Ito's lemma explained term by term, and what it does to a share price

CM2 · Financial Engineering & Loss Reserving · 12 min read

What the lemma is for

You know the chain rule. If x changes with time and y = f(x), then dy = f′(x) dx. Everything in ordinary calculus that involves a function of a changing quantity rests on it.

Now make x random — a share price, say, which does not change smoothly but jitters. The ordinary chain rule breaks, and it breaks for a reason worth understanding rather than accepting. A Taylor expansion of f gives

In ordinary calculus the second term is discarded, because dx is of the order of dt and so (dx)2 is of the order of dt2, which vanishes faster than everything else as the step shrinks. For a random path that argument fails. A Brownian path moves by an amount of the order of √dt over a step of length dt, so (dx)2 is of the order of dt — the same order as the first term, and far too big to throw away.

Ito's lemma is the chain rule with that second term kept. That is all it is, and if you remember only that, you will be able to reconstruct it.

Standard Brownian motion

Everything in this topic is driven by one process, written Wt and defined by four properties:

• W0 = 0;
• the path is continuous;
• increments are independent — what happens next does not depend on what has happened so far;
• the increment over an interval of length t is normally distributed with mean 0 and variance t, so Wt − Ws ~ N(0, t − s).

The variance growing in proportion to time, rather than to time squared, is the property that does all the work. It is why the standard deviation of the movement grows like √t, and therefore why a step of length dt produces a movement of size √dt.

The one new fact: (dW)² = dt

This is the single piece of machinery that distinguishes stochastic calculus from the calculus you already have, so it is worth stating precisely rather than waving at.

Over a step of length dt, the increment dW is normal with mean 0 and variance dt. So its square has expectation E[(dW)2] = dt, and — this is the part that matters — its variance is 2(dt)2, which is of a smaller order than dt itself. As the step shrinks, the square of the increment therefore stops being random relative to its own size: it collapses onto its mean. In the limit it can be treated as the deterministic quantity dt.

The other products behave as you would hope: dt · dt and dW · dt are both of smaller order than dt and are discarded. So in any expansion:

Those three substitutions turn a Taylor expansion into Ito's lemma, mechanically.

The lemma

Let X_t follow the stochastic differential equation

and let f(Xt, t) be a smooth function of it. Then

In words: the change in f is the ordinary chain-rule term, plus the change caused by the passage of time, plus a correction equal to half the second derivative multiplied by the variance rate of the process.

Substituting for dX and collecting the dt terms gives the form that is easier to read off:

Three observations make this easier to use under exam pressure. The dW coefficient is just the chain rule — no correction there. The whole of the correction sits in the dt coefficient. And the correction is zero whenever f is linear in x, because then the second derivative is zero, which is exactly why you never noticed any of this in ordinary calculus with straight lines.

The sign of the correction, and what it means

The correction term carries the sign of ∂2f/∂x2. So:

• For a concave function — a logarithm, a square root — the second derivative is negative and the correction reduces the drift.
• For a convex function — a square, an exponential — it is positive and increases it.

That is not a technicality; it is Jensen's inequality appearing in the calculus. Randomness pushes the average of a convex function above the function of the average, and pulls the average of a concave function below it. Ito's lemma is the precise, differential statement of that fact, and the ½σ2f″ term is the price of the curvature.

The application that everything else uses

A share price is modelled as geometric Brownian motion:

In words: the share has an expected proportional growth rate of μ a year, and its proportional movements have a standard deviation of σ a year. The St multiplying both terms is what makes it proportional — a ₹1,000 share and a ₹10 share move by the same percentage, not the same number of rupees. That is both realistic and the reason the price can never go negative.

Now take f = ln S, so f′ = 1/S and f″ = −1/S2, and there is no explicit time dependence. Substituting into the lemma:

Every S has cancelled. The logarithm of the price follows Brownian motion with constant drift and constant volatility, which means it can be integrated by inspection:

The share price is lognormally distributed. That single result is the foundation of the Black-Scholes formula, and it was obtained in three lines from Ito's lemma.

Why the log drifts slower than the price grows

The −½σ2 is the part that puzzles people, and it is worth settling once, because it is also the most commonly dropped term in the paper.

The price grows at an expected rate of μ. The logarithm of the price drifts at μ − ½σ2, which is less. There is no contradiction. The exponential function is convex, so the average price is dragged above the price corresponding to the average log. The two ½σ2 terms are the same quantity seen from the two ends, and they cancel exactly when you take the expectation:

So the price does grow at μ on average, as the model intends. What the correction tells you is that the typical outcome is worse than the average one: the median price is S0e(μ−½σ2)t, strictly below the mean, and the gap widens with volatility and with time. A high-volatility share can have a healthy expected return and a median outcome that loses money. That is a real feature of lognormal growth and not an artefact of the mathematics.

Where marks are lost

Dropping the correction term. The whole point of the lemma. If an answer looks like the ordinary chain rule, the ½σ2f″ term has been forgotten.

Using σ² when the process has σS in front of dW. The σ in the lemma is the whole coefficient of dW in the stochastic differential equation. For geometric Brownian motion that coefficient is σS, so the correction carries (σS)2, not σ2. Getting this wrong leaves a stray S in an answer that should have none, which is a useful signal.

Forgetting the explicit time derivative. If f depends on t as well as on x — and every option price does, through its time to maturity — the ∂f/∂t term is there and is usually the largest of the three.

Confusing μ with the expected return on the log. μ is the drift of the price. The expected continuously compounded return over a year is μ − ½σ2. Both are called "the expected return" in ordinary speech, which is exactly why a question will be specific and your answer should be too.

Questions students ask

Do I have to be able to derive it?

