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.

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 X_t is an Itô drift-diffusion process that satisfies the stochastic differential equation

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

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

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

Substituting X_t for x and therefore μ_t dt + σ_t dB_t 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 dB_t tend to zero faster than dB², which is O(dt). Setting the dt² and dt dB_t terms to zero, substituting dt for dB² (due to the quadratic variation of a Wiener process), 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,X_t) 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

${\displaystyle {\begin{aligned}df(t,\mathbf {X} _{t})&={\frac {\partial f}{\partial t}}\,dt+\left(\nabla _{\mathbf {X} }f\right)^{T}\,d\mathbf {X} _{t}+{\frac {1}{2}}\left(d\mathbf {X} _{t}\right)^{T}\left(H_{\mathbf {X} }f\right)\,d\mathbf {X} _{t},\\&=\left\{{\frac {\partial f}{\partial t}}+\left(\nabla _{\mathbf {X} }f\right)^{T}{\boldsymbol {\mu }}_{t}+{\frac {1}{2}}\operatorname {Tr} \left[\mathbf {G} _{t}^{T}\left(H_{\mathbf {X} }f\right)\mathbf {G} _{t}\right]\right\}dt+\left(\nabla _{\mathbf {X} }f\right)^{T}\mathbf {G} _{t}\,d\mathbf {B} _{t}\end{aligned}}}$

where ${\displaystyle \nabla _{\mathbf {X} }f}$ 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 + Δt] is hΔt 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

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

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 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 Y_t, the left limit in t is denoted by Y_t−, which is a left-continuous process. The jumps are written as ΔY_t = Y_t − Y_t−. Then, Itô's lemma states that if X = (X¹, X², ..., 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

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

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(X_t).

Multiple non-continuous jump processes[]

^{[citation needed]}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:

${\displaystyle {\begin{aligned}f(t,X_{t}^{1},\ldots ,X_{t}^{d})={}&f(0,X_{0}^{1},\ldots ,X_{0}^{d})+\int _{0}^{t}{\dot {f}}({s_{-}},X_{s_{-}}^{1},\ldots ,X_{s_{-}}^{d})d{s}\\&{}+\sum _{i=1}^{d}\int _{0}^{t}f_{i}({s_{-}},X_{s_{-}}^{1},\ldots ,X_{s_{-}}^{d})\,dX_{s}^{(c,i)}\\&{}+{\frac {1}{2}}\sum _{i_{1},\ldots ,i_{d}=1}^{d}\int _{0}^{t}f_{i_{1},\ldots ,i_{d}}({s_{-}},X_{s_{-}}^{1},\ldots ,X_{s_{-}}^{d})\,dX_{s}^{(c,i_{1})}\cdots X_{s}^{(c,i_{d})}\\&{}+\sum _{0<s\leq t}\left[f(s,X_{s}^{1},\ldots ,X_{s}^{d})-f({s_{-}},X_{s_{-}}^{1},\ldots ,X_{s_{-}}^{d})\right]\end{aligned}}}$

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 ${\displaystyle dS_{t}=\sigma S_{t}\,dB_{t}+\mu S_{t}\,dt}$, for a Brownian motion B. Applying Itô's lemma with ${\displaystyle f(S_{t})=\log(S_{t})}$ gives

${\displaystyle {\begin{aligned}df&=f^{\prime }(S_{t})\,dS_{t}+{\frac {1}{2}}f^{\prime \prime }(S_{t})(dS_{t})^{2}\\&={\frac {1}{S_{t}}}\,dS_{t}+{\frac {1}{2}}(-S_{t}^{-2})(S_{t}^{2}\sigma ^{2}\,dt)\\&={\frac {1}{S_{t}}}\left(\sigma S_{t}\,dB_{t}+\mu S_{t}\,dt\right)-{\frac {1}{2}}\sigma ^{2}\,dt\\&=\sigma \,dB_{t}+\left(\mu -{\tfrac {\sigma ^{2}}{2}}\right)\,dt.\end{aligned}}}$

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 concave (or convex upwards), 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.

