Itô's lemma

In mathematics, Itô's lemma is an identity used in Itô 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. It can be heuristically derived by forming the Taylor series expansion of the function up to its second derivatives and retaining terms up to first order in the time increment and second order in the Wiener process increment. 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.

Itô's lemma, which is named after Kiyosi Itô, is occasionally referred to as the Itô–Doeblin theorem in recognition of posthumously discovered work of Wolfgang Doeblin.[1]

Note that while Itô's lemma was proved by Kiyosi Itô, 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 Itô's lemma by expanding a Taylor series and applying the rules of stochastic calculus.

Assume Xt is a Itô drift-diffusion process that satisfies the stochastic differential equation

${\displaystyle dX_{t}=\mu _{t}\,dt+\sigma _{t}\,dB_{t},}$

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

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

Substituting Xt for x and therefore μtdt + σtdBt for dx gives

${\displaystyle df={\frac {\partial f}{\partial t}}\,dt+{\frac {\partial f}{\partial x}}(\mu _{t}\,dt+\sigma _{t}\,dB_{t})+{\frac {1}{2}}{\frac {\partial ^{2}f}{\partial x^{2}}}\left(\mu _{t}^{2}\,dt^{2}+2\mu _{t}\sigma _{t}\,dt\,dB_{t}+\sigma _{t}^{2}\,dB_{t}^{2}\right)+\cdots .}$

In the limit dt → 0, the terms dt2 and dt dBt tend to zero faster than dB2, which is O(dt). Setting the dt2 and dt dBt terms to zero, substituting dt for dB2, and collecting the dt and dB terms, we obtain

${\displaystyle df=\left({\frac {\partial f}{\partial t}}+\mu _{t}{\frac {\partial f}{\partial x}}+{\frac {\sigma _{t}^{2}}{2}}{\frac {\partial ^{2}f}{\partial x^{2}}}\right)dt+\sigma _{t}{\frac {\partial f}{\partial x}}\,dB_{t}}$

as required.

Mathematical formulation of Itô's lemma

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

Itô drift-diffusion processes (due to: Kunita-Watanabe)

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

${\displaystyle dX_{t}=\mu _{t}\,dt+\sigma _{t}\,dB_{t}}$

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

${\displaystyle df(t,X_{t})=\left({\frac {\partial f}{\partial t}}+\mu _{t}{\frac {\partial f}{\partial x}}+{\frac {\sigma _{t}^{2}}{2}}{\frac {\partial ^{2}f}{\partial x^{2}}}\right)dt+\sigma _{t}{\frac {\partial f}{\partial x}}\,dB_{t}.}$

This immediately implies that f(t,Xt) is itself an Itô drift-diffusion process.

In higher dimensions, if ${\displaystyle \mathbf {X} _{t}=(X_{t}^{1},X_{t}^{2},\ldots ,X_{t}^{n})^{T}}$ is a vector of Itô processes such that

${\displaystyle d\mathbf {X} _{t}={\boldsymbol {\mu }}_{t}\,dt+\mathbf {G} _{t}\,d\mathbf {B} _{t}}$

for a vector ${\displaystyle {\boldsymbol {\mu }}_{t}}$ and matrix ${\displaystyle \mathbf {G} _{t}}$, Itô's lemma then states that

{\displaystyle {\begin{aligned}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{aligned}}}

where X f is the gradient of f w.r.t. X, HX 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 + Δt] is hΔt plus higher order terms. h could be a constant, a deterministic function of time, or a stochastic process. The survival probability ps(t) is the probability that no jump has occurred in the interval [0, t]. The change in the survival probability is

${\displaystyle dp_{s}(t)=-p_{s}(t)h(t)\,dt.}$

So

${\displaystyle p_{s}(t)=\exp \left(-\int _{0}^{t}h(u)\,du\right).}$

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

${\displaystyle d_{j}S(t)=\lim _{\Delta t\to 0}(S(t+\Delta t)-S(t^{-}))}$

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

${\displaystyle E[d_{j}S(t)]=h(S(t^{-}))\,dt\int _{z}z\eta (S(t^{-}),z)\,dz.}$

Define ${\displaystyle dJ_{S}(t)}$, a compensated process and martingale, as

${\displaystyle dJ_{S}(t)=d_{j}S(t)-E[d_{j}S(t)]=S(t)-S(t^{-})-\left(h(S(t^{-}))\int _{z}z\eta \left(S(t^{-}),z\right)\,dz\right)\,dt.}$

Then

${\displaystyle d_{j}S(t)=E[d_{j}S(t)]+dJ_{S}(t)=h(S(t^{-}))\left(\int _{z}z\eta (S(t^{-}),z)\,dz\right)dt+dJ_{S}(t).}$

Consider a function ${\displaystyle g(S(t),t)}$ of the jump process dS(t). If S(t) jumps by Δs then g(t) jumps by Δg. Δg is drawn from distribution ${\displaystyle \eta _{g}()}$ which may depend on ${\displaystyle g(t^{-})}$, dg and ${\displaystyle S(t^{-})}$. The jump part of ${\displaystyle g}$ is

