Part I · Returns and risk · Chapter 1C

Expected value and edge

Why winning often is not the same as making money

Edge is not a feeling, it is not a story, and it is not "I really think this stock is going up". Edge is a number: the expected value of a trade or a strategy, net of all costs, after appropriate discounting for risk. Taking expected value seriously has one consequence more important than any other, previewed in Chapter 1B. A strategy that wins most of the time can still lose money, and a strategy that wins less than half the time can make money.

Win rate is correlated with edge but is not the same thing. The retail bias toward strategies with a high "probability of profit" is a bias toward shapes that feel like edge but may not contain any.

1 · The definition of edge

For a trade with a list of possible outcomes, the expected value is the probability-weighted average of the outcomes, as in Chapter 1B:

Here $x_i$ is the profit or loss of outcome $i$ and $p_i$ its probability. Most of the bets in this chapter have two outcomes. A binary bet wins an amount $G$ with probability $p$, loses an amount $L$ with probability $1 - p$, and costs $c$ up front. Weighting each outcome by its probability and subtracting the cost gives its expected value:

The trade has edge if and only if $\mathbb{E}[X] > 0$. If $\mathbb{E}[X] = 0$ the trade is fair, and if $\mathbb{E}[X] < 0$ you are paying for the privilege of playing.

Setting the expected value to zero and solving for $p$ gives the breakeven win rate, the win rate a bet must beat to have edge:

The breakeven win rate depends only on the sizes of the win, the loss and the cost. A bet whose loss is large relative to its win needs a very high win rate just to break even.

2 · Two famous coin flips

These two examples are foundational in quantitative trading curricula, and they are worth committing to memory.

The figure below builds a two-outcome bet from three sliders: the probability of winning, P(win), from 10% to 90%; the size of the win; and the size of the loss. It starts on coin flip 1, and its three bars show what the wins bring on average ($p \times G$), what the losses cost on average ($(1-p) \times L$), and the balance between them, the expected value per round. Press the two presets, "the high-win-rate trap" and "the low-win-rate edge", to load the two coin flips.

P(win)

55%

win

$1.00

loss

$1.25

preset the high-win-rate trap the low-win-rate edge

P(loss) = 45% E[X] = −$0.012 per round

Negative EV. Do not play. The losses accumulate.

On the first preset the loss bar is slightly longer than the win bar, and the expected value is a sliver below zero, about a penny a round. On the second, a win bar of $8.75 dwarfs a loss bar of $0.75. Now try a bet the presets do not cover. Set P(win) to 90%, the win to $1 and the loss to $25: the expected value is $0.9 - 2.5 = -\$1.60$ a round, although you win nine times in ten. The verdict line judges the expected value only. How much of your account to put on a bet with edge is a separate question, which the last part of this chapter begins to answer.

3 · Win rate is not edge

Consider two coin flips with the same lopsided odds but opposite roles.

Scenario A: the insurance buyer. With probability 95% you pay $1, and with probability 5% you receive $25. Your win rate is 5%, because you "win" only when the rare event hits, and your expected value is $0.95 \times (-1) + 0.05 \times 25 = -0.95 + 1.25 = +\$0.30$ a round. This is the structural shape of a tail-hedge buyer, who pays small premiums most of the time and collects a large payoff when the rare event arrives. It has positive expected value, a low win rate and positive skew.

Scenario B: the premium seller. Now flip Scenario A around, so that with probability 95% you receive $1, and with probability 5% you pay $25. Your win rate is now 95%, and the expected value is $0.95 \times 1 - 0.05 \times 25 = +0.95 - 1.25 = -\$0.30$ a round. By (1C.3), this bet needs a win rate of $25/26 = 96.2\%$ to break even.

You win 19 times out of 20 and still lose 30 cents a round on average. This is the trap. It shows, as plainly as anything can, why the "feel" of trading ("I won 19 trades in a row") can systematically diverge from the actual profit and loss.

The same shape, 95% small wins and 5% catastrophic losses, describes every undisciplined premium-selling strategy in existence. Sell a deep out-of-the-money put, and in 95% of months the premium expires worthless and you keep it. In the other 5% the underlying blows through your strike and erases a year of accumulated premium. The structural shape is identical to the coin flip, and only the magnitudes change. Win rate is not edge; expected value is.

