Introduction to Stochastic Calculus (Ito's Lemma)

Quants and risk managers who implement stochastic models of options and stock prices use Ito's lemma to check what a model's equations mean and to read where the partial differential equation for valuation comes from.

What Ito's lemma expresses

Brownian motion is written $W_t$. The introduction of Brownian motion is covered on the Random Walks and Brownian Motion page.

Ito's lemma is a formula for the change in a function when the ordinary chain rule cannot be used. A path of Brownian motion is continuous but not differentiable at any point, so the quantity $dW_t/dt$ does not exist. The chain rule $df = f'(W_t)\,(dW_t/dt)\,dt$ therefore has no meaning¹.

You might try writing $df = f'(W_t)\,dW_t$ instead, but that is not enough. The Taylor expansion of a function around $x$ is $f(x + \Delta x) - f(x) = f'(x)\Delta x + \tfrac{1}{2}f''(x)(\Delta x)^2 + \cdots$. When $x$ is Brownian motion, the expected value of $(\Delta W_t)^2$ is $\Delta t$, which is as large as the time step. The second-order term therefore cannot be ignored, and exactly one second-order term remains¹. The following equation is the simplest form of Ito's lemma, for one variable.

$$df(W_t) = f'(W_t)\,dW_t + \tfrac{1}{2}f''(W_t)\,dt$$

Compared with the ordinary chain rule, there is one extra deterministic term, $\tfrac{1}{2}f''\,dt$. When $f'' > 0$, this term makes the expected movement of the function larger than the change you would get from the ordinary rules of differentiation.

Stochastic differential equations and the Ito integral

A stochastic differential equation is an equation that expresses the small change in a stochastic process $X_t$ as a term proportional to time plus a term proportional to the increment of Brownian motion. In the form below, $\mu_t$ is called the drift, and $\sigma_t$ is the coefficient of the $dW_t$ term². For geometric Brownian motion, $\sigma_t$ takes the form of the volatility $\sigma$ multiplied by the price ($\sigma S_t$).

$$dX_t = \mu_t\,dt + \sigma_t\,dW_t$$

$\mu_t$ and $\sigma_t$ may be constants, or functions of time $t$ or of the price $X_t$. If they are constants, $X_t = \mu t + \sigma W_t$, which is Brownian motion with drift. Geometric Brownian motion is the case $\mu_t = \mu S_t$ and $\sigma_t = \sigma S_t$, written as follows.

$$dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$$

This equation is shorthand for the movement over a short time, and its meaning is fixed only once it is rewritten in integral form. The integral form is as follows.

$$X_T = X_0 + \int_0^T \mu_s\,ds + \int_0^T \sigma_s\,dW_s$$

The last term is the Ito integral. Define $F$ with differential $dF = f\,dW + g\,dt$ as the sum of the integrals of $f\,dW$ and $g\,dt$, with the integral taken as the inverse of the differential². The Ito integral requires the integrand (here $\sigma_s$) to be adapted, meaning that its value at each time is determined only by the movement of Brownian motion up to that time. This corresponds to the fact that you cannot see future prices when deciding how much stock to trade³.

The Ito integral of an adapted integrand has two convenient properties. First, if the integrand satisfies conditions such as square integrability, the integral is a martingale (a process whose expected future value equals its current value)³. Second, the Ito isometry says that, for an adapted integrand $\Delta_s$, the expected value of the square of the integral ($E$ denotes the expected value) equals the expected value of the integral of the squared integrand³.

$$E\left[\left(\int_0^t \Delta_s\,dW_s\right)^2\right] = E\left[\int_0^t \Delta_s^2\,ds\right]$$

Consider a hypothetical example in which the integrand is the constant $\Delta_s = 0.2$ and $t = 1$ year. The right-hand side is $0.2^2 \times 1 = 0.04$. Because the mean is 0, the variance of the integral is 0.04 and its standard deviation is 0.2 (20%). If the integrand is a fixed function of time alone, the integral follows a normal distribution³.

