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5. Ito Calculus

Types of derivatives

Consider a function F (S t, t) depending on two

variables S t (say, price) and time t, where

variable S t itself varies with time t.

In standard calculus there are three types of

derivatives:

Partial derivative:

F s = ∂F (S t, t)

∂S t, F t =

∂F (S t, t)

∂t .(1)

Total derivative:

dF = F s dS t + F t dt.(2)

Chain rule:

dF (S t, t)

dt = F s

dS t

dt + F t.(3)

1

Partial derivative are abstractions. Usually

they are called multipliers or marginal effects

(cf. the Greeks in option theory).

Total derivative describes the total change or

response in F as time t and S t change

The chain rule indicates the chain effects in

the change of the price S t and F as time

changes.

We will consider in this section stochasticcounterparts for the total differential and chain

rule. Essentially as we will see the major dif-

ferences are that we have to interpret the dif-

ferential in stochastic processes via the stochas-

tic integral and that the second order term

(dS t)2 is not negligible as in standard calcu-

lus.

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Ito’s stochastic differential equation

Ito’s lemma gives the stochastic version for

the chain rule.

Let

dS t = a(S t, t) dt + σ(S t, t) dW t.(4)

where a(S t, t) and σ(S t, t) are nonanticipating

and W t is the standard Brownian motion.

3

We interpret dS t via the stochastic integral

such that t

0dS u = S t − S 0,(5)

so that

S t = S 0 + t

0

dS u = t

0

a(S u, u) du + t

0

σ(S u, u) dW u,(6)

where the first integral is the usual Riemann

integral and the second one is the Ito inte-

gral.

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Consider a function F (S t, t) (e.g. derivative

of a stock)

As time t changes by dt what is the total

effect on F (S t, t). The change is

t →

W t →

S t →

F (S t, t).(7)

The interest is dF (S t, t).

5

Suppose the observation interval of S t is [0, T ].

Let 0 = t0 < t1 < · · · < tn = T be a partition

with

h = tk − tk−1 = T

n k = 1, . . . , n ,(8)

so that T = nh.

Consider the finite difference representation

of dS t

∆S k = akh + σk∆W k, k = 1, . . . , n ,(9)

with ∆S k

= S tk

−S tk−1

, ak

= a(S tk−1

, tk

), σk

=

σ(S k−1, tk), and ∆W k = W tk − W tk−1

.

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Ito formula is derived using the Taylor expan-

sion of a ”smooth” function.

If f (x) is such a function the Taylor expansion

around x0 becomes

f (x) = f (x0) + f (x0)(x − x0) + 1

2f (x0)(x − x0)2 + R,

(10)

where R is the remainder.

7

For a function with two variables the Taylor

expansion is

F (S tk, tk) = F (S tk−1

, tk) + F s∆S k + F th + 12

F ss (∆S k)2

+12

F tt h2 + F st h ∆S k + R.

(11)

Arranging terms

∆F k = F s∆S k + F th + 12F ss (∆S k)2

+12F tt h2 + F st h ∆S k + R,

(12)

where

∆F k = F (S tk, tk) − F (S tk−1

, tk−1).(13)

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As n → ∞, h = tk −tk−1 → dt, ∆S k → dS , and

∆F k → dF , and R → 0 because it consists of

terms (∆tk)m and (∆W k)m with m ≥ 3.

So we get

dF (S t, t) = F s dS t + F t dt + 12

F ss (dS t)2

+12

F tt (dt)2 + F st dtdS t.(14)

Using the calculation rules for the differen-

tials, we obtain

dtdS t = dt (a(S t, t)dt + σ(S t, t) dW t)

= a(S t, t)(dt)2 + σ(S t, t) dtdW t

= 0,

(15)

because (dt)2 = 0 and dtdW t = 0.

So we get

dF (S t, t) = F s dS t + F t dt + 1

2F ss(dS t)2(16)

9

Remark 5.1: If S t is non-stochastic then (dS t)2 = 0

and the above formula is just the total derivative dF =

F s dS + F t dt.

Replacing dS t with its Ito representation, wehave

dF (S t, t) = F s · [a(S t, t) dt + σ(S t, t) dW t] + F t dt

+12

F ss [a(S t, t) dt + σ(S t, t) dW t]2 .

(17)

Using the infinitesimal calculation rules again

yields

(dS t)2 = σ2(S t, t) dt.(18)

Arranging terms, we obtain finally the fa-

mous Ito’s differential formula

dF =

F t + a(S t, t)F t +

1

2σ2(S t, t)F ss

dt + σ(S t, t) dW t.