${\displaystyle g(t)-g(t^{-})=h(t)\,dt\int _{\Delta g}\,\Delta g\eta _{g}(\cdot )\,d\Delta g+dJ_{g}(t).}$

If ${\displaystyle S}$ contains drift, diffusion and jump parts, then Itô's Lemma for ${\displaystyle g(S(t),t)}$ is

${\displaystyle dg(t)=\left({\frac {\partial g}{\partial t}}+\mu {\frac {\partial g}{\partial S}}+{\frac {\sigma ^{2}}{2}}{\frac {\partial ^{2}g}{\partial S^{2}}}+h(t)\int _{\Delta g}\left(\Delta g\eta _{g}(\cdot )\,d{\Delta }g\right)\,\right)dt+{\frac {\partial g}{\partial S}}\sigma \,dW(t)+dJ_{g}(t).}$

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

Non-continuous semimartingales

Itô'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 Itô's lemma. For any cadlag process Yt, the left limit in t is denoted by Yt−, which is a left-continuous process. The jumps are written as ΔYt = YtYt−. Then, Itô's lemma states that if X = (X1, X2, ..., Xd) is a d-dimensional semimartingale and f is a twice continuously differentiable real valued function on Rd then f(X) is a semimartingale, and

{\displaystyle {\begin{aligned}f(X_{t})&=f(X_{0})+\sum _{i=1}^{d}\int _{0}^{t}f_{i}(X_{s-})\,dX_{s}^{i}+{\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\leq t}\left(\Delta f(X_{s})-\sum _{i=1}^{d}f_{i}(X_{s-})\,\Delta X_{s}^{i}-{\frac {1}{2}}\sum _{i,j=1}^{d}f_{i,j}(X_{s-})\,\Delta X_{s}^{i}\,\Delta X_{s}^{j}\right).\end{aligned}}}

This differs from the formula for continuous semi-martingales 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(Xt).

Multiple Non-continuous Jump-Processes

There is also a version of this for a twice-continuously differentiable in space once in time function f evaluated at (potentially different) non-continuous semi-martingales which may be written as follows:

{\displaystyle {\begin{aligned}f(t,X_{t}^{1},...,X_{t}^{d})&=f(0,X_{0}^{1},...,X_{0}^{d})+\int _{0}^{t}{\dot {f}}({s_{-}},X_{s_{-}}^{1},...,X_{s_{-}}^{d})d{s}\\&+\sum _{i=1}^{n}\int _{0}^{t}f_{i}({s_{-}},X_{s_{-}}^{1},...,X_{s_{-}}^{d})\,dX_{s}^{(c,i)}\\&+{\frac {1}{2}}\sum _{i_{1},..,i_{d}=1}^{d}\int _{0}^{t}f_{i_{1},..,i_{d}}({s_{-}},X_{s_{-}}^{1},...,X_{s_{-}}^{d})\,dX_{s}^{(c,i_{1})}...X_{s}^{(c,i_{d})}\\&+\sum _{0

where ${\displaystyle X^{c,i}}$ denotes the continuous part of the ith semi-martingale.

Examples

Geometric Brownian motion

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

{\displaystyle {\begin{aligned}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{aligned}}}

It follows that

${\displaystyle \log(S_{t})=\log(S_{0})+\sigma B_{t}+\left(\mu -{\tfrac {\sigma ^{2}}{2}}\right)t,}$

exponentiating gives the expression for S,

${\displaystyle S_{t}=S_{0}\exp \left(\sigma B_{t}+\left(\mu -{\tfrac {\sigma ^{2}}{2}}\right)t\right).}$

The correction term of σ2/2 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 σ2/2 appears in the d1 and d2 auxiliary variables of the Black–Scholes formula, and can be interpreted as a consequence of Itô's lemma.

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 Y0 = 1. It is sometimes denoted by Ɛ(X). Applying Itô's lemma with f(Y) = log(Y) gives

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

Exponentiating gives the solution

${\displaystyle Y_{t}=\exp \left(X_{t}-X_{0}-{\tfrac {1}{2}}[X]_{t}\right).}$

Black–Scholes formula

Itô's lemma can be used to derive the Black–Scholes equation for an option.[3] 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, St), Itô's lemma gives

${\displaystyle df(t,S_{t})=\left({\frac {\partial f}{\partial t}}+{\frac {1}{2}}\left(S_{t}\sigma \right)^{2}{\frac {\partial ^{2}f}{\partial S^{2}}}\right)\,dt+{\frac {\partial f}{\partial S}}\,dS_{t}.}$

The term f/S dS represents the change in value in time dt of the trading strategy consisting of holding an amount f/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

${\displaystyle dV_{t}=r\left(V_{t}-{\frac {\partial f}{\partial S}}S_{t}\right)\,dt+{\frac {\partial f}{\partial S}}\,dS_{t}.}$

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

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