Brandon Lee
15.450
Recitation 3
Brandon Lee
Stochastic Calculus

Defining properties of Brownian motion Z t are
1
2
3
4
Z
0
= 0.
It has continuous paths.
It has independent increments: if t
1
Z t
2
− Z t
1 and Z s
2
− Z s
1
< t
2
≤ s
1
< s
2
, then are independent random variables.
Its increments are normally distributed: Z t
2
− t
1
∼ N (0 , t
2
− t
1
).
Brownian motion is the continuous time analogue of random walk in discrete time. Brownian motion is the basic building block that we use to model uncertatinty or random movements in our continous time finance models.
Brandon Lee
Stochastic Calculus

An Ito process X t is of the form
X t
Z t
=
µ s
0 ds +
Z
0 t
σ s dZ s o r in differential form dX t
=
µ t dt +
σ t dZ t
H euristically speaking, over a very small time interval ∆, the in crement X t +∆
− X t has normal distribution with mean
µ t
∆ a nd variance
σ t
2
∆. Call
µ t the instantaneous drift and
σ t th e in stantaneous volatility or diffusion coefficient.
Brandon Lee
Stochastic Calculus

Suppose X t is an Ito process. Ito’s Lemma states that f ( t , X is an Ito process as well and shows how to compute the drift t
) and diffusion coefficient of df ( t , X t
).
Ito’s Lemma: Suppose dX t
=
µ t dt +
σ t dZ t then d ( f ( t , X t
)) =
∂ f ( t , X t
)
+
∂ t
∂ f ( t , X t
)
µ t
∂ X t
+
1
2
∂
2 f ( t , X t
)
σ t
2
∂ X t
2
Remember the rule
( dt ) = o ( dt ) dt · dZ t
= o ( dt )
( dZ t
) = dt dt +
∂ f ( t , X t
)
∂ X t
σ t dZ t
In other words, the first two are of smaller order than dt , and we can ignore them in Ito’s lemma.
Brandon Lee
Stochastic Calculus

Suppose dS t
S t
= rdt +
σ dZ t and define Y t
= f
Use’s Ito’s lemma:
( t , S t
) = e
− rt
S t
. Calculate dY t
.
dY t
=
∂ f ( t , S t
)
∂ f ( t , S t
)
∂ t
+
∂
S t rS t
+
1
2
∂
2 f ( t , S t
)
∂
S t
2
σ
2
S t
2
= − re
− rt
S t
+ re
− rt
S t
+ 0 dt + e
− rt
σ
S t dZ t
=
σ
Y t dZ t dt +
∂ f ( t , S t
)
σ
S t dZ t
∂
S t
Why does Y t have zero drift?
Brandon Lee
Stochastic Calculus

Suppose
η t is some stochastic process. Define
ξ t
= exp −
Z t
0
η s dZs −
Z t
0
η
2 s ds
2
Calculate d
ξ t
.
Define X t
= R
0 t
η s dZs + R t η
2 s
0 2 ds . Note that dX t
=
η t
2 dt +
η t dZ t
2 and
ξ t
= exp ( − X t
).
Apply Ito’s Lemma and we get d
ξ t
= −
ξ t dX t
+
ξ t
( dX t
)
2
= −
ξ t
η t dZ t
Practice and make sure you can do those calculations!
Brandon Lee
Stochastic Calculus

Suppose dX t
X t dY t
Y t
=
µ x dt +
σ x dZ t
1
=
µ y dt +
σ y dZ t
2 where dZ t
1
· dZ
2 t
=
ρ dt
Define
X t
M t
=
Y t
What is the instantaneous volatility of dM t
M t
?
Brandon Lee
Stochastic Calculus

Let M t
= f ( X t
, Y t
) =
X t
Y t
. Then by Ito’s Lemma, dM t
=
+
1
2
∂ f ( X t
, Y t
) dX t
+
∂
X t
∂
2 f ( X t
, Y t
)
∂
X t
2
∂ f
( dX t
(
∂
X t
Y t
, Y t
∂
)
2 dY t
∂ f (
X t
X
∂ t
, Y
Y t t
) dX t
· dY t
+
∂
2 f ( X t
, Y t
)
∂
Y t
2
( dY t
)
2
Carry out the calculation and arrive at dM t
= M t dM t
M t
µ x
−
µ y
−
ρ σ x σ y
+
σ
=
µ x
−
µ y
−
ρ σ x σ y
+
σ
2 y
Note that y
2 dt dt
+
σ
+ x
M t dZ t
1
σ x dZ t
1
− M
−
σ y dZ t
2 t σ y dZ t
2
2 dM t
M t
=
σ x dZ t
1 −
σ y d Z t
2
2
=
σ x
2
+
σ y
2
−
ρ σ x σ y dt so that instantaneous volatility of dM t
M t is q
σ x
2 + σ y
2 −
ρ σ x
σ y
Brandon Lee
Stochastic Calculus

MIT OpenCourseWare http://ocw.mit.edu
15.450 Analytics of Finance
Fall 2010
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