Chapter 10 · Itô Calculus

The Itô Integral

Welcome to the engine room — the mathematics of profit and loss in a market that never stops wiggling.
The engine room

So far we have described how prices move — randomly, continuously, jaggedly. But we have not yet answered the trader's real question: how do I calculate my profit and loss while the market is moving? This is what the Itô Integral solves. It is the mathematical machinery that converts a strategy — a sequence of positions held over time — into a running bank account balance, tick by tick, as the market wiggles beneath you.

The Ito Integral as a running bank account — positions times price moves equals PnL Three boxes connected by multiplication and equals signs: position size delta on the left, times price move dW in the middle, equals running PnL I of t on the right Position Size Δ(t) How many shares you hold × Price Move dW(t) The random market wiggle = Running P&L I(t) Your bank account balance

The Itô Integral in one line: position size × price move = running P&L. Summed continuously over time.

1 · The simple strategy: trading once an hour

To build intuition, we start with the simplest possible strategy. Imagine you are a trader who makes exactly one decision per hour. At 10:00 AM you choose how many shares to hold. You cannot change your mind until 11:00 AM — no matter what happens in between. This disciplined, rigid approach is what mathematicians call a Simple Process.

Simple process — step function showing position size held constant in each hour-long block A step function chart where position size is constant across each one-hour interval — 50 shares from 10 to 11, 20 shares from 11 to 12, 35 shares from 12 to 1 50 35 20 0 10 AM 11 AM 12 PM 1 PM Δ = 50 shares Δ = 20 shares Δ = 35 shares Shares held

A Simple Process: position size is fixed within each hour. Decisions can only use information from before that hour began.

The "Adapted" rule is critical: the decision at 10:00 AM can use any news from 9:59 AM, but it cannot use news from 10:01 AM. This is the Filtration (ℱₜ) from Chapter 8 — your strategy cannot be a time-traveller. A backtest that accidentally uses 10:01 AM data to make a 10:00 AM decision is lying to you.

2 · The Itô integral: your running bank account

The Itô Integral I(t) is simply the accumulated profit and loss of your strategy up to time t. At each moment, it multiplies the number of shares you are holding by the price move that just happened, and adds it to the running total. Think of it as a bank account that updates every tick.

I(t) = ∫₀ᵗ Δ(u) dW(u)
Δ(u) — Your position at time u (how many shares you hold)
dW(u) — The infinitesimal price move at time u (the market's wiggle)
∫₀ᵗ — Sum all these tiny position × move products from time 0 to time t

In plain English: "Add up (shares held × price move) for every instant you were in the market."

Here is a concrete three-hour example. Trace through the numbers and you will see that the Itô Integral is nothing more than a glorified running ledger:

Time period Shares held (Δ) Price move (dW) P&L this period Running total I(t)
10:00 → 11:00 50 shares +$2.00 ↑ +$100 $100
11:00 → 12:00 20 shares −$1.00 ↓ −$20 $80
12:00 → 1:00 35 shares +$0.50 ↑ +$17.50 $97.50
Final Itô Integral I(T) $97.50
Running PnL chart — bank account balance growing and falling with each trade outcome A step chart showing the running PnL balance: starts at zero, jumps to 100 after the first hour, drops to 80 after the second, rises to 97.50 after the third $120 $100 $80 $0 10 AM 11 AM 12 PM 1 PM +$100 profit −$20 loss +$17.50 $97.50

The Itô Integral is the yellow staircase — your bank account balance updating after every period's P&L is settled.

3 · Theorem 10.1 — the integral is a martingale

Here is a profound realisation. We proved in Chapter 6 that Brownian Motion is a Martingale — a fair game with no predictable drift. Theorem 10.1 extends this to your entire strategy.

THEOREM 10.1
The Itô Integral is a Martingale
𝔼[ I(t) | ℱ(s) ] = I(s) for all s ≤ t
If the market itself is a fair game, then any strategy you build on it is also a fair game. Your expected future bank balance, given everything you know today, is simply your bank balance today — not higher, not lower.
Coin flip analogy — betting different amounts on a fair coin never changes your expected outcome from zero Three columns showing different bet sizes on a fair coin — $1, $100, and alternating — all with expected profit equal to zero Bet $1 every flip Bet $100 every flip Skip every other flip Win: +$1 (p=½) Lose: −$1 (p=½) Win: +$100 (p=½) Lose: −$100 (p=½) Bet or skip, up to you Same 50/50 on every flip 𝔼[P&L] = 0 𝔼[P&L] = 0 𝔼[P&L] = 0 Size doesn't help Size doesn't help Timing doesn't help

Changing your bet size or timing cannot change the expected outcome if the underlying game is fair. The Martingale property is preserved.