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

${\displaystyle {\begin{aligned}d\log(Y)&={\frac {1}{Y}}\,dY-{\frac {1}{2Y^{2}}}\,d[Y]\\[6pt]&=dX-{\tfrac {1}{2}}\,d[X].\end{aligned}}}$

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.^[1] 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), 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

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

Product rule for Itô processes[]

Let ${\displaystyle \mathbf {X} _{t}}$ be a two-dimensional Ito process with SDE:

${\displaystyle d\mathbf {X} _{t}=d{\begin{pmatrix}X_{t}^{1}\\X_{t}^{2}\end{pmatrix}}={\begin{pmatrix}\mu _{t}^{1}\\\mu _{t}^{2}\end{pmatrix}}dt+{\begin{pmatrix}\sigma _{t}^{1}\\\sigma _{t}^{2}\end{pmatrix}}\,dB_{t}}$

Then we can use the multi-dimensional form of Ito's lemma to find an expression for ${\displaystyle d(X_{t}^{1}X_{t}^{2})}$.

We have ${\displaystyle \mu _{t}={\begin{pmatrix}\mu _{t}^{1}\\\mu _{t}^{2}\end{pmatrix}}}$ and ${\displaystyle \mathbf {G} ={\begin{pmatrix}\sigma _{t}^{1}\\\sigma _{t}^{2}\end{pmatrix}}}$.

We set ${\displaystyle f(t,\mathbf {X} _{t})=X_{t}^{1}X_{t}^{2}}$ and observe that ${\displaystyle {\frac {\partial f}{\partial t}}=0,\ (\nabla _{\mathbf {X} }f)^{T}=(X_{t}^{2}\ \ X_{t}^{1})}$ and ${\displaystyle H_{\mathbf {X} }f={\begin{pmatrix}0&1\\1&0\end{pmatrix}}}$

Substituting these values in the multi-dimensional version of the lemma gives us:

${\displaystyle {\begin{aligned}d(X_{t}^{1}X_{t}^{2})&=df(t,\mathbf {X} _{t})\\&=0\cdot dt+(X_{t}^{2}\ \ X_{t}^{1})\,d\mathbf {X} _{t}+{\frac {1}{2}}(dX_{t}^{1}\ \ dX_{t}^{2}){\begin{pmatrix}0&1\\1&0\end{pmatrix}}{\begin{pmatrix}dX_{t}^{1}\\dX_{t}^{2}\end{pmatrix}}\\&=X_{t}^{2}\,dX_{t}^{1}+X_{t}^{1}dX_{t}^{2}+dX_{t}^{1}\,dX_{t}^{2}\end{aligned}}}$

This is a generalisation of Leibniz's product rule to Ito processes, which are non-differentiable.

Further, using the second form of the multidimensional version above gives us

${\displaystyle {\begin{aligned}d(X_{t}^{1}X_{t}^{2})&=\left\{0+(X_{t}^{2}\ \ X_{t}^{1}){\begin{pmatrix}\mu _{t}^{1}\\\mu _{t}^{2}\end{pmatrix}}+{\frac {1}{2}}\operatorname {Tr} \left[(\sigma _{t}^{1}\ \ \sigma _{t}^{2}){\begin{pmatrix}0&1\\1&0\end{pmatrix}}{\begin{pmatrix}\sigma _{t}^{1}\\\sigma _{t}^{2}\end{pmatrix}}\right]\right\}\,dt+(X_{t}^{2}\sigma _{t}^{1}+X_{t}^{1}\sigma _{t}^{2})\,dB_{t}\\[5pt]&=\left(X_{t}^{2}\mu _{t}^{1}+X_{t}^{1}\mu _{t}^{2}+\sigma _{t}^{1}\sigma _{t}^{2}\right)\,dt+(X_{t}^{2}\sigma _{t}^{1}+X_{t}^{1}\sigma _{t}^{2})\,dB_{t}\end{aligned}}}$

so we see that the product ${\displaystyle X_{t}^{1}X_{t}^{2}}$ is itself an Itô drift-diffusion process.

See also[]

• Wiener process
• Itô calculus
• Feynman–Kac formula

Notes[]

References[]

• Kiyosi Itô (1944). Stochastic Integral. Proc. Imperial Acad. Tokyo 20, 519–524. This is the paper with the Ito Formula; Online
• Kiyosi Itô (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.
• Philip E Protter (2005). Stochastic Integration and Differential Equations, 2nd edition. Springer. ISBN 3-662-10061-4. Section 2.7.

External links[]

• Derivation, Prof. Thayer Watkins
• Informal proof, optiontutor