(19)

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The result is summarized as Ito’s Lemma:

Lemma 5.1: (Itˆ o’s Lemma) Let F (S t, t) be a twice-differentiable function of t and of the random process S t with Ito differential equation

dS t = at dt + σt dW t, t ≥ 0,

with at = a(S t, t) and σt = σ(S t, t) continuously twice-

differentiable (real valued) functions. Then

dF = F s dS t + F t dt + 1

2F ssσ2

t dt,(20)

or, after substituting for the right hand side of dS tabove

dF =

F s at + F t +

1

2F ssσ2

dt + F sσt dW t,(21)

where

F s = ∂F

∂S t, F t =

∂F

∂t , and F ss =

∂ 2F

∂S 2t.(22)

11

The major usage of the Ito formula in fi-

nance is to find the (Ito) stochastic differen-

tial equation (SDE) for the financial deriva-

tive once the (Ito) SDE of the underlying

asset is given.

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Ito’s formula can be used also in some cases

to find the stochastic integral itself.

Example 5.1: Let

F (W t, t) = W 2t .(23)

Using formula (16) with S t = W t we obtain

F w = ∂ W 2/∂W = 2W ,(24)

and

F ww = ∂ 2F/∂W 2 = 2.(25)

Then because F t = 0,

dF (W t, t) = F w dW t + 12

F ww(dW t)2

= dt + 2W t dW t.

(26)

Thus the drift of F is a(F, t) = 1 and diffusion param-

eter is σ(F, t) = 2W t.

13

Example 5.2:

F (W t, t) = 3 + t + eW t.(27)

Using again Ito’s formula (16) with S t = W t

dF (W t, t) = F t dt + F w dW t + 12

F ww (dW t)2

= dt + eW t dW t + 12

eW t dt

=

1 + 12

eW t

dt + eW t dW t.

(28)

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Example 5.3: Consider the geometric Brownian mo-

tionS t = S 0e{(µ− 1

2σ2)t+σ W t},(29)

where S 0 is a constant. Then using again Ito withformula (20), and noting that σ(W t, t) = 1, we get

dS t = ∂S t∂W t

dW t + ∂S t∂t

dt + 12

∂ 2S t∂W 2

t

dt

= S 0σe{(µ− 1

2σ2)t+σ W t}

dW t + (µ − 1

2σ2

)S 0e(µ− 1

2σ2)t+σ W t

dt

+12

σ2S 0e(µ− 1

2σ2)t+σ W t dt

= S tσ dW t + (µ − 12

σ2)S t dt + 12

σ2S t dt

= S t(µ dt + σ dW t),(30)

ordS t

S t= µ dt + σdW t,(31)

or

dS t = µS tdt + σS tdW t.(32)

Remark 5.2: Comparing to the general formula dS t =

a(S t, t)dt + σ(S t, t)dW t we find that in (32) a(S t, t) =

µ S t, and σ(S t, t) = σ S t.

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Ito’s formula as an integration tool

Suppose our task is to evaluate t

0W s dW s.(33)

Make a guess

F (W t, t) = 1

2W 2t .(34)

Then using Ito

dF (W t, t) = W t dW t + 1

2dt.(35)

The integral form is1

2W 2t = F (W t, t) =

t

0

dF (W s, s) =

t

0

W s dW s + 1

2

t

0

ds.

(36)

So

t

0W s dW s =

1

2W 2

t −

1

2t.(37)

The start off point here is to make a “good

guess”.

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Example 5.4: Consider Ito integral

t

0

s dW s.(38)

Make a start off guess

F (W t, t) = tW t.(39)

Then

dF (W t, t) = W t dt + t dW t.(40)and

tW t =

t

0

dF (W s, s) =

t

0

W s ds +

t

0

s dW s.(41)

So

t

0

s dW s = tW t − t

0

W s ds.(42)

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Example 5.5: Consider

dS t

S t= µ dt + σ dW t.(43)

Let

F (S t, t) = log S t.(44)

Then

dF (S t, t) = F t dt + F s dS t + 1

2

F ss(dS t)2

= 1S t

dS t − 12

1S 2t

(dS t)2

= µ dt + σ dW t − 12

1S 2t

σ2S 2t dt

= (µ − 12

σ2) dt + σ dW t.

