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. 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–Doeblin Theorem in recognition of the recently discovered work of Wolfgang Doeblin.[1]

Informal derivation[edit]

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

 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

 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 μtdt + σtdBt for dXt 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 → 0, the terms dt2 and dt dBt tend to zero faster than dB2, which is O(dt). Setting the dt2 and dt dB2 terms to zero, substituting dt for dB2, 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[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 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,Xt) 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, then 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 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[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 + 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

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^-) - \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 Δs then g(t) jumps by Δg. Δg 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[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 = YtYt−. 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\\
&\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(Xt).

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 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 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 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 dY = Y 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 - \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[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/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

 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.

See also[edit]

Notes[edit]

References[edit]

External links[edit]