5. differentiation integration

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Agenda

1

• Differentiation

• Taylor's approximation

• Application of Derivatives in Finance

• Partial Differentiation

• Integral calculus

• Optimization

• Modified Duration of Bonds


Agenda

2

• Differentiation

• Taylor's approximation

• Application of Derivatives in Finance

• Partial Differentiation

• Integral calculus

• Optimization

• Modified Duration of Bonds


Differential and Integral calculus

• Differentiation

– Measure of how a function changes, with respect to its input

– Applications

• Calculation of marginal utility

• Measuring the rate of change of the price of a bond as a result of a change in the yield

on that bond

• In structuring of a portfolio of risky assets in order to maximize the portfolio return for a

given level of portfolio risk. i.e. optimization

• Integration

– Known as summation/ anti-derivative, finds area under the curve or surface

– Applications

• Finding the expected value of a European option at expiry

• Finding the probability in a given range of outcomes by finding the area under a

probability density function.

3


Differentiation

• First derivative

– Measures how fast the dependent variable is changing with respect to independent

variable

– Denoted by or f'(x)

– For a line Y=mX + c, m represents the slope of the line :change of vertical coordinate

with respect to horizontal coordinate and is given by dx

dy

dx

dy

4


Differentiation (Cont…)

Let a, b, c be the constants then

• Example: Differentiate

• Example : Differentiate

0)(

dx

cd

adx

baxd

)(

1)( nn

nxdx

xd

45

5)(

xdx

xd

)5

6(5

1

5

1)(

x

dx

xd

5x

51

x

5



• Product of two function

– If y = u.v, then

• Examples: Differentiate the following

a)

Using summation rule of differentiation we get,

b)

232

1583)5()4()3(

xxdx

xd

dx

xd

dx

xd

dx

dy

32 543 xxx

)24( xv

xdx

dv43

dx

du

xu 3

dx

dvu

dx

duv

dx

dy

6243)24()4(3 xxxdx

dy

6



• Rules

• Example

• Chain rule

– If y = f(u) and u = f(x), then

• Differentiate

556

)43(18)3()43(6)43(

xxdx

xd

dx

du

du

dy

dx

dy*

4uy

34udu

dy 3

dx

du

))('())(())(( 1 xfxfc

dx

xfd cc

4)43( x

43 xu

33 )43(1212 xudx

dy

7



• Rules

• Example :Differentiate

xx

edx

de

xdx

xd 1ln

)3ln( 25 xe x

xe

dx

dyhence

x

xfxdx

dv

edx

du

xveu

x

x

x

25,

2

)3('3

1

5

)3ln(,

5

2

2

5

25

8


Higher order derivatives

• Second order derivative

– Derivative of a derivative

– Rate of change of slope

– Denoted by

• Third order derivative =

• Further higher order derivatives are also calculated in the similar way

• Example: Find 2nd order and 3rd order derivatives of

dx

dx

dyd )(

2

2

)("dx

ydorxf

)(2

2

dx

yd

dx

d

3)43( x

xdx

yd

xxfxdx

yd

xdx

dy

xy

162

)43(54)43('*)43(18

)43(9

)43(

3

3

2

2

2

3

9

5. differentiation integration

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