Itō's lemma

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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. It is named after Kiyoshi Itō. It is best memorized using the Taylor series expansion of the function up to its second derivatives and identifying the square of an increment in the stochastic 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. Ito's Lemma is occasionally referred to as the Itō–Doeblin Theorem in recognition of the recently discovered work of Wolfgang Doeblin.[1]

Mathematical formulation of Itō's lemma[edit]

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

Itō drift-diffusion processes[edit]

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 ƒ(tx) 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{1}{2}\sigma_t^2\frac{\partial^2f}{\partial x^2}\right)dt+ \sigma_t \frac{\partial f}{\partial x}\,dB_t

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

In higher dimensions, Ito's lemma states


df(t,\mathbf{X}_t) = \frac{\partial f}{\partial t} dt + (\boldsymbol{\nabla}_\mathbf{X}^{\mathsf T} f) d\mathbf{X}_t + \tfrac{1}{2} (d\mathbf{X}_t^\mathsf{T}) (\nabla_\mathbf{X}^2 f) d\mathbf{X}_t,

where \mathbf{X}_t^\mathsf{T} = (X_{t,1}, X_{t,2}, \ldots, X_{t,n}) is a vector of Itō processes, \boldsymbol{\nabla}_\mathbf{X} f = \partial f/\partial \mathbf{X} is the gradient of ƒ w.r.t. X, and \nabla_\mathbf{X}^2 f = \boldsymbol{\nabla}_\mathbf{X}\boldsymbol{\nabla}_\mathbf{X}^\mathsf{T} f is the Hessian matrix of ƒ w.r.t. X.

Continuous semimartingales[edit]

More generally, the above formula also holds for any continuous d-dimensional semimartingale X = (X1,X2,…,Xd), and twice continuously differentiable and real valued function f on Rd. Some people prefer to present the formula in another form with cross variation shown explicitly as follows, f(X) is a semimartingale satisfying

df(X_t) = \sum_{i=1}^d f_{i}(X_t)\,dX^i_t + \frac{1}{2}\sum_{i,j=1}^df_{i,j}(X_t)\,d[X^i,X^j]_t.

In this expression, the term fi represents the partial derivative of f(x) with respect to xi, and [Xi,Xj ] is the covariation process of Xi and Xj.

Poisson jump processes[edit]

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 + \Delta t] is h \Delta 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.

So

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^-)-(h(S(t^-)) \int_z z \eta(S(t^-),z) \, dz) \, dt.

Then

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

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


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

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


\begin{align}
d g(t) & = \left( \frac{\partial g}{\partial t}+\mu \frac{\partial g}{\partial S}+\frac{1}{2} \sigma^2 \frac{\partial^2 g}{\partial S^2}+h(t)\int_{\Delta g} (\Delta g \eta_g(\cdot) \, d{\Delta}g) \, \right) dt + \frac{\partial g}{\partial S} \sigma \, d W(t) + d J_g(t).
\end{align}

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[edit]

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 Yt, the left limit in t is denoted by Yt-, which is a left-continuous process. The jumps are written as ΔYt = Yt - Yt-. Then, Itō'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


\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\\
&{} + \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(Xt).

Informal derivation[edit]

A formal proof of the lemma requires us to take the limit of a sequence of random variables, which is not done here. 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 the Itō process is in the form of

 dx= a\,dt + b\,dB.\! (dB, Brownian motion)

Expanding f(xt) in a Taylor series in x and t we have

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

and substituting a dt + b dB for dx gives

 df = \frac{\partial f}{\partial x}(a\,dt + b\,dB) + \frac{\partial f}{\partial t}\,dt + \frac{1}{2}\frac{\partial^2 f}{\partial x^2}(a^2\,dt^2 + 2ab\,dt\,dB + b^2\,dB^2) + \cdots.

In the limit as dt tends to 0, the dt2 and dt dB terms disappear but the dB2 term tends to dt. The latter can be shown if we prove that

 dB^2 \rightarrow E(dB^2), since  E(dB^2) = dt \, ; see Wiener process# Basic properties.

Deleting the dt2 and dt dB terms, substituting dt for dB2, and collecting the dt and dB terms, we obtain

 df = \left(a\frac{\partial f}{\partial x} + \frac{\partial f}{\partial t} + \frac{1}{2}b^2\frac{\partial^2 f}{\partial x^2}\right)dt + b\frac{\partial f}{\partial x}\,dB

as required.

The formal proof is somewhat technical and is beyond the current scope of this article.

Examples[edit]

Geometric Brownian motion[edit]

A process S is said to follow a geometric Brownian motion with volatility σ and drift μ if it satisfies the stochastic differential equation dSS(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 +(\mu-\sigma^2/2)\,dt.
\end{align}

It follows that log(St) = log(S0) + σBt + (μ − σ2/2)t, and exponentiating gives the expression for S,

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

The correction term of -\sigma^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 \sigma^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.

The Doléans exponential[edit]

The Doléans exponential (or stochastic exponential) of a continuous semimartingale X can be defined as the solution to the SDE dYY dX with initial condition Y0 = 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 - \frac{1}{2}\,d[X].
\end{align}

Exponentiating gives the solution

Y_t = \exp\left(X_t-X_0-[X]_t/2\right).

Black–Scholes formula[edit]

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,St), 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 (∂f/∂SdS 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

 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{1}{2}\sigma^2S^2\frac{\partial^2 f}{\partial S^2} + rS\frac{\partial f}{\partial S}-rf = 0.

See also[edit]

Notes[edit]

References[edit]

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