1

Application of Moment Expansion Method to Options

Square Root ModelYun Zhou

Advisor: Dr. Heston

2

Approach

• Heston (1993) Square Root Model

s vt tdW dW dt

st t t t tdS S dt v S dW

( ) vt t t tdv k v dt v dW

vt is the instantaneous variance μ is the average rate of return of the asset. θ is the long variance, or long run average price varianceκ is the rate at which νt reverts to θ. is the vol of vol, or volatility of the volatility

3

Approach Continued• Heston closed form solution

based on two parts

1st part: present value of the spot asset before optimal exercise 2nd part: present value of the strike-price payment

P1 and P2 satisfy the backward equation, thus their characteristic function also satisfy the backward equation

• Measure the distance of these two solutions to determine the highest moment need to use

1 2( , , ) ( , )C s v t SP KP t T P ( )( , ) r T tP t T e

4

Approach Continued

• Moment Expansion

terminal condition• With the SDE of volatility, formulate the backward equation

• Solve the coefficients C

( , ,0, ) nM x v n x

12( ) s

tdx v dt vdW

ln ( )x S t

21 1 12 2 2( ) ( )xx xv vv x vvM vM vM v M k v M M

0 0

( , , , ) ( )n n i

n i jij

i j

M x v n C x v

5

Approach Continued

• To get coefficients C 1) Backward equations give a group of linear

ODEs: Initial conditions:

' 1, , 2, 1 1,2

21 1, 1 1, 12 2

( 2)( 1) ( ( 1) ( 1))

( ( 1) ( 1)) ( 1)i j i j i j i j

i j i j

C kjC i j C i j i C

j j k j C i C

0

0

( ,0) | 1( , ) | 0C nC i j

6

Approach Continued

2) Recall matrix exponential

has solution with initial value b

'( ) ( )y t A y t

( ) Aty t e b

7

Moments• 1-3rd Moments

• E(x^3)=6. mu3-9. mu2 v+k2 x (v (3. -1.5 x)+theta (-3.+1.5 x))+v (4.5 rho sigma v-0.75 v2+sigma2 (-3.-3. rho2+1.5 x))+k (v (v (9. -4.5 x)+theta (-9.+4.5 x))+rho sigma (theta (3. -3. x)+v (-6.+6. x)))+mu (v (-9. rho sigma+4.5 v)+k (theta (9. -9. x)+v (-9.+9. x)))

• Due to computation limit, only 3rd moments get in Mathematica, will use MATLAB in numerical way.

2 2 2( ) ( ) (2 2 ( ) ( ) 0.5 ( ) (1 ( ))( ( )))( )E x v t v t v t v t k x t v t T t

12( ) ( ) ( )( )E x x t v T t

8

Option Price from Moments

• European call option payoff, K is the strike price

• Corrado and Su (1996) used Skewness and Kurtosis to adjust Black-Scholes option price on a Gram-Charlier density expansion

3 4 33 4 23! 4!

20

( ) ( ) 1 ( 3 ) ( 6 3)

ln( / ) ( / 2)t

g z n z z z z z

S S r tzt

( , ( )) ( ( ) )C T S T S T K

n(z) is probability density function

9

Black-Scholes Solution with Adjustment

3 3 4 4

213 03!

2 3 3/ 214 04!

( 3)

((2 ) ( ) ( ))

(( 1 3 ( )) ( ) ( ))

GC BSC C Q Q

Q S t t d n d tN d

Q S t d t d t n d t N d

0 ( ) ( )rtBSC S N d Ke N d t

GCCBase on Gram-Charlier density expansion, the option price is denoted as (1)

Where is the Black-Scholes option Pricing formula, N(d) is cumulative distribution function

represent the marginal effect of nonnormal skewness and Kurtosis

3 4,Q Q

10

Implement the Moments

( ) ( , , , ) | 0nE x M x v n

33 ( ) ( , , ,3) | 0E x M x v

• Get moments from the Moment Function by

•After get (2)

(3)

•Implement (2) and (3) into (1), will get Option price based on Gram-Charlier Expansion

44 ( ) ( , , , 4) | 0E x M x v

11

Heston Closed Form Solution Test

70 80 90 100 110 120 130 140-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Asset Price

Opt

ion

pric

e di

ffere

nce

European Call Option Price difference between Heston and BS

rho=0.5rho=-0.5

Parameters

Mean reversion k=2Long run variance theta =0.01Initial variance v(0) = 0.01Correlation rho = +0.5/-0.5Volatility of Volatility = 0.1Option Maturity = 0.5 yearInterest rate mu = 0Strike price K = 100All scales in $

12

European Call Option Price

Same parameters on p11

70 80 90 100 110 120 130 1400

5

10

15

20

25

30

35

40

45

Asset Price

Opt

ion

pric

e European Call Option Price between Heston and BS

13

Test on Volatility of Volatility

70 80 90 100 110 120 130 140-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Asset Price

Opt

ion

pric

e di

ffere

nce

European Call Option Price difference between Heston and BS

sigma=0.1sigma=0.2 Parameters

Mean reversion k=2Long run variance theta =0.01Initial variance v(0) = 0.01Correlation rho = 0Option Maturity = 0.5 yearInterest rate mu = 0Strike price K = 100All scales in $

14

Test on Corrado’s Results

0(%) 100rt

rt

Ke SMoneynessKe

Parameters

Initial Stock Price = $100Volatility = 0.15Option Maturity = 0.25 yInterest rate = 0.04Strike Price = $(75~125)

Red line represent -Q3Blue line represent Q4In equation (1) (p9)

-40 -30 -20 -10 0 10 20-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Option Moneyness (%)

Pric

e A

djus

tmen

t

Price Adjustment by Kurtosis and Skewness

-SkewnessKurtosis

15

Next Semester

• Implement the moments into European Call Option Price (Corrado and Su)

• Continue work on solving ODEs for any order of n

• Test on Heston FFT solution• Compare moment solution with FFT

solution

16

References

• Corrado, C.J. and T. Su, 1996, Skewness and Kurtosis in S&P 500 Index Returns Implied by Option Prices, Journal of Financial Research 19, 175-92.

• Heston, 1993, A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, The Review of Financial Studies 6, 2,327-343