(45)

We get

log S t = log S 0 + t

0dF (S u, u)

= log S 0 + t

0(µ − 1

2σ2)du +

t0

σ dW u

= log S 0 + (µ − 12

σ2)t + σW t.

(46)

So

S t = S 0e(µ−1

2 σ2

)t+σW t.(47)

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Integral form of Ito’s Lemma

Integrating both sides of (21) yields Ito’s for-mula in integral form:

F (S t, t) = F (S 0, 0) +

t

0

F 2(S u, u) +

1

2F 11(S u, u)σ2

u

du

+

t

0 F s dS u.(48)where

F 1(x, y) = ∂F (x, y)

∂x ,F 2(x, y) =

∂F (x, y)

∂y ,(49)

and

F 11(x, y) = ∂ 2F (x, y)∂x2

,(50)

and we have used t

0dF (S u, u) = F (S t, t) − F (S 0, 0).(51)

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Remark 5.3: Rearranging terms in the Ito’s integral form yields t

0

F sdS u = [F (S t, t) − F (S 0, 0)]

−

t

0

F 2(S u, u) +

1

2F 11(S u, u)σ2

u

du,

(52)

which is a representation of the stochastic integral as a function

of integrals with respect to time.

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Multivariate Ito formula

(53) dS 1(t)dS 2(t)

=

a1(t)a2(t)

dt+

σ11(t) σ12(t)σ21(t) σ22(t)

dW 1(t)dW 2(t)

or

(54)

dS 1(t) = a1(t) dt + σ11(t) dW 1(t) + σ12(t) dW 2(t)

dS 2(t) = a2(t) dt + σ21(t) dW 1(t) + σ22(t) dW 2(t),

where it is assumed that Wiener processes

W 1(t) and W 2(t) are independent.

21

Suppose F (S 1(t), S 2(t), t) is a twice differen-

tiable real valued function.

Use of the Taylor expansion and taking limit

in the same manner as in the univariate case,

yields (with (dt)2 = 0, dt dS 1 = 0, and dt dS 2 =

0)

(55)

dF = F t dt + F s1dS 1 + F s2

dS 2

+

1

2

F s1,s1(dS 1)

2

+ F s2,s2(dS 2)

2

+ 2F s1,s2dS 1dS 2

.

The independence of W 1 and W 2 implies that

dW 1 dW 2 = 0 (otherwise if they were cor-

related with correlation ρ, then dW 1 dW 2 =

ρ dt).

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Then

(dS 1)2 = (σ211 + σ2

12) dt,(56)

(dS 2)2 = (σ221 + σ2

22) dt,(57)

and

dS 1 dS 2 = (σ11σ21 + σ12σ22)dt.(58)

Using these in the multivariate Ito gives a

formula as a function of dW 1 and dW 2.

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Example 5.6: Interest rate derivatives. Assume that

the yield curve depends on two state variables, shortrate rt, and long rate Rt. Denote the price of thederivative as F (rt, Rt, t). Assume

drt = a1(t) dt + σ11(t) dW 1(t) + σ12(t) dW 2(t),(59)

and

dRt = a2(t) dt + σ21(t) dW 1(t) + σ22(t)dW 2(t).(60)

Straightforward application of the Ito formula gives(61)

dF = F t dt + F r drt + F R dRt

+12

F rr(σ2

11 + σ212) + F RR(σ2

21 + σ222) + 2F rR(σ11σ21 + σ12σ22)

dt,

which indicates how the price of an interest rate deriva-

tive will change during a small interval dt.

Remark 5.4:

Cov(drt, dRt) = [σ11(t)σ21(t) + σ12(t)σ22(t)] dt.(62)

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Example 5.7: Total value of wealth

Y (t) =

ni=1

N i(t)P i(t),(63)

where N i(t) is units of the ith asset and P i(t) the price.Increment of wealth as time passes

dY (t) = ∂Y

∂t dt +

n

i=1

∂Y

∂N idN i(t) +

n

i=1

∂Y

∂P idP i(t)

+1

2

ni=1

∂ 2Y

∂N 2i(dN i(t))2 +

1

2

ni=1

∂ 2Y

∂P 2i(dP i(t))2

+

ni=1

∂ 2Y

∂N i∂P idN i(t) dP i(t)

=

ni=1

P i(t)dN i(t) +

ni=1

N i(t)dP i(t)

+

ni=1

dN i(t) dP i(t).

(64)

In the standard calculations the last term would not

be present.

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