Expected Value (EV)
What you make on average per trade if it were repeated infinitely — the central concept on which all professional trading is built.
The Mathematical Definition
Expected Value (EV) is one of the most powerful and most abused ideas in mathematics applied to trading. Formally, for a random variable X with possible outcomes x₁, x₂, …, xₙ and probabilities p₁, p₂, …, pₙ (where Σpᵢ = 1), expected value is defined as EV = Σ(pᵢ × xᵢ). That is, the weighted sum of every possible outcome multiplied by its respective probability. It is what you would average if you could repeat the same experiment infinitely. For trading, the "outcomes" are the possible P&Ls of a position, and the "probabilities" come from the model (POP to win, 1−POP to lose) or from historical analysis. A simple example: a fair coin flip where you win $10 on heads and lose $5 on tails. EV = (0.5 × 10) + (0.5 × −5) = 5 − 2.50 = +$2.50. On average, each flip makes you $2.50, even though a single flip can only produce +$10 or −$5. The law of large numbers guarantees that if you repeat the experiment enough times, your average P&L converges to $2.50 per flip. This concept is so fundamental that casinos are literally built on it: every game has a negative EV for the player (and positive for the house), and the law of large numbers makes the house systematically make money at scale.
Positive vs Negative EV in Trading
The only universal rule of professional trading is: only trade with positive EV. It sounds obvious but it is surprisingly hard in practice. Many popular technical patterns have an expected value near zero once costs are deducted; many strategies "proven" in backtests have positive EV on historical data and zero in forward testing because of overfitting; almost all popular lotteries and many retail derivative structures have negative EV for the buyer. Identifying positive EV requires a real edge: asymmetry in information (hard, and increasingly rare), asymmetry in execution such as that of market makers, asymmetry in costs (low fees), or structural asymmetry in the market (like the variance risk premium in options). A positive-EV trade can lose most of the time and still be systematically profitable — buying OTM puts in macro risk-off trades has historically had positive EV even failing 7 times out of 10, because the 3 winners pay multiples. Conversely, a negative-EV trade can win most of the time and still lose money systematically — selling far OTM options without management discipline can win 90% of the time, but the 10% of losses are catastrophic. The most dangerous cognitive bias in trading is counting wins rather than analysing EV; never evaluate a strategy by its hit rate in isolation.
A Coin Flip in Disguise
An example with realistic figures clarifies why EV matters. You buy a long call with: a 60% probability of expiring worthless (a $200 premium loss), a 25% probability of expiring near breakeven (a $50 gain), and a 15% probability of a large rally (an $800 gain). EV = (0.60 × −200) + (0.25 × 50) + (0.15 × 800) = −120 + 12.50 + 120 = +$12.50. Positive, but barely — it hardly covers commissions. Now consider the same call with a better outcome skew: 55% probability of total loss (−$200), 25% near breakeven (+$60), 20% large rally (+$900). EV = −110 + 15 + 180 = +$85. The second trade has only a slightly better hit rate but a much better expected value, because the large rally pays more and is more likely. In practice, analysing any potential trade this way — estimating the probability distribution of outcomes and calculating EV honestly — is more useful than memorising technical patterns. The difficulty is that estimating the real probabilities requires solid historical data or well-calibrated models; in practice, the inputs to the calculation are approximate, and much of the edge consists of being systematically more accurate than the consensus in those estimates.
EV in Options: The Fair Value Question
In the world of options, EV is intimately tied to the question of fair value. An option is at fair value when its market price equals its mathematical expected value under the pricing model. If you buy an option for $2.00 and its Black-Scholes fair value is $2.10, your expected value is +$10 per contract, before commissions. If you pay $2.20 for that same option, your expected value is −$10. The market tends to keep options near fair value through arbitrage, but with predictable deviations: (1) relatively overpriced options are those with elevated IV (OTM puts in equities, OTM calls in commodities in backwardation), which gives negative expected value to the buyer and positive to the seller; (2) underpriced options are rare but occur in situations of illiquid demand or extreme sentiment. The variance risk premium — the phenomenon whereby implied volatility typically sits 3% to 5% above the volatility that subsequently materialises — is the structural reason systematic premium-selling strategies have positive expected value over the long run. Rigorous academic studies (Bakshi & Kapadia 2003, Carr & Wu 2009, among many others) have documented this premium in SPX and quantified it at roughly 2-4% annualised, risk-free. It is not a free-money trade, because the accumulated drawdowns can be severe, but it is a positive and persistent expected value.
EV and the Casino House
The best illustration of why EV matters more than hit rate is the casino. In American roulette (with 0 and 00), the player has roughly a 47% chance of winning a red/black bet (18 red numbers out of 38 total). The house wins slightly more than half the time — 52.6% against the player’s 47.4% — which translates into an edge of 5.26% on every dollar wagered. But the house wins every year, every month, every day. Why? Because: (1) the edge is consistent on every spin; (2) the house runs thousands of spins a day; (3) the law of large numbers guarantees the observed result converges to the theoretical EV; (4) the house has the capital advantage to tolerate the variance. This is exactly what a professional trader does: find a small but consistent statistical edge, apply it in volume, manage risk so as not to be wiped out by a bad run, and collect the accumulated EV over time. Failing traders do the opposite: they chase occasional big wins, trade setups with expected value that is barely positive or outright negative, and change strategy after every bad run. The moral: trading successfully means scaling a small edge, not finding the perfect trade. If your strategy has an EV of +$10 per trade and you make 500 trades a year, you make $5,000 before commissions. At 2,000 trades, $20,000. If you multiply size by five with discipline and without blowing up, $100,000. The path to wealth in trading is the arithmetic of compounded EV, not brilliant directional speculation.
Estimating EV Honestly: Common Traps
Estimating EV correctly is harder than it looks, and there are several systematic traps. (1) Overconfidence in probabilities: we tend to overestimate the probability of events that confirm our thesis. Countermeasures: use objective models (Black-Scholes POP), historical frequencies from empirical data, and critical review of the premises. (2) Ignoring fat tails: normal models underestimate extreme events (Black Monday, flash crashes). Add 10-20% to the probability of maximum loss as a conservative adjustment. (3) Not counting costs: spread, commissions, slippage and margin financing all reduce expected value. An option bought at the ask and sold at the bid loses the spread; in high-frequency trading that can turn positive EV negative. (4) Selection bias in backtests: those that exclude bear markets underestimate drawdowns, and those covering one specific period can produce a very different expected value from the one the next will offer. (5) Regime changes: a strategy with positive expected value in bull markets can have negative EV in bears, and vice versa. (6) Ignoring correlation: several trades with individually positive expected value can produce a lower combined EV if they are correlated. Professional practice: calculate expected value with conservative assumptions, include all costs, adjust for fat tails, diversify across genuinely independent trades, and re-evaluate regularly.