# Stochastic Volatility Models in Option Pricing

## Full text

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### Michael Wennermo

Department of Mathematics and Physics

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### Comprising: 30 hp

Department of Mathematics and Physics

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NTRODUCTION

PTION

RICING

SER

S GUIDE

ATA

ENERATION

OMPARISON OF

UTPUT

ONCLUSIONS

ART

ILE

ART

LACK

CHOLES

ART

ORMAL

ISTRIBUTION

ART

OWER

ERIES

ART

ODEL

ART

ODEL

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( ) 2 1

t

rTt

x

1

2

t

2 2 1

2

1 2

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2, …

1

∞ →

n n

1 2

### X

2, …, with finite mean

2

1 2

n n

n

n→∞

### family of IID random variables X

1, X2, … and a real-valued function h(x), the function h(X) is

1

2

1

n n

∞ →

1

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∞ ∞ −

1

n n

1

2

1

2

2

1

n

n

2 2

α

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### X

2 generated by Monte Carlo simulation. The estimate is equal to (X1+ X2)/2 and the variance

1

2

1

2

1

2

1

2

X

Y

Y

2

2

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2

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2

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2

2

− −

### )

T t t T T T t T r t t

2

( )

2

2

t

T

t

t2

T

T

2

r(T t)

t t T

T

T

2

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## ]

t

t2

r(Tt)

T

T

T

t2

T

0 2

t

t2

t2

2

### =

4 2 4 3 2 1 1 2 2

k

### ]

5 2 2 2 1 2 1 2 1 1

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3 3 2 3 6

k k

2

3

*

*

0

2 2 2

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2 2 2

2 2 2

2

T t T t T t

2

2

T

t

### −

) W ξ(W t) )(T ξ V(t) V(T) T t

### =

+ − − + − 2 2 1 ln ln ) W ξ(W t) )(T ξ (µ T t

− − + − 2 2 1

T

t

### and letting z be a standard normal variable we get:

z t ξ t ) ξ (µ i i

− ∆ + ∆ 2 2 1 1

i

### we arrive at the formula for the volatility:

        ∆ + ∆ − −

### =

t ξ v t ) ξ (µ i i i

1 2 1 2 1

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i

i

0

1

2

i

i

1

2

0 1 1

t

0 2

2 2 2

t t 2 1 2 1 1

− −

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1 1 1

t

t

2 2 2

2 2 2 2

2 2

2 2

1 1 1

t

t

1

2

1

2

t1

− − 2 1 1 2 1 2

t t

1

2

2 1 2 2

2 1 1

t

t

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T t T t T t T t

2

2 1 2

T t T t t t T t

1

1 2

2 2 1T 1t 2T 2t

1 1 2T 2t t t

### −

) ( ) ( 1 ) )( 2 1 ( ) ( ln ) ( ln 2 2 1 1 2 2 t T t T Z Z Z Z t T t X T X

### =

+ µ− ξ − + −ρ ξ − +ρξ − ) ( ) )( 2 1 ( ) ( ln ) ( ln 1 1 2 2 t T t t T t V Z Z V r t Y T Y

### =

+ − − − + − − ) ( ) ( 1 ) )( 2 1 ( 2 2 1 1 2 2

### (

t T t T Z Z Z Z t T

### ×

µ− ξ − + −ρ ξ − +ρξ − ) ( ) )( 2 ( 1 2 2

### (

t T t t T t V Z Z V r

− − + − −

### Rewriting we arrive at the formulas for stock price and volatility evolution:

t V u t V r i i i i i

− ∆ + − ∆ −

1 1 ) 2 ( 1

### (Eq. 2.10 – SSV Model)

t v t u t i i i i

### =

− ∆ + ∆ + − 2 ∆ 2 2 2 2) 1 2 1 ( 1 ξ ρ ξ ρ ξ µ

0

0

i

i

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n

1

) (

n t T r

− −

) ( n t T r

− −

2

i

i

3

i

i

i

4

i

i

1

2

i

i

0

1 3 1

2 4 2

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1

1

1

2

2

2

3

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### The stochastic volatility found using the SV model:

        ∆ + ∆ − −

### =

t ξ v t ) ξ (µ i i i

1 2 1 2 1

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1*

1

1*

1*

1

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1*

1

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1

1

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## References

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