automatically establish an advantage after fees.
These intervals describe a statistical procedure under stated assumptions. They are not guarantees about the next group of trades.
Variation around positive expectancy
Return variability remains substantial even if the probability is known. For a $100 stake paying $80 profit on success, the two net outcomes are $180 apart.
At a 60% win probability, the standard deviation of one trade’s result is $180 × √(0.60 × 0.40), approximately $88.18. Across 100 independent, identical trades, the standard deviation of total profit is ten times that amount, approximately $881.82.
Expected total profit is $800, but the scale of variation around that expectation is large. A positive model does not imply a smooth sequence of gains or a profitable result over every short sample.
Independence and selection affect the evidence
Several contracts on the same asset with nearly identical expiries may respond to one price movement. Counting them as independent observations exaggerates the amount of information collected.
A strategy chosen after testing hundreds of indicator combinations also needs fresh evaluation. The best result from that search benefits from having been selected for looking good. Testing it again on the same historical period does not provide independent confirmation.
The record should preserve the offered payout, accepted entry, expiry condition, settlement result and stake for every eligible trade. A chart-based signal can appear successful while the associated contracts lose money because the actual payouts were lower or entries occurred later.
A later period reserved for evaluation is useful only if the trading rules remain fixed during that test. Changing the rules after inspecting results turns the period into further development data.
Stake Size, Compounding and the Kelly Criterion
A stake determines how much capital is exposed to the contract’s losing outcome. In a simple fully paid contract, that exposure is the stake plus any additional charges. There is no need for a large underlying price move to lose that amount.
If a $10,000 account commits $100 to a contract, it exposes 1% of its starting equity before separate fees. Ten identical simultaneous positions can expose $1,000. Their individual limits do not prevent a 10% combined loss if they all settle unsuccessfully.
Fixed stakes and fixed fractions
With a fixed $100 stake, ten consecutive losses reduce a $10,000 account to $9,000 before fees. The stake then represents approximately 1.11% of remaining capital rather than its original 1%.
If each stake instead equals 1% of current equity, ten losses leave $10,000 × 0.99 to the power of 10, approximately $9,043.82. At 5% of current equity, ten losses leave approximately $5,987.37.
These percentages are illustrations of account behaviour, not suggested stake sizes. Minimum contract sizes and charges can prevent exact fractional sizing as the account becomes smaller.
For a constant fraction f and winning profit rate r, a win multiplies equity by 1 + rf. A loss multiplies it by 1 − f. At a 10% stake fraction and an 80% profit rate, one win followed by one loss multiplies capital by 1.08 × 0.90, or 0.972. The account loses 2.8%.
With unchanged payout and fractional sizing, the final balance for a given number of wins and losses is independent of their order. The drawdown experienced along the way is not. A run of losses concentrated early or late can produce a different peak-to-trough decline.
What Kelly sizing assumes
The growth criterion associated with J. L. Kelly’s original paper maximises expected logarithmic capital growth. For repeated independent binary outcomes with known, constant probability p, winning profit rate r and no fees, the unconstrained optimal stake fraction is:
f = [p(1 + r) − 1] ÷ r.
At p = 60% and r = 80%, this gives 10%. If the true probability is 56%, it gives only 1%. A small estimation error therefore changes the calculated exposure sharply.
Under this model, using the 10% fraction when the true probability is 56% produces negative expected logarithmic growth, despite slightly positive arithmetic expectancy. Excessive sizing can undermine a strategy that has a small advantage.
Kelly calculations do not establish the probability being entered. They also do not incorporate personal loss tolerance or an obligation to preserve capital over a short period. Where the probability is estimated poorly, the apparent precision of the answer is misleading. A negative calculated fraction provides no case for staking on that side under the assumed model.
Losing Sequences and Martingale Systems
A sequence of losses can occur even when individual contracts have positive expectancy. Its probability depends on the outcome model and the number of opportunities for the sequence to appear.
If trades are independent and each has a 60% win probability, the probability that the next six all lose is 0.40 to the power of six, or 0.4096%.
That is the probability for one specified block of six trades. It is not the probability of encountering any six-loss sequence during a long trading record. Repeated trading creates many possible starting points for such a run.
