Expected Value (EV)
Expected value answers one question: if I made this exact trade thousands of times, would I come out ahead or behind?
It is the single most important concept in trading. Not technical analysis. Not chart patterns. Not "conviction." Every profitable trading desk in the world, from market makers to hedge funds, operates by finding and repeating positive EV trades.
EV is what you expect to make (or lose) on average per trade, calculated by multiplying each possible outcome by its probability and adding them up.
Start Simple: A Coin Flip
Before options, think about a bet. Someone offers you a coin flip:
- Heads: you win $200
- Tails: you lose $100
Should you take it? Your gut might say "it's 50/50, risky." But do the math:
The EV is +$50 per flip. You should take this bet every time it is offered. You will lose individual flips, but the wins are large enough to more than cover the losses.
Win rate and EV are not the same thing. A trade that wins 80% of the time can have negative EV. A trade that loses 60% of the time can have positive EV. What matters is how much you win vs. how much you lose, weighted by how often each happens.
The Formula
Hover each term for a definition. "loss" is negative, so the second term subtracts from the first.
That is the whole formula. Everything else on this page is applying it.
Why Win Rate Lies to You
This is the trap that catches most beginners. Look at these two traders:
Trader A: 80% win rate
Sells OTM puts
- Wins 80 out of 100 trades
- Average win: +$50
- Average loss: -$400
- Total: (80 × $50) + (20 × -$400) = -$4,000
- Net result: LOST $4,000
Trader B: 35% win rate
Buys OTM calls
- Wins 35 out of 100 trades
- Average win: +$600
- Average loss: -$100
- Total: (35 × $600) + (65 × -$100) = +$14,500
- Net result: MADE $14,500
Trader A looks like a genius 80% of the time. Trader B looks like a loser 65% of the time. But Trader B is the one making money. This is why EV, not win rate, is the number that matters.
Trader A's strategy is the classic "picking up pennies in front of a steamroller." Small, frequent wins feel good. But when the loss comes, it is catastrophic relative to the wins. Option sellers who ignore EV fall into this trap constantly.
EV Calculator
Plug in your own numbers. Try different combinations to build intuition for how probability and payoff size interact.
Applying EV to Options
Now let's use a real options trade. You buy a BTC call option for $200 premium. You estimate:
- 40% chance BTC rallies enough for the option to be worth 500**)
- 60% chance the option expires worthless (you lose your $200 premium)
The EV is +$80 per trade. You will lose money on 60% of these trades. That can feel terrible. But the 40% that win pay enough to more than compensate.
Over 100 trades: 40 wins at +500 = 20,000. 60 losses at -200 = -12,000. Net profit: 8,000 dollars.
More Than Two Outcomes
Most option trades have more than "win or lose." A call could expire deep ITM, slightly ITM, or worthless. The EV formula handles this naturally by adding more rows:
Same logic, more granularity. The deep ITM outcome is rare but large, and it contributes meaningfully to total EV.
When Is Buying vs. Selling Positive EV?
Neither buying nor selling options is inherently positive or negative EV. It depends on whether the market is pricing the option correctly.
EV Is Only as Good as Your Probability Estimates
The math is easy. The hard part is estimating the probabilities. The market's option prices already embed a consensus probability distribution (via implied vol). To have positive EV, you need to believe that consensus is wrong. If your estimate is no better than the market's, your EV after fees is slightly negative.
See It Over Time: The Convergence Simulation
This is where EV becomes intuitive. The chart below simulates 200 trades using the probabilities and payoffs you set. The blue line is your actual running P&L (random). The dashed green line is where EV says you should end up.
Click Re-shuffle multiple times. Notice how the blue line wanders randomly but keeps ending up near the dashed EV projection. That is the law of large numbers at work.
Now try these experiments:
Variance Is Not the Same as Negative EV
A positive EV strategy can lose money for 20, 50, or even 100 trades in a row by pure chance. That does not mean the strategy is broken. It means the sample size is too small. This is why professional traders care about volume: they need enough trades for the math to work out. If your strategy only produces 5 trades per month, it may take years before you can distinguish skill from luck.
Why Your P&L Chart Lies to You
There is a counterintuitive result from probability theory called the ArcSine law (first described by Paul Levy; Taleb applies it to trading P&L in Dynamic Hedging, Chapter 3). In a fair game (EV = 0), you would expect to spend roughly half the year in profit and half in loss. That is wrong. The most likely outcome is spending almost the entire year on one side: either 11 months in profit and 1 in loss, or 11 months in loss and 1 in profit. Spending exactly 6 months on each side is the least likely outcome.
Probability of time spent in profit (fair game, 1 year)
This means: a trader who has been profitable for 10 straight months is not necessarily skilled. A trader who has been losing for 10 straight months is not necessarily unskilled. Cumulative P&L charts over short periods are deeply misleading. The only reliable signal is a large number of independent trades, which is why the convergence chart above uses 200.
Track Records Are Mostly Noise
If you put infinite monkeys in front of trading screens, one of them will produce a 10-year track record that looks like Warren Buffett. That does not mean the monkey has edge. The ratio of luck to skill in a track record decreases with transaction frequency: for a market maker doing thousands of trades per day, a year of profitability is strong evidence of edge. For a fund manager making 10 trades per year, even a decade of profits could be chance. Always ask: how many independent bets does this track record represent?
This framing comes from Taleb's application of the Borel-Cantelli lemma to trader track records (Dynamic Hedging, Chapter 3). The original mathematical result proves that with enough trials, even extremely unlikely events become certain.
What Is "Edge"?
Edge is whatever makes your EV positive. Without edge, your EV on any trade is zero or slightly negative (after fees). Edge means you know something the market price does not fully reflect.
Real Edge
Where positive EV comes from
- Better volatility estimates (you believe realized vol will differ from IV)
- Faster information (you react to on-chain data or news before prices adjust)
- Structural advantages (cheaper funding, more efficient margining, ability to warehouse risk)
- Market making (you capture the bid-ask spread on each trade)
Not Edge
Things people mistake for edge
- "I feel strongly that BTC will go up" (conviction is not analysis)
- "This back-tested strategy returned 300%" (overfitting to historical data)
- "Options usually expire worthless" (the premium already prices that in)
- "I follow a whale wallet" (by the time you see it, so does everyone else)
Common EV Traps
Thinking in EV
The most important shift a trader can make is moving from judging trades by their outcome to judging trades by their process.
Outcome Thinking
How beginners evaluate
- "That trade lost money, so it was a bad trade"
- "I made money, so my strategy works"
- "I should stop selling puts, I got burned last month"
- Judges the decision by the result
EV Thinking
How professionals evaluate
- "That trade had positive EV. The loss was expected variance."
- "I made money, but was the trade actually positive EV or did I get lucky?"
- "The put sale had positive EV. I'll keep selling, but size smaller."
- Judges the decision by the process
EV Across Different Domains
💡 Tip: Try answering each question yourself before revealing the answer.
Related:
- Payoff vs P&L - The difference between what you receive and what you keep
- Implied Volatility - The market's expected move, and the main input for options EV
- Basic Strategies - Simple option trades and when they have positive EV
- Delta Hedging - Isolating vol exposure to capture EV from vol mispricing
- Beta & Alpha - Alpha is persistent positive EV after accounting for market exposure
- Common Mistakes - Pitfalls that destroy EV