The trap is Scenario B, not premium selling in general. The CBOE PUT Index of Chapter 1B, which writes at-the-money cash-secured puts on SPX mechanically each month from 1986 forward, has produced positive long-run risk-adjusted returns over four decades. Premium selling can have positive or negative expected value, depending on strike selection, sizing, roll discipline, and whether the premium collected actually compensates for the tail risk taken. The point is not that selling premium is bad. The point is that a 95% win rate tells you nothing about which kind of premium selling you are doing.

Why the trap feels safe

This is also why the brokerage statements of new premium sellers look magnificent for their first six to twelve months. The first batch of options expires worthless, and so does the next. The trader concludes that they have discovered a free money machine.

The 5% catastrophe has not happened yet, because a 5% event happens about once every twenty trials. On a monthly cadence that means roughly twenty months between hits. The chance of getting through $n$ months with no hit is $0.95^n$, which after 12 months is still $0.95^{12} = 54\%$. More often than not, a new seller's whole first year is clean. The trader is sampling from a distribution whose left tail they have not yet seen, and concluding that the left tail does not exist.

Ask what the losers lose

Anyone who tells you that their options-selling strategy has "an 85% win rate" or "a 92% probability of profit" is telling you about the shape, not the edge. Always ask what the average losing trade loses, in dollars, relative to what the average winning trade wins. Set that ratio against the win rate and you know the sign of the expected value.

The breakeven formula makes this a one-line check: at a win rate $p$, a bet with no costs breaks even when the average loss is $p/(1-p)$ times the average win. An 85% win rate is profitable only if the average loss is less than $0.85/0.15 = 5.7$ times the average win. A 92% win rate is profitable only if it is less than $0.92/0.08 = 11.5$ times.

4 · A more realistic version: selling a put

Real trades have more than two outcomes, and the losses vary in size. Here is a simplified short put, with the average loss standing in for the whole loss side of the distribution.

Now change only the loss side, and suppose the realized average loss is $3 instead of $2, because losses cluster in stress, when implied volatility expands and puts get dearer. The mean falls to $+\$0.45 - \$0.30 = +\$0.15$. Suppose next that you cannot hold to expiry because of margin. A broker's margin call can force you to close at the worst mark-to-market loss along the way. The realized loss can then be $4 to $4.50 even when the loss at expiry would have been $2, and the mean falls to $+\$0.45 - \$0.40 = +\$0.05$, and to zero at $4.50.

The win rate is 90% in every row, yet the realized edge depends heavily on the loss distribution and on the path the trade takes, not just on the win rate. Chapter 1E develops the path, and the way margin turns a paper loss into a realized one.

5 · Edge comes before everything

Risk management, position sizing and discipline, the subjects of Part XIII, keep you in the game, but they cannot manufacture edge. If your underlying expected value is negative, perfect discipline just means losing money more methodically. It is the single most important sentence in this book, and later chapters repeat it in different forms.

The machinery of later chapters, from dealer mechanics and regime classification to conviction scoring and kill rules, assumes that you have a process with positive expected value. It exists to keep an edge from being destroyed by behavior, fees, taxes and bad sizing, and it does not create edge where none exists.

The two claims inside a short put

When you sell a deep out-of-the-money index put, you have placed a structural bet. You are betting that the equity skew and the variance risk premium together compensate you enough for the rare tail outcome. The equity skew is the richer pricing of index puts than of calls, which Chapter 5G describes. You are also betting that you can size and roll the position so that the tail outcome does not end the game. Notice which of these assumptions is doing the work.

• The premium has to be enough. This is an empirical claim about the market, sometimes true and sometimes not, and Chapter 10C examines the evidence for it.
• Your sizing has to survive the tail. This is a discipline claim about you, and it is almost always the binding constraint.

The slide from a profitable premium seller into Scenario B is rarely about a sudden change in market behavior. It is about a sudden change in the trader's behavior, such as:

• a doubled position size after a winning streak;
• a strike moved one band closer to at-the-money "because vol is high right now";
• a stop level that was never pre-committed in writing.