Quadratic variation and the sum of squared increments

The extra term in Ito's lemma arises because the square of a Brownian increment is replaced by the deterministic time step $dt$. This property is expressed by quadratic variation. Split the interval $[0, T]$ into small subintervals and let the largest width of the partition approach 0. The sum of the squared Brownian increments then converges to $T$ with probability 1⁴.

$$\lim_{|\Pi| \to 0} \sum_i \left(W_{t_{i+1}} - W_{t_i}\right)^2 = T$$

Here $\Pi$ is the partition of the interval and $|\Pi|$ is its largest width. For a smooth function $g$, the same sum approaches 0. Each increment shrinks in proportion to the step, so its square is a small value proportional to the square of the step⁴. The table below compares the sums when the interval $[0, 1]$ is divided into equal parts.

The rightmost column follows from the fact that each increment of $g(t) = t$ is $\Delta t$, so the sum is $n \times (\Delta t)^2 = \Delta t$. For Brownian motion, the sum of squares stays at 1 however fine the steps become, while for the smooth function it approaches 0.

The Brownian motion column of the table uses a random walk with increments of $\pm\sqrt{\Delta t}$ as an approximation. Each squared increment is exactly $\Delta t$, so the sum is 1 on every path. For Brownian motion itself, each squared increment is a random value with mean $\Delta t$, and their sum approaches $T$ by the law of large numbers.

This relation can be summarized as rules for multiplying infinitesimal quantities. $(dW)^2$ is replaced with $dt$, and the $dt\,dW$ and $(dt)^2$ terms are negligible¹.

The formula of Ito's lemma

Let the function $f(t, x)$ be smooth and let $X_t$ follow $dX_t = \mu_t\,dt + \sigma_t\,dW_t$. The small change in $f(t, X_t)$ is then given by the following equation¹.

$$df(t, X_t) = \left( \frac{\partial f}{\partial t} + \mu_t \frac{\partial f}{\partial x} + \frac{1}{2}\sigma_t^2 \frac{\partial^2 f}{\partial x^2} \right) dt + \sigma_t \frac{\partial f}{\partial x}\,dW_t$$

$\partial f/\partial t$ is the change due to time and $\partial f/\partial x$ is the first-order change due to $X$. Both terms also appear in the ordinary chain rule. The term $\tfrac{1}{2}\sigma_t^2\,\partial^2 f/\partial x^2$ is the one that Ito's lemma adds. If $f$ is linear ($\partial^2 f/\partial x^2 = 0$), this term vanishes and the result has the same form as the ordinary chain rule. The greater the curvature of $f$, the larger this term becomes.

Deep diveOutline of the derivation of Ito's lemma

Expand $f(t, X_t)$ in a Taylor series and write the change up to second order. The second-order terms that contain $dt$ ($dt\,dX_t$ and $(dt)^2$) are 0 by the multiplication rules in the previous section, so they are omitted.

$$df = \frac{\partial f}{\partial t}\,dt + \frac{\partial f}{\partial x}\,dX_t + \frac{1}{2}\frac{\partial^2 f}{\partial x^2}\,(dX_t)^2$$

Squaring $dX_t = \mu_t\,dt + \sigma_t\,dW_t$ gives $\mu_t^2 (dt)^2 + 2\mu_t\sigma_t\,dt\,dW_t + \sigma_t^2 (dW_t)^2$. By the multiplication rules in the previous section, the first two terms are 0 and the last term becomes $\sigma_t^2\,dt$. Hence $(dX_t)^2 = \sigma_t^2\,dt$. Substituting this into the equation above and collecting the $dt$ terms and the $dW_t$ terms gives the formula in the main text. The proof follows the same sequence of calculations¹. Terms of third order and higher are even smaller relative to the time step and are ignored.

Worked examples: W squared and the logarithm of a stock price

The first example is $f(x) = x^2/2$. Since $f' = x$ and $f'' = 1$, Ito's lemma gives $d(W_t^2/2) = W_t\,dW_t + \tfrac{1}{2}dt$. Integrating both sides gives the following relation¹.

$$\int_0^T W_t\,dW_t = \frac{1}{2}W_T^2 - \frac{T}{2}$$

In ordinary calculus, $\int x\,dx = x^2/2$, and the $-T/2$ term does not appear. Consider a hypothetical example in which a path has $T = 1$ and $W_T = 1.5$. The ordinary calculus value is $1.5^2/2 = 1.125$, and the Ito integral is $1.125 - 0.5 = 0.625$. Since $E[W_T^2] = T$, the expected value of the Ito integral is $T/2 - T/2 = 0$, which is consistent with the martingale property mentioned earlier³.

