In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in
French literature) is an
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), an ...
used in
Itô calculus to find the
differential of a time-dependent function of a
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
. It serves as the
stochastic calculus counterpart of the
chain rule. It can be heuristically derived by forming the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
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
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...
increment. The
lemma
Lemma may refer to:
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* Lemma (morphology), the canonical, dictionary or citation form of a word
* Lemma (psycholinguistics), a mental abstraction of a word about to be uttered
Science and mathematics
* Lemma (botany), ...
is widely employed in
mathematical finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
In general, there exist two separate branches of finance that require ...
, and its best known application is in the derivation of the
Black–Scholes equation In mathematical finance, the Black–Scholes equation is a partial differential equation (PDE) governing the price evolution of a European call or European put under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE ...
for option values.
Motivation
Suppose we are given the stochastic differential equation
where is a
Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...
and the functions
are deterministic (not stochastic) functions of time. In general, it's not possible to write a solution
directly in terms of
However, we can formally write an integral solution
This expression lets us easily read off the mean and variance of
(which has no higher moments). First, notice that every
individually has mean 0, so the expectation value of
is simply the integral of the drift function:
Similarly, because the
terms have variance 1 and no correlation with one another, the variance of
is simply the integral of the variance of each infinitesimal step in the random walk:
However, sometimes we are faced with a stochastic differential equation for a more complex process
in which the process appears on both sides of the differential equation. That is, say
for some functions
and
In this case, we cannot immediately write a formal solution as we did for the simpler case above. Instead, we hope to write the process
as a function of a simpler process
taking the form above. That is, we want to identify three functions
and
such that
and
In practice, Ito's lemma is used in order to find this transformation. Finally, once we have transformed the problem into the simpler type of problem, we can determine the mean and higher moments of the process.
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.
Suppose is an
Itô drift-diffusion process that satisfies the
stochastic differential equation
:
where is a
Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...
.
If is a twice-differentiable scalar function, its expansion in a
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
is
:
Substituting for and therefore for gives
:
In the limit , the terms and tend to zero faster than , which is . Setting the and terms to zero, substituting for (due to the quadratic variation of a
Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...
), and collecting the and terms, we obtain
:
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
:
and any twice
differentiable scalar function of two real variables and , one has
:
This immediately implies that is itself an Itô drift-diffusion process.
In higher dimensions, if
is a vector of Itô processes such that
:
for a vector
and matrix
, Itô's lemma then states that
:
where
is the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of w.r.t. , is the
Hessian matrix of w.r.t. , and is the
trace operator.
Poisson jump processes
We may also define functions on discontinuous stochastic processes.
Let be the jump intensity. The
Poisson process model for jumps is that the probability of one jump in the interval is plus higher order terms. could be a constant, a deterministic function of time, or a stochastic process. The survival probability is the probability that no jump has occurred in the interval . The change in the survival probability is
:
So
:
Let be a discontinuous stochastic process. Write
for the value of ''S'' as we approach ''t'' from the left. Write
for the non-infinitesimal change in as a result of a jump. Then
:
Let ''z'' be the magnitude of the jump and let
be the
distribution Distribution may refer to:
Mathematics
* Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
*Probability distribution, the probability of a particular value or value range of a vari ...
of ''z''. The expected magnitude of the jump is
:
Define
, a
compensated process and
martingale, as
:
Then
:
Consider a function
of the jump process . If jumps by then jumps by . is drawn from distribution
which may depend on
, ''dg'' and
. The jump part of
is
:
If
contains drift, diffusion and jump parts, then Itô's Lemma for
is
:
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 -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 , the left limit in is denoted by , which is a left-continuous process. The jumps are written as . Then, Itô's lemma states that if is a -dimensional semimartingale and ''f'' is a twice continuously differentiable real valued function on then ''f''(''X'') is a semimartingale, and
:
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 is Δ''f''(''X
t'').
Multiple non-continuous jump processes
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:
:
where
denotes the continuous part of the ''i''th 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 , for a Brownian motion ''B''. Applying Itô's lemma with
gives
:
It follows that
:
exponentiating gives the expression for ''S'',
:
The correction term of corresponds to the difference between the median and mean of the
log-normal distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a norma ...
, 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 In mathematical finance, convexity refers to non-linearities in a financial model. In other words, if the price of an underlying variable changes, the price of an output does not change linearly, but depends on the second derivative (or, loosely spe ...
. 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 appears in the ''d''
1 and ''d''
2 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 with initial condition . It is sometimes denoted by .
Applying Itô's lemma with ''f''(''Y'') = log(''Y'') gives
:
Exponentiating gives the solution
:
Black–Scholes formula
Itô's lemma can be used to derive the
Black–Scholes equation In mathematical finance, the Black–Scholes equation is a partial differential equation (PDE) governing the price evolution of a European call or European put under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE ...
for an
option.
Suppose a stock price follows a
geometric Brownian motion given by the stochastic differential equation . Then, if the value of an option at time is ''f''(''t'', ''S
t''), Itô's lemma gives
:
The term represents the change in value in time ''dt'' of the trading strategy consisting of holding an amount 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
:
This strategy replicates the option if ''V'' = ''f''(''t'',''S''). Combining these equations gives the celebrated Black–Scholes equation
:
Product rule for Itô processes
Let
be a two-dimensional Ito process with SDE:
:
Then we can use the multi-dimensional form of Ito's lemma to find an expression for
.
We have
and
.
We set
and observe that
and
Substituting these values in the multi-dimensional version of the lemma gives us:
:
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
:
so we see that the product
is itself an
Itô drift-diffusion process.
See also
*
Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...
*
Itô calculus
*
Feynman–Kac formula
*
Euler–Maruyama method
Notes
References
*
Kiyosi Itô
was a Japanese mathematician who made fundamental contributions to probability theory, in particular, the theory of stochastic processes. He invented the concept of stochastic integral and stochastic differential equation, and is known as the fo ...
(1944). Stochastic Integral. ''Proc. Imperial Acad. Tokyo'' 20, 519–524. This is the paper with the Ito Formula
''Online''*
Kiyosi Itô
was a Japanese mathematician who made fundamental contributions to probability theory, in particular, the theory of stochastic processes. He invented the concept of stochastic integral and stochastic differential equation, and is known as the fo ...
(1951). On stochastic differential equations. ''Memoirs, American Mathematical Society'' 4, 1–51
''Online''*
Bernt Øksendal
Bernt Karsten Øksendal (born 10 April 1945 in Fredrikstad) is a Norwegian mathematician. He completed his undergraduate studies at the University of Oslo, working under Otte Hustad. He obtained his PhD from University of California, Los Angele ...
(2000). ''Stochastic Differential Equations. An Introduction with Applications'', 5th edition, corrected 2nd printing. Springer. . Sections 4.1 and 4.2.
*Philip E Protter (2005). ''Stochastic Integration and Differential Equations'', 2nd edition. Springer. . Section 2.7.
External links
Derivation Prof. Thayer Watkins
optiontutor
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