Itō's lemma

From Wikipedia, the free encyclopedia

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

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

Itō drift-diffusion processes

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(t, x) of two real variables t and x, one has

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

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

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

$df(X_{t})=\sum _{{i=1}}^{d}f_{{i}}(X_{t})\,dX_{t}^{i}+{\frac {1}{2}}\sum _{{i,j=1}}^{d}f_{{i,j}}(X_{t})\,d[X^{i},X^{j}]_{t}.$

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

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 $[t,\ t+\Delta t]$ 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

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

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

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

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

Then

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

Consider a function $g(S(t),t)$ of jump process $dS(t)$ . 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

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

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

${\begin{aligned}dg(t)&=\left({\frac {\partial g}{\partial t}}+\mu {\frac {\partial g}{\partial S}}+{\frac {1}{2}}\sigma ^{2}{\frac {\partial ^{2}g}{\partial S^{2}}}+h(t)\int _{{\Delta g}}(\Delta g\eta _{g}(\cdot )\,d{\Delta }g)\,\right)dt+{\frac {\partial g}{\partial S}}\sigma \,dW(t)+dJ_{g}(t).\end{aligned}}$

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

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 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, Itō'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

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

This differs from the formula for continuous 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(X_t).

Informal derivation

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(x, t) 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

Geometric Brownian motion

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

${\begin{aligned}d\log(S)&=f^{\prime }(S)\,dS+{\frac {1}{2}}f^{{\prime \prime }}(S)S^{2}\sigma ^{2}\,dt\\&={\frac {1}{S}}\left(\sigma S\,dB+\mu S\,dt\right)-{\frac {1}{2}}\sigma ^{2}\,dt\\&=\sigma \,dB+(\mu -\sigma ^{2}/2)\,dt.\end{aligned}}$

It follows that log(S_t) = log(S₀) + σB_t + (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

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

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

Exponentiating gives the solution

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

Black–Scholes formula

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,S_t), 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

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

Notes

↑ "Stochastic Calculus :: Itô–Döblin formula", Michael Stastny

References

Kiyoshi Itō (1944). Stochastic Integral. Proc. Imperial Acad. Tokyo 20, 519-524. This is the paper with the Ito Formula; Online
Kiyoshi Itō (1951). On stochastic differential equations. Memoirs, American Mathematical Society 4, 1–51. Online
Hagen Kleinert (2004). Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th edition, World Scientific (Singapore); Paperback ISBN 981-238-107-4. Also available online: PDF-files. This textbook also derives generalizations of Itō's lemma for non-Wiener (non-Gaussian) processes.
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.
Domingo Tavella (2002). Quantitative Methods in Derivatives Pricing: An Introduction to Computational Finance, John Wiley and Sons. ISBN 978-0-471-27479-7. Pages 36–39.

External links

Derivation, Prof. Thayer Watkins
Informal proof, optiontutor

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.