The second example is the solution of geometric Brownian motion. Let $f(S) = \ln S$. Then $\partial f/\partial S = 1/S$ and $\partial^2 f/\partial S^2 = -1/S^2$. Substitute $\mu_t = \mu S_t$ and $\sigma_t = \sigma S_t$ into Ito's lemma⁵.

$$d\ln S_t = \left( \mu S_t \cdot \frac{1}{S_t} - \frac{1}{2}\sigma^2 S_t^2 \cdot \frac{1}{S_t^2} \right) dt + \sigma S_t \cdot \frac{1}{S_t}\,dW_t = \left( \mu - \frac{\sigma^2}{2} \right) dt + \sigma\,dW_t$$

The coefficients on the right-hand side are constants, so integrating and taking the exponential gives the solution for $S_t$.

$$S_T = S_0 \exp\left( \left( \mu - \frac{\sigma^2}{2} \right) T + \sigma W_T \right)$$

The drift of the logarithm, $\mu - \sigma^2/2$, is smaller than $\mu$ by $\sigma^2/2$ because the extra term in Ito's lemma comes from the curvature of $\ln S$ (its second derivative is negative). Consider a hypothetical example in which $S_0 = 100$, $\mu = 5\%$, $\sigma = 20\%$, and $T = 1$ year.

The difference between the two prices in the table shows up in the distribution in Figure 1. The price after 1 year follows a lognormal distribution. The price of 103.05, where the log return equals its mean, is the median (the price below which half the probability lies), and the expected value of 105.13 lies to its right.

The expected value is proportional to $e^{\mu T}$, which follows from the expected value of the lognormal distribution. Ito's lemma explains why the drift of the logarithm is $\mu - \sigma^2/2$.

Figure description

The figure shows the distribution of the stock price after 1 year for geometric Brownian motion with $S_0 = 100$, $\mu = 5\%$, $\sigma = 20\%$, and $T = 1$ year. It corresponds to the logarithm of the price following a normal distribution with mean $\ln 100 + 0.03$ and standard deviation 0.20. The vertical axis is the probability density, and area represents probability.

The distribution has a long right tail, and the vertical line marks the expected value, 105.13. The median, 103.05, lies about 2.08 to the left of the line, and the peak of the curve (about 99) lies further left still.

In this example, the mean log return is 0.03, and the difference from the drift-only value of 0.05 is $\sigma^2 T/2 = 0.02$. As the interval is divided more finely, the sum of squared increments of the log price approaches $\sigma^2 T = 0.04$. The simulator below lets you check both.

An entry point to the Black-Scholes equation

The option price $C(S, t)$ is a function of the underlying price $S_t$ and time $t$, so Ito's lemma can describe its movement. When $S_t$ follows geometric Brownian motion and the interest rate $r$ is constant, the result is as follows⁶.

$$dC = \left( \mu S \frac{\partial C}{\partial S} + \frac{\partial C}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} \right) dt + \sigma S \frac{\partial C}{\partial S}\,dW_t$$

Next, consider a portfolio of $x_t$ units of cash (the value of cash grows as $e^{rt}$) and $y_t$ units of the underlying, with no money added or withdrawn along the way. Choose $x_t$ and $y_t$ so that this portfolio replicates the movement of the option. Set the initial value $P_0 = C_0$ and equate the $dW_t$ terms and the $dt$ terms of the two equations so that $P_t = C_t$. The movement of the portfolio value $P_t$ is as follows⁶.

$$dP_t = \left( r x_t e^{rt} + y_t \mu S_t \right) dt + y_t \sigma S_t\,dW_t$$

Equating the $dW_t$ terms gives $y_t = \partial C/\partial S$. Equating the $dt$ terms gives $r x_t e^{rt} = \partial C/\partial t + \tfrac{1}{2}\sigma^2 S^2\,\partial^2 C/\partial S^2$.

Now substitute and rearrange. Since $P_t = C_t$, the portfolio value $x_t e^{rt} + y_t S_t$ equals $C$, so $x_t e^{rt} = C - S\,\partial C/\partial S$. Putting this into the equation for the $dt$ terms gives $r\left(C - S\,\partial C/\partial S\right) = \partial C/\partial t + \tfrac{1}{2}\sigma^2 S^2\,\partial^2 C/\partial S^2$, and rearranging gives the following Black-Scholes partial differential equation⁷.

$$\frac{\partial C}{\partial t} + rS\frac{\partial C}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} - rC = 0$$

To solve the equation you need boundary conditions such as the condition at maturity ($C(S, T) = \max(S - K, 0)$ for a call)⁷. The solution formula and its use are covered on The Black-Scholes Model page.

