Itō's lemma

In mathematics, Itō's lemma is an identity used in Itō calculus to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule. Typically, it is memorized by forming the Taylor series expansion of the function up to its second derivatives and identifying the square of an increment in the Wiener process with an increment in time. The lemma is widely employed in mathematical finance, and its best known application is in the derivation of the Black–Scholes equation for option values.

Itō's lemma, which is named after Kiyoshi Itō, is occasionally referred to as the Itō–Döblin theorem in recognition of the recently discovered work of Wolfgang Döblin.^[1]

Note that while Ito's lemma was proved by Kiyoshi Itō, Itô's theorem, a result in group theory, is due to Noboru Itô.^[2]

Informal derivation

A formal proof of the lemma relies on taking the limit of a sequence of random variables. This approach is not presented here since it involves a number of technical details. Instead, we give a sketch of how one can derive Itō's lemma by expanding a Taylor series and applying the rules of stochastic calculus.

Assume $X t$ is a Itō drift-diffusion process that satisfies the stochastic differential equation

dX_t= \mu_t \, dt + \sigma_t \, dB_t,

where $B t$ is a Wiener process. If $f (t, x)$ is a twice-differentiable scalar function, its expansion in a Taylor series is

df = \frac{\partial f}{\partial t}\,dt + \frac{\partial f}{\partial x}\,dx + \frac{1}{2}\frac{\partial^2 f}{\partial x^2}\,dx^2 + \cdots .

Substituting $X t$ for $x$ and $μ t dt + σ t dB t$ for $dX t$ gives

df = \frac{\partial f}{\partial t}\,dt + \frac{\partial f}{\partial x}(\mu_t\,dt + \sigma_t\,dB_t) + \frac{1}{2}\frac{\partial^2 f}{\partial x^2} \left (\mu_t^2\,dt^2 + 2\mu_t\sigma_t\,dt\,dB_t + \sigma_t^2\,dB_t^2 \right ) + \cdots.

In the limit as $dt \to 0$ , the terms $dt 2$ and $dt dB t$ tend to zero faster than $dB 2$ , which is $O (dt)$ . Setting the $dt 2$ and $dt dB t$ terms to zero, substituting $dt$ for $dB 2$ , and collecting the $dt$ and $dB$ terms, we obtain

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

as required.

Mathematical formulation of Itō's lemma

In the following subsections we discuss versions of Itō's lemma for different types of stochastic processes.

Itō drift-diffusion processes

In its simplest form, Itō's lemma states the following: for an Itō drift-diffusion process

dX_t= \mu_t \, dt + \sigma_t \, dB_t

and any twice differentiable scalar function $f (t, x)$ of two real variables $t$ and $x$ , one has

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

This immediately implies that $f (t, X t)$ is itself an Itō drift-diffusion process.

In higher dimensions, if $\mathbf{X}_t = (X^1_t, X^2_t, \ldots, X^n_t)^T$ is a vector of Itō processes such that

d\mathbf{X}_t = \boldsymbol{\mu}_t\, dt + \mathbf{G}_t\, d\mathbf{B}_t

for a vector $\boldsymbol{\mu}_t$ and matrix $\mathbf{G}_t$ , Itō's lemma then states that

\begin{align} df(t,\mathbf{X}_t) &= \frac{\partial f}{\partial t}\, dt + \left (\nabla_\mathbf{X} f \right )^T\, d\mathbf{X}_t + \frac{1}{2} \left(d\mathbf{X}_t \right )^T \left( H_\mathbf{X} f \right) \, d\mathbf{X}_t, \\ &= \left\{ \frac{\partial f}{\partial t} + \left (\nabla_\mathbf{X} f \right)^T \boldsymbol{\mu}_t + \frac{1}{2}\text{Tr}\left[ \mathbf{G}_t^T \left( H_\mathbf{X} f \right) \mathbf{G}_t \right] \right\} dt + \left (\nabla_\mathbf{X} f \right)^T \mathbf{G}_t\, d\mathbf{B}_t \end{align}

where $\nabla X f$ is the gradient of $f$ w.r.t. $X$ , $H X f$ is the Hessian matrix of $f$ w.r.t. $X$ , and $Tr$ is the trace operator.

