Itô's lemma

When ${\displaystyle X_{t+dt}}$ is a gaussian random variable, ${\displaystyle f(X_{t+dt})}$ is also approximately gaussian random variable, but its mean ${\displaystyle E[f(X_{t+dt})]}$ differs from ${\displaystyle f(E[X_{t+dt}])}$ by a factor proportional to ${\displaystyle f''(E[X_{t+dt}])}$ and the variance of ${\displaystyle X_{t+dt}}$.

Suppose we know that ${\displaystyle X_{t},X_{t+dt}}$ are two jointly-gaussian distributed random variables, and ${\displaystyle f}$ is nonlinear but has continuous second derivative, then in general, neither of ${\displaystyle f(X_{t}),f(X_{t+dt})}$ is gaussian, and their joint distribution is also not gaussian. However, since ${\displaystyle X_{t+dt}|X_{t}}$ is gaussian, we might still find ${\displaystyle f(X_{t+dt})|f(X_{t})}$ is gaussian. This is not true when ${\displaystyle dt}$ is finite, but when ${\displaystyle dt}$ becomes infinitesimal, this becomes true.

The key idea is that ${\displaystyle X_{t+dt}=X_{t}+\mu _{t}dt+dW_{t}}$ has a deterministic part and a noisy part. When ${\displaystyle f}$ is nonlinear, the noisy part has a deterministic contribution. If ${\displaystyle f}$ is convex, then the deterministic contribution is positive (by Jensen's inequality).

To find out how large is the contribution, we write ${\displaystyle X_{t+dt}=X_{t}+\mu _{t}dt+\sigma _{t}{\sqrt {dt}}z}$, where ${\displaystyle z}$ is a standard gaussian, then perform Taylor expansion.

$${\displaystyle {\begin{aligned}f(X_{t+dt})&=f(X_{t})+f'(X_{t})\mu _{t}dt+f'(X_{t})\sigma _{t}{\sqrt {dt}}z+{\frac {1}{2}}f''(X_{t})(\sigma _{t}^{2}z^{2}dt+\mu _{t}\sigma _{t}zdt^{3/2}+\mu _{t}^{2}dt^{2})+o(dt)\\&=\left(f(X_{t})+f'(X_{t})\mu _{t}dt+{\frac {1}{2}}f''(X_{t})\sigma _{t}^{2}dt+o(dt)\right)+\left(f'(X_{t})\sigma _{t}{\sqrt {dt}}z+{\frac {1}{2}}f''(X_{t})\sigma _{t}^{2}(z^{2}-1)dt+o(dt)\right)\end{aligned}}}$$

We have split it into two parts, a deterministic part, and a random part with mean zero. The random part is non-gaussian, but the non-gaussian parts decay faster than the gaussian part, and at the ${\displaystyle dt\to 0}$ limit, only the gaussian part remains. The deterministic part has the expected ${\displaystyle f(X_{t})+f'(X_{t})\mu _{t}dt}$, but also a part contributed by the convexity: ${\displaystyle {\frac {1}{2}}f''(X_{t})\sigma _{t}^{2}dt}$.

To understand why there should be a contribution due to convexity, consider the simplest case of geometric Brownian walk (of the stock market): ${\displaystyle S_{t+dt}=S_{t}(1+dB_{t})}$. In other words, ${\displaystyle d(\ln S_{t})=dW_{t}}$. Let ${\displaystyle X_{t}=\ln S_{t}}$, then ${\displaystyle S_{t}=e^{X_{t}}}$, and ${\displaystyle X_{t}}$ is a Brownian walk. However, although the expectation of ${\displaystyle X_{t}}$ remains constant, the expectation of ${\displaystyle S_{t}}$ grows. Intuitively it is because the downside is limited at zero, but the upside is unlimited. That is, while ${\displaystyle X_{t}}$ is normally distributed, ${\displaystyle S_{t}}$ is log-normally distributed.

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.

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 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.

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).

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.

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.

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

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

Itô's lemma can be used to derive the Black–Scholes equation for an option.[2] 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

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

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.