This equation does not contain $\mu$. Setting $\mu = r$ gives the same equation, which is why the method came to be called risk-neutral valuation⁷.

There are two simple solutions you can check by hand⁸. One is $V = S$ (the underlying itself). Putting $\partial V/\partial t = 0$, $\partial V/\partial S = 1$, and $\partial^2 V/\partial S^2 = 0$ into the equation gives $rS \times 1 - rS = 0$. The other is $V = K e^{-r(T-t)}$ (the value of a promise to receive $K$ with certainty at maturity). Since $\partial V/\partial t = rV$, the result is $rV - rV = 0$. Both are values that can be built from cash and the underlying alone, and both satisfy the equation.

Assumptions and limitations

Ito's lemma assumes that the function is smooth and that the process follows a stochastic differential equation of the form above¹. The convention for calculus is also not unique. Another integral can be defined with a term different from Ito's ($-\tfrac{1}{2}f''\,dt$). The Ito integral matches the premise that the future cannot be seen, and the source calls it a natural way to bring in time⁹. When you read a source that uses another convention, check the sign of the correction term and the definition of the integral.

The Black-Scholes partial differential equation was derived assuming geometric Brownian motion with constant volatility and a portfolio, with no money added or withdrawn, that can be rebalanced continuously. In real markets these assumptions do not hold. The fact that volatility differs by strike is one example.

What this article does not cover

The rigorous definition of the Ito integral (the limit of sums built from the value at the left end of each subinterval), the exact conditions on functions for which Ito's lemma holds, multiple variables, and processes whose paths are not continuous are not covered on this page.

Notes

1. Massachusetts Institute of Technology, "Lecture 18: Itō Calculus" (18.S096, Fall 2013), section 1 (pages 1–3), equation (1.2), Theorem 1.1 with its proof, and Example 1.3 (ii). The source's $B_t$ is replaced with $W_t$. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷

2. Same, Definition 1.2 and Example 1.3 (i) (pages 2–3). The integral form ($X_T = X_0 + \int \mu\,ds + \int \sigma\,dW_s$) is notation written from this definition. ↩ ↩²

3. Same, section 2 (pages 4–5), Definition 2.1, Theorem 2.3, Example 2.4, and Theorem 2.5. The variance 0.04 is the calculation $0.2^2 \times 1$ on the right-hand side of the isometry. ↩ ↩² ↩³ ↩⁴ ↩⁵

4. Massachusetts Institute of Technology, "Lecture 17: Stochastic Processes II" (18.S096, Fall 2013), Theorem 2.5 (page 4) and the paragraph after it (page 5). The values in the table are hand calculations for a random walk with increments of $\pm\sqrt{\Delta t}$ and for $g(t) = t$. ↩ ↩²

5. A calculation that puts $f = \ln S$ and similar values into the formula of Theorem 1.1 of the source (Lecture 18, page 2). The values in the table are $e^{0.03}$ and $e^{0.05}$ multiplied by 100 and rounded to two decimal places. ↩

6. Martin Haugh, "Black-Scholes and the Volatility Surface" (Columbia University, Fall 2013), section 1 (page 1), equations (1)–(5). The source covers a call on a stock paying no dividends, and writes the portfolio value as $P_t$. ↩ ↩²

7. Same, "Black-Scholes and the Volatility Surface", section 1 (page 2), equations (6)–(8) and the paragraphs that follow. The boundary condition for a call, and the fact that the same equation results when $\mu = r$, are also there. ↩ ↩² ↩³

8. A calculation that puts $V = S$ and $V = K e^{-r(T-t)}$ into equation (8) of the source (Haugh, page 2). For the latter, the partial derivative with respect to $t$ is $rV$. ↩

9. Massachusetts Institute of Technology, "Lecture 18: Itō Calculus", the Remark after Example 1.3 (pages 3–4). The example of another convention is $df = f'(B_t)\,dB_t - \tfrac{1}{2}f''(B_t)\,dt$. ↩