Poisson jump processes

We may also define functions on discontinuous stochastic processes.

Let $h$ be the jump intensity. The Poisson process model for jumps is that the probability of one jump in the interval $[t, t + Δ t]$ is $h Δ t$ plus higher order terms. $h$ could be a constant, a deterministic function of time, or a stochastic process. The survival probability $p s (t)$ is the probability that no jump has occurred in the interval $[0, t]$ . The change in the survival probability is

d p_s(t) = -p_s(t) h(t) \, dt.

p_s(t) = \exp \left(-\int_0^t h(u) \, du \right).

Let $S (t)$ be a discontinuous stochastic process. Write $S(t^-)$ for the value of S as we approach t from the left. Write $d_j S(t)$ for the non-infinitesimal change in $S (t)$ as a result of a jump. Then

d_j S(t)=\lim_{\Delta t \to 0}(S(t+\Delta t)-S(t^-))

Let z be the magnitude of the jump and let $\eta(S(t^-),z)$ be the distribution of z. The expected magnitude of the jump is

E[d_j S(t)]=h(S(t^-)) \, dt \int_z z \eta(S(t^-),z) \, dz.

Define $d J_S(t)$ , a compensated process and martingale, as

d J_S(t)=d_j S(t)-E[d_j S(t)]=S(t)-S(t^-) - \left ( h(S(t^-))\int_z z \eta \left (S(t^-),z \right) \, dz \right ) \, dt.

Then

d_j S(t) = E[d_j S(t)] + d J_S(t) = h(S(t^-)) \left (\int_z z \eta(S(t^-),z) \, dz \right ) dt + d J_S(t).

Consider a function $g(S(t),t)$ of jump process $dS (t)$ . If $S (t)$ jumps by $Δ s$ then $g (t)$ jumps by $Δ g$ . $Δ g$ is drawn from distribution $\eta_g()$ which may depend on $g(t^-)$ , dg and $S(t^-)$ . The jump part of $g$ is

g(t)-g(t^-) =h(t) \, dt \int_{\Delta g} \, \Delta g \eta_g(\cdot) \, d\Delta g + d J_g(t).

If $S$ contains drift, diffusion and jump parts, then Itō's Lemma for $g(S(t),t)$ is

d g(t) = \left( \frac{\partial g}{\partial t}+\mu \frac{\partial g}{\partial S}+\frac{\sigma^2}{2} \frac{\partial^2 g}{\partial S^2} + h(t) \int_{\Delta g} \left (\Delta g \eta_g(\cdot) \, d{\Delta}g \right ) \, \right) dt + \frac{\partial g}{\partial S} \sigma \, d W(t) + d J_g(t).

Itō's lemma for a process which is the sum of a drift-diffusion process and a jump process is just the sum of the Itō's lemma for the individual parts.

Non-continuous semimartingales

Itō's lemma can also be applied to general $d$ -dimensional semimartingales, which need not be continuous. In general, a semimartingale is a càdlàg process, and an additional term needs to be added to the formula to ensure that the jumps of the process are correctly given by Itō's lemma. For any cadlag process $Y t$ , the left limit in $t$ is denoted by $Y t-$ , which is a left-continuous process. The jumps are written as $Δ Y t = Y t - Y t-$ . Then, Itō's lemma states that if $X = (X 1, X 2, ..., X d)$ is a $d$ -dimensional semimartingale and f is a twice continuously differentiable real valued function on $R d$ then f(X) is a semimartingale, and

\begin{align} f(X_t) &= f(X_0) +\sum_{i=1}^d\int_0^t f_{i}(X_{s-})\,dX^i_s + \frac{1}{2}\sum_{i,j=1}^d \int_0^t f_{i,j}(X_{s-})\,d[X^i,X^j]_s\\ &\qquad+ \sum_{s\le t} \left(\Delta f(X_s)-\sum_{i=1}^df_{i}(X_{s-})\,\Delta X^i_s -\frac{1}{2}\sum_{i,j=1}^d f_{i,j}(X_{s-})\,\Delta X^i_s \, \Delta X^j_s\right). \end{align}

This differs from the formula for continuous semimartingales by the additional term summing over the jumps of X, which ensures that the jump of the right hand side at time $t$ is Δf(X_t).