You have to be able to apply it accurately and to say where the extra term comes from. Set out the function, write down its first and second derivatives, identify the drift and the diffusion coefficient of the process, substitute, and collect the dt terms. Doing it that way on paper, every time, is far more reliable than trying to remember the final form of any particular case.

What happens with a mean-reverting process?

Nothing changes about the lemma. An Ornstein-Uhlenbeck process, dX = −α(X − θ)dt + σdW, pulls back towards θ at a speed set by α, and the lemma applies to a function of it exactly as written. Because the diffusion coefficient is the constant σ rather than σX, the process can go negative — which is the standard objection to using it, in the Vasicek form, as a model of an interest rate.

Why does the model use the logarithm and not the price itself?

Because the logarithm is what has constant drift and constant volatility, so it is the quantity that can be integrated. Taking logs turns a process whose coefficients depend on its own level into one whose coefficients are numbers. It is the same instinct as taking logs to turn a product into a sum.

Is a lognormal share price realistic?

Partly. It gets the important structural facts right: the price stays positive and moves in proportional terms. It gets the tails wrong, because real returns show more extreme moves than a normal distribution allows, and it assumes volatility is constant, which it is not. Those two failures are the starting point for most of the models that come after this one.

A worked example

A share currently trades at ₹500 and is modelled as geometric Brownian motion with a drift of μ = 12% a year and a volatility of σ = 25% a year.

Find the distribution of the price in three years, its median, its mean, and the probability that it is above ₹700.

The share and both parameters are invented for this example. They are not a real company and not a forecast.

Step 1: apply the lemma to the logarithm

With f = ln S: ∂f/∂S = 1/S, ∂2f/∂S2 = −1/S2, and no explicit t.

Step 2: the distribution after three years

So ln S3 ~ N(6.4808581, 0.1875), and S₃ is lognormal.

Step 3: the median

Because the logarithm is normal and therefore symmetric, the median of S₃ is the exponential of the mean of the log:

Check it the other way: 500 × e0.26625 = 500 × 1.3050613 = ₹652.53. The same figure, since multiplying by the exponential of the drift is what exponentiating the sum amounts to.

Step 4: the mean, two independent ways

Route one, from the lognormal formula E[S] = exp(m + ½v) using the two parameters just found:

Route two, straight from the drift of the price, which never involved the lemma at all:

The two agree to the paisa, and the reason they must is the whole content of this page. Route one subtracted ½σ2t = 0.09375 inside the drift of the log and then added it back in the lognormal mean. Route two never subtracted it. If your two routes disagree by a factor of e0.09375, or about 9.8%, the correction term has been applied once instead of twice or twice instead of once.

Reading the gap between the mean and the median

₹716.66 against ₹652.53 — the mean is about 9.8% above the median. The ratio is exactly e½σ2t = e0.09375 = 1.0982, which depends only on the volatility and the time and not at all on the drift.

Put in words a client would understand: this share is expected to be worth ₹716.66 in three years, but there is a better-than-even chance it is worth less than ₹652.53. Both statements are true at once, and the distance between them is produced entirely by volatility. That is why a projection quoting only an expected value is a misleading projection.

Step 5: the probability of exceeding ₹700

Convert to a statement about the log, which is normal.

About a 43.6% chance, which is below a half even though ₹700 is below the expected price of ₹716.66 — exactly the asymmetry described above, now as a probability rather than as a gap between two averages.

A second application of the lemma, as a check on the method

Take the simplest possible case: Yt = Wt2, a function of standard Brownian motion itself, so the drift is 0 and the diffusion coefficient is 1. With f = x2, f′ = 2x and f″ = 2:

Now check it against something already known. Taking expectations, the dW term has expectation zero, so d E[Y] = dt, which integrates to E[Wt2] = t. And E[Wt2] is the variance of Wt, which the definition of Brownian motion says is t. At t = 3 both give 3.

If the correction term had been dropped, the lemma would have given dY = 2W dW, whose expectation is zero, implying the variance of Brownian motion never grows. The correction term is not a refinement; without it the result is not merely imprecise but false.

Where this fits in CM2

This page explains one idea from CM2 Financial Engineering & Loss Reserving. The full CM2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

Other CM2 topics explained

• Utility theory and risk aversion: non-satiation, A(w) and R(w)
• Measures of investment risk: variance, semi-variance, VaR and Tail VaR
• CAPM and the security market line: beta and the market price of risk
• The chain ladder method: estimating outstanding claims from a run-off triangle
• The Bornhuetter-Ferguson method: the chain ladder with a second opinion
• Binomial option pricing: where the risk-neutral probability comes from
• The Black-Scholes formula and put-call parity, read term by term
• Ruin theory: the adjustment coefficient and Lundberg bound explained
• Mean-variance portfolio theory: the efficient frontier and why diversification works
• The term structure: spot rates, forward rates and par yields
• Short-rate models: Vasicek, Cox-Ingersoll-Ross and Hull-White compared
• The Merton model: company equity as an option, and where a credit spread comes from
• The Greeks: delta, gamma, vega, theta and rho, and how a hedge decays
• The inflation-adjusted chain ladder: taking past inflation out and putting a future assumption in
• The average cost per claim method: developing numbers and severity separately
• Valuing a maturity guarantee by simulation, and checking it against the formula
• The Efficient Markets Hypothesis: the three forms, and how each one is tested
• Single-index and multifactor models: getting a covariance matrix out of a few betas
• Excess-of-loss reinsurance: the retained claim, and choosing the retention
• State price deflators: pricing with real-world probabilities
• What makes a risk insurable, and what to do when it almost is
• Two-state and intensity credit models: backing a default rate out of a bond price
• The martingale representation theorem, and why hedging is possible at all

Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.