This theorem is the mathematical foundation of the Efficient Market Hypothesis. If prices are a fair Martingale, no trading strategy — however clever its position sizing or timing — can generate a positive expected profit. Beating a truly random market through position management alone is mathematically impossible. Strategies that appear to generate alpha must either be taking on hidden risk, or the market is not perfectly Martingale in that regime.

4 · Theorem 10.2 — the Itô isometry: risk is a choice

If Theorem 10.1 is the bad news (no free profit), Theorem 10.2 is the empowering insight. While you cannot control your expected P&L, you have complete control over your risk. The size of your bets directly and precisely determines the volatility of your wealth.

THEOREM 10.2 — THE ITÔ ISOMETRY
Risk equals the sum of squared positions
𝔼[ I²(t) ] = 𝔼[ ∫₀ᵗ Δ²(u) du ]
The variance (expected squared P&L) of your bank account equals the integral of your squared position sizes over time. You do not need to know which way the market will move. You only need to know how big your bets were. Larger positions → larger potential swings.
Isometry — larger position sizes produce proportionally larger PnL variance Three scenarios showing position size of 1, 10, and 50 shares and the corresponding multiplication of PnL variance by 1, 100, and 2500 The Isometry in action: Δ² scales your risk Δ = 1 share Δ² = 1 Risk × 1 Δ = 10 shares Δ² = 100 Risk × 100 Δ = 50 shares Δ² = 2,500 Risk × 2,500

The Isometry: doubling your position size quadruples your risk. Squaring punishes large bets disproportionately.

This is why risk managers use position limits and Value-at-Risk models. The Itô Isometry makes the relationship precise: if a trader doubles their position size, their P&L variance does not double — it quadruples. The squaring is not an approximation; it is exact. Risk grows as the square of position size.

5 · Theorem 10.3 — quadratic variation of your wealth

The final theorem connects the Itô Integral back to our Quadratic Variation concept from Chapter 4. Your portfolio's wealth is not smooth — it accumulates its own jaggedness as the market moves. The rate at which it jiggles is precisely controlled by the size of your positions.

THEOREM 10.3
QV of the Itô Integral
[I, I](t) = ∫₀ᵗ Δ²(u) du
The Quadratic Variation of your bank account accumulates at a rate equal to your squared position size at every moment. If the market wiggles by 1 unit and you hold Δ shares, your account wiggles by Δ units. When we square those wiggles to compute QV, your account is Δ² times more jagged than the market itself.
QV amplification — market wiggles by 1 unit but portfolio with 10 shares wiggles by 10 units, QV amplified by 100 Two panels side by side showing market QV accumulation at rate 1 and portfolio QV accumulation at rate delta-squared equals 100 for a 10-share position Market: [W,W](t) = t Portfolio (Δ=10): [I,I](t) = 100t Rate = 1 (gentle slope) Time → 100× steeper! Rate = Δ² = 100 QV accumulates steadily QV accumulates 100× faster

Same market, different position. Holding 10 shares makes your wealth 100× more jagged than the market itself.

This is the mathematical reason why leverage amplifies risk non-linearly. A trader with 10× leverage does not have 10× the risk — they have 100× the Quadratic Variation. The jiggle in their account is 100 times more intense than the market's jiggle, even though every market move is the same. The squaring in Δ² is not a technicality. It is the precise mechanism by which leverage destroys unprepared traders.

Try it — build and run your own strategy

Set your position size for each of three trading periods. A random market will wiggle up or down in each period. Watch your running Itô Integral — your bank account — update after each period. Run it many times and observe that, on average, the expected P&L is always zero — confirming Theorem 10.1. But your risk grows with Δ².

50
20
35
15%
This run I(T)
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500-trial avg
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500-trial std dev
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Predicted std dev (Isometry)
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Green bars = profit periods  |  Red bars = loss periods  |  Gold line = running I(t). Run 500 trials to verify the Martingale property.

Three theorems — three insights:

The key insight: Brownian Motion cannot be differentiated — it has no instantaneous speed. Itô solved this by working with the integral (the accumulation) instead of the derivative (the speed). The Itô Integral is how calculus is done in a world of randomness. Next: the Itô-Doeblin Formula — which tells you how any function of price changes as the underlying Brownian Motion moves. This is the bridge to the Black-Scholes equation.

Next Chapter: Itô-Doeblin Formula →