Examples

Geometric Brownian motion

A process S is said to follow a geometric Brownian motion with volatility σ and drift μ if it satisfies the stochastic differential equation $dS = S (σdB + μdt)$ , for a Brownian motion B. Applying Itō's lemma with f(S) = log(S) gives

\begin{align} d\log(S) & = f^\prime(S)\,dS + \frac{1}{2}f^{\prime\prime} (S)S^2\sigma^2 \,dt \\ & = \frac{1}{S} \left( \sigma S\,dB + \mu S\,dt\right) - \frac{1}{2}\sigma^2\,dt \\ &= \sigma\,dB +\left (\mu-\tfrac{\sigma^2}{2} \right )\,dt. \end{align}

It follows that

\log (S_t) = \log (S_0) + \sigma B_t + \left (\mu-\tfrac{\sigma^2}{2} \right )t,

exponentiating gives the expression for S,

S_t=S_0\exp\left(\sigma B_t+ \left (\mu-\tfrac{\sigma^2}{2} \right )t\right).

The correction term of $- σ 2 / 2$ corresponds to the difference between the median and mean of the log-normal distribution, or equivalently for this distribution, the geometric mean and arithmetic mean, with the median (geometric mean) being lower. This is due to the AM–GM inequality, and corresponds to the logarithm being convex down, so the correction term can accordingly be interpreted as a convexity correction. This is an infinitesimal version of the fact that the annualized return is less than the average return, with the difference proportional to the variance. See geometric moments of the log-normal distribution for further discussion.

The same factor of $σ 2 / 2$ appears in the d₁ and d₂ auxiliary variables of the Black–Scholes formula, and can be interpreted as a consequence of Itō's lemma.

Doléans-Dade exponential

The Doléans-Dade exponential (or stochastic exponential) of a continuous semimartingale X can be defined as the solution to the SDE $dY = Y dX$ with initial condition $Y 0 = 1$ . It is sometimes denoted by $Ɛ (X)$ . Applying Itō's lemma with f(Y) = log(Y) gives

\begin{align} d\log(Y) &= \frac{1}{Y}\,dY -\frac{1}{2Y^2}\,d[Y] \\ &= dX - \tfrac{1}{2}\,d[X]. \end{align}

Exponentiating gives the solution

Y_t = \exp\left(X_t-X_0-\tfrac{1}{2} [X]_t\right).

Black–Scholes formula

Itō's lemma can be used to derive the Black–Scholes equation for an option. Suppose a stock price follows a geometric Brownian motion given by the stochastic differential equation $dS = S (σdB + μ dt)$ . Then, if the value of an option at time $t$ is f(t, S_t), Itō's lemma gives

df(t,S_t) = \left(\frac{\partial f}{\partial t} + \frac{1}{2}\left(S_t\sigma\right)^2\frac{\partial^2 f}{\partial S^2}\right)\,dt +\frac{\partial f}{\partial S}\,dS_t.

The term $\partial f / \partial S dS$ represents the change in value in time dt of the trading strategy consisting of holding an amount $\partial f / \partial S$ of the stock. If this trading strategy is followed, and any cash held is assumed to grow at the risk free rate r, then the total value V of this portfolio satisfies the SDE

dV_t = r\left(V_t-\frac{\partial f}{\partial S}S_t\right)\,dt + \frac{\partial f}{\partial S}\,dS_t.

This strategy replicates the option if V = f(t,S). Combining these equations gives the celebrated Black–Scholes equation

\frac{\partial f}{\partial t} + \frac{\sigma^2S^2}{2}\frac{\partial^2 f}{\partial S^2} + rS\frac{\partial f}{\partial S}-rf = 0.

Notes

References

Kiyoshi Itō (1944). Stochastic Integral. Proc. Imperial Acad. Tokyo 20, 519-524. This is the paper with the Ito Formula; Online
Kiyoshi Itō (1951). On stochastic differential equations. Memoirs, American Mathematical Society 4, 1–51. Online
Bernt Øksendal (2000). Stochastic Differential Equations. An Introduction with Applications, 5th edition, corrected 2nd printing. Springer. ISBN 3-540-63720-6. Sections 4.1 and 4.2.

External links

Derivation, Prof. Thayer Watkins
Informal proof, optiontutor