Ito引理证明

Now consider another variable C, such as the price of a call option, which is a function of S and t, say C = f (S, t). Because
C is a function of the stochastic variable S, C will have a stochastic component as well as a deterministic component. C will
of w2 from its expected value of 1; i.e., the additional term
1 2
(
∂2f ∂S2
)Hale Waihona Puke b2v2w2dt).
(11)
However the variance of this additional term is proportional to (dt)2 whereas the variance of the stochastic term given in
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2
An immediate question is whether is an extension of Ito’s Lemma for stable distributions of z other than the normal distrib-
ution. This question is investigated in a page on stable distributions.
dS = adt + bdz,
(1)
where a and b may be functions of S and t as well as other variables; i.e., dS = a(S, t, x)dt + b(S, t, x)dz.
The expected value of dz is zero so the expected value of dS is equal to the deterministic component, adt.
(10) is proportional to (dt). Thus the stochastic term given in (11) vanishes in comparison with the stochastic term given in
(10).
Ito’s Lemma is essential in the derivation of Black and Scholes Equation.
Taking into account the infinitesimal nature of dt so that dt to any power higher than unity vanishes, (7) reduces to:
dC
=
∂f (
∂t
)dt
+
∂f (
∂S
)adt
+
∂f (
∂S
)bvw(dt)2
(4)
∂S
1
Ito’s Lemma is crucial in deriving differential equations for the value of derivative securities such as stock options. The Taylor series for f (S, t) gives the increment in C as:
dC
=
∂f (
)dt
+
∂f (
)adt
+
∂f (
1
)bvw(dt) 2
+
1 (∂2f /∂S2)(a2dt2
+
3
2abvw(dt) 2
+
b2v2w2dt)
∂t
∂S
∂S
2
+(
∂2f
)(a(dt)2
+
bvw(dt)
3 2
)
+
∂S∂t
1 2
(
∂2f ∂t2
)(dt)2
+
higher
order
terms.
(7)
Ito’s Lemma and its Derivation
Huiwei Wang April 22, 2009
Ito’s Lemma is named for its discoverer, the brilliant Japanese mathematician Kiyoshi Ito. The human race lost this extraordinary individual on November 10, 2008. He died at age 93. His work created a field of mathematics that is a calculus of stochastic variables.
dS = adt + bdz.
(3)
Ito’s Lemma gives the answer. The deterministic and stochastic components of dC are given by:
p
=
∂f ∂t
∂f + ( )a +
∂S
1 2
(
∂2f ∂S2
)b2
∂f
q = ( )b.
mean or expected value of dz is zero. The variance of a random variable which is the accumulation of independent effects over
an interval of time is proportional to the length of the interval, in this case dt. The standard deviation of dz is thus proportional
The random variable dz represents an accumulation of random influences over the interval dt. The Central Limit Theorem
then implies that dz has a normal distribution and hence is completely characterized by its mean and standard deviation. The
+
1 ∂2f 2(∂S2
)(b2v2w2dt).
(8)
Noting that the expected value of w2 is unity, the expected value of dC is:
∂f [
∂t
∂f + ( )a +
∂S
1 2
(
∂2f ∂S2
)b2]dt.
(9)
This is the deterministic component of dC. The stochastic component is the term that depends upon dz, which in (8) is
to
the
square
root
of
dt,(dt)
1 2
.
All
of
this
means
that
the
random
variable
dz
is
equivalent
to
a
random
variable
w(dt)
1 2
,
where
w is a standard normal variable with mean zero and standard deviation equal to unity.
dC
=
∂f (
∂t
)dt
+
∂f ( )adt
∂S
+
∂f (
∂S
1
)bvw(dt) 2
+
1 ∂2f 2 ( ∂S2 )(adt
+
bv
w(dt)
1 2
)2
∂2f
1
+( )(adt + bvw(dt) 2 )(dt) +
∂S∂t
1 2
(
∂2f ∂t2
)(dt)2
+
higher
order
terms.
(6)
With the expansion of the squared term and the product term the result is:
Ito’s lemma (also known as Ito-Doeblin Theorem) is a theorem in stochastic calculus. It tells you that if you have a random walk, in Y , say, and a function of that randomly walking variable, call it F (Y, t), then you can easily write an expression for the random walk in F . A function of a random variable is itself random in general.