Probability of Profit & Expected Value

There are two questions you can ask about any options structure, and they are not the same question. The first is: how often will this make money? The second is: how much will it make on average? Beginners collapse these into one. Professionals keep them rigorously apart, because a structure can win the great majority of the time and still lose you money in the long run, and another can lose more often than it wins and still be the better trade. To see why, and to compute both numbers honestly, you need the object that the previous chapter built toward: the risk-neutral density.

Recall what that density is. From the prices of options across strikes, the market hands you an entire probability distribution over where the underlying might land at expiry. Call that density f of S, where S is the terminal price. It is the market's own picture of the future — not your forecast, but the consensus encoded in option prices. Everything in this chapter is just two different integrals against that one curve.

Start with intuition. Picture the density as a hill of probability sitting over the price axis, fattest near the current forward and thinning out into both tails. Now lay your structure's payoff diagram underneath it. Somewhere there is a price — or a set of prices — where you break even, where the payoff exactly equals what you paid to put the trade on. To the profitable side of that break-even, you keep money. To the other side, you lose. The probability of profit is simply the total weight of the density sitting over the profitable region. You are asking: of all the futures the market thinks possible, what fraction land where I win?

Formally, the probability of profit is the integral of f of S over exactly the region where your terminal payoff exceeds the premium you paid. For a long call struck at K with premium c, you profit whenever the terminal price clears K plus c — the strike plus the cost. So the POP is the integral of f of S from K-plus-c out to infinity. For a structure with two break-evens, like a long strangle, the profitable region is two disjoint tails and you add both pieces. The shape of the payoff dictates the limits of integration; the density supplies the weight. That is the whole mechanism.

Now the second question, and here is where most intuition fails. Probability of profit tells you how often, but it says nothing about how much. A far out-of-the-money option you sell might profit ninety-five times out of a hundred — a beautiful POP — yet the five losses are so large they swamp the ninety-five small wins. To capture magnitude you need the expected value: not the chance of winning, but the average dollar outcome, weighting every possible terminal payoff by how likely the market thinks it is.

Expected value is the integral of your terminal payoff, payoff of S, multiplied by the density f of S, integrated across all prices from zero to infinity, and then you subtract the premium. Written as the platform writes it: EV equals the integral of payoff times f, with respect to S, minus the premium. The integral is the average gross payoff at expiry — every outcome scored by its probability mass. Subtracting the premium converts that gross figure into net profit or loss. If the result is positive, the market's own density says this structure pays more, on average, than it costs. If it is negative, you are paying more than the trade is worth under the consensus distribution.

Hold those two formulas side by side, because their difference is the lesson. POP integrates the density alone over a region. EV integrates the payoff weighted by density over everything. POP is blind to magnitude; EV is built from magnitude. That is precisely why a high POP can sit on top of a negative EV — the wins are frequent but small, the losses rare but enormous, and only EV sees the size of the losses. Conversely, a lottery-like long option may show a low POP and still carry a positive EV if the rare payoff is large enough.

This is also where the chapter ties back to the throughline of this institute: knowing when a number is genuine evidence. A POP of seventy percent feels like evidence the trade is good. It is not — by itself it is half a sentence. Only when you read it alongside EV do you know whether the seventy percent is earning its keep or quietly setting you up for a fat-tailed loss. The discipline is to never let one number masquerade as the verdict.

One honest caveat. Both integrals are taken against the risk-neutral density, which is the market's risk-adjusted picture, not the real-world frequency of outcomes. A risk-neutral EV near zero is exactly what an efficiently priced option should show; the interesting signal is the deviation, and the gap between the risk-neutral and your real-world view is where an edge, if you have one, actually lives. We sharpen that distinction in the next chapter.