Part XIII (Discipline) returns to this in detail, because it is the central practical issue, while Part V (Options) is the technical preamble.

6 · Expected utility: why dollars are not enough

The plain expected value calculation assumes that you value money linearly: that gaining $1,000 and losing $1,000 have equal and opposite impact. This is approximately true for bets that are small relative to your wealth and approximately false for large ones.

Real people have concave utility, which bends downward the way the logarithm does in Chapter 1A. A $1,000 gain when you have $50,000 in the bank matters more than a $1,000 gain when you have $5 million. Equivalently, a $1,000 loss hurts more than a $1,000 gain feels good. That is just the concavity of utility expressed in symmetric form.

The mathematical formalization is expected utility theory (von Neumann and Morgenstern, 1944), which says that a rational decision-maker maximizes the probability-weighted average of the value placed on each outcome:

where $W_i$ is your wealth in outcome $i$ and $U$ is some concave utility function, the value you place on each level of wealth. The most-used utility function in finance is the logarithm:

With log utility the $1,000 example becomes arithmetic: on $50,000, a gain is worth $\ln(51{,}000/50{,}000) = +0.0198$ and a loss costs $\ln(49{,}000/50{,}000) = -0.0202$, slightly more. On $5 million, the same gain is worth only $\ln(5{,}001{,}000/5{,}000{,}000) = +0.0002$.

Log utility has a special property: the bet size that maximizes expected log utility is the bet size of the Kelly criterion, which Chapter 3D develops. Log utility is also the natural utility function for an investor who compounds, because maximizing expected log return is equivalent to maximizing the expected compound growth rate. For an investor with a multi-decade horizon and no need to spend the money soon, log utility is approximately the right utility function.

Why the expected log return is the growth rate

Step 1: compound many rounds. Take $n$ independent rounds of the same bet, with simple returns $R_1, \ldots, R_n$ on the account, so that wealth after $n$ rounds is $W_n = W_0 (1+R_1)(1+R_2)\cdots(1+R_n)$.

Step 2: take logarithms. The product becomes a sum, as in Chapter 1A:

$$\ln\frac{W_n}{W_0} = \sum_{t=1}^{n} \ln(1 + R_t)$$ (1C.8)

Step 3: average. Divide both sides by $n$. By the law of large numbers, the average of many independent draws approaches their expected value:

$$\frac{1}{n}\ln\frac{W_n}{W_0} \to \mathbb{E}[\ln(1+R)]$$ (1C.9)

Step 4: read the result. For large $n$, wealth is about $W_n \approx W_0\, e^{\,n\,\mathbb{E}[\ln(1+R)]}$, so the account grows at a rate of $\mathbb{E}[\ln(1+R)]$ per round. Maximizing that expectation maximizes the long-run growth rate, and if it is negative the account shrinks toward zero, whatever the dollar expected value.

Compute expected log return, not expected dollars

The practical implication is direct: when you compute expected value for sizing purposes, compute the expected log return on the portfolio, not the expected dollar return. Log utility automatically penalizes catastrophic outcomes, and dollar expected value does not.

7 · Where this connects to options

Every options strategy can, and must, be evaluated by its expected value, decomposed by component. A short at-the-money put has an expected value of the gross premium, minus the expected payout at exercise, minus transaction costs, minus the expected cost of financing its margin. A long calendar spread has an expected gain from the re-pricing of forward volatility, minus the cost of setting it up, minus the expected loss from the underlying moving against the position. Every line in a brokerage statement is a sample drawn from a distribution, and the question is always the same: is the mean of that distribution positive after costs?

This is also why most options education content fails. It teaches you the shape of a strategy, its payoff diagram, without teaching you how to evaluate whether the strategy has positive expected value in the current market. On its own, a payoff diagram tells you nothing about edge.

Take a short iron condor and its mirror image, the long iron condor. Both have limited risk and limited reward, in opposing geometries: one wins small and often, the other wins large and rarely. Whether the short or the long one has positive expected value depends on whether implied volatility is over-pricing or under-pricing the volatility that actually occurs. Edge is what determines the outcome, while geometry tells you about the path, not about the edge.

8 · Check yourself