The same calculation is also invalid if the loss probability changes during the sequence. A strategy exposed to a market condition it handles poorly may produce clustered losses rather than independent outcomes.
Why simple doubling fails at an 80% profit rate
A martingale system increases the stake after losses, aiming to recover them with a subsequent win. The familiar doubling version relies on assumptions that do not match a contract paying less than 100% net profit.
Start with a $10 stake. After a loss, stake $20. A win at an 80% profit rate earns $16, leaving $6 after recovering the first $10 loss. That is below the $8 earned by a first-trade win.
After three losses at stakes of $10, $20 and $40, cumulative losses are $70. A winning $80 stake earns $64. The sequence still loses $6 despite the final win.
Doubling does not guarantee recovery at this payout. The shortfall grows as the sequence extends.
Calculating the actual recovery requirement
To recover accumulated losses L and earn a target profit G at a winning profit rate r, the next stake must satisfy rS = L + G. Therefore:
S = (L + G) ÷ r.
With a target of $8 and an 80% profit rate, the first stake is $10. If it loses, the next required stake is $22.50. Further losses require approximately $50.63, $113.91, $256.29 and $576.65, with these figures displayed after rounding an otherwise unrounded calculation.
Each required stake is 2.25 times the previous one. Six consecutive losses consume approximately $1,029.97, and the seventh required stake is approximately $1,297.46. The initial $10 commitment gives little indication of the capital the recovery rule can demand.
Platform limits, available funds and changing payouts can prevent the next stake from being placed. Stopping the progression limits further exposure but leaves the accumulated loss realised.
Changing stakes does not create an advantage
Suppose each new contract has the same negative expected return per dollar, regardless of earlier outcomes. Increasing the stake after a loss increases the expected monetary loss on the next contract. It does not change the sign of expectancy.
A losing streak alone does not make the next independent outcome more likely to win. Any claim that probability has improved requires evidence about the market or signal, rather than the account’s recent losses.
Progressive staking can produce many small completed recoveries and occasional large failures. Evaluating only the frequency of profitable sequences misses the size of those failures. The full distribution of cash outcomes is the relevant measurement.
Evaluating a Binary Options Strategy and Its Provider
A useful evaluation begins with the exact cash flows and then tests whether the probability estimate is credible.
Consider a hypothetical $50 stake with an 80% winning profit rate and a $0.50 fee charged on every contract. At an estimated 58% win probability, expected profit is 0.58 × $40 − 0.42 × $50 − $0.50, or $1.70.
The break-even win probability is $50.50 divided by $90, approximately 56.11%. The estimate clears that threshold by less than two percentage points. If the true win probability is 55%, expected profit becomes negative $1.
The decision therefore depends more on the quality of the probability estimate than on the simplicity of the payout. A calculation producing $1.70 to two decimal places can still rest on an uncertain input.
Separate contract mathematics from legal availability
For UK readers using binaryoptions.co.uk guides to compare terminology and contract formats, legal availability requires a separate check against primary sources. The FCA’s current binary options rules prohibit firms within their scope from marketing, distributing or selling these products to retail clients. The permanent prohibition took effect on 2 April 2019.
A UK-facing website or an overseas account accepting registrations does not establish that a provider is authorised to offer the product to UK retail customers. Rules elsewhere depend on the jurisdiction, contract and provider.
Settlement reliability is also an assumption in every expected-value calculation. The CFTC documents complaints about binary options platforms involving withheld funds and manipulated software. A profitable theoretical contract cannot compensate for a payment that is not made.
Reconcile the model with actual results
The trading record should connect each stake to the terms accepted at entry and the cash ultimately credited or debited. Refunds, early closures and separate charges need their own treatment rather than being forced into a simple win-or-loss label.
Results should be assessed using the payouts available when the trades occurred. Applying today’s highest advertised rate to past signals can manufacture a profit that the historical contracts would not have delivered.
The central calculation remains the same: probability-weighted receipts minus the amount paid and all applicable costs. The harder work is establishing whether the probabilities, contract terms and settlement assumptions used in that calculation are supported by evidence.