Transcript Slide 1

Application of Moment
Expansion Method to Options
Square Root Model
Yun Zhou
Advisor: Dr. Heston
1
Approach
• Heston (1993) Square Root Model
dSt   St dt  vt St dWt
s
dvt  k (  vt )dt   vt dWt v
dWt s dWt v   dt
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 variance
κ is the rate at which νt reverts to θ.
 is the vol of vol, or volatility of the volatility
2
Approach Continued
• Heston closed form solution
 based on two parts
C(s, v, t )  SP1  KP(t , T ) P2
P(t , T )  e r (T t )
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
3
Approach Continued
dx  (  12 v)dt 
x  ln S (t )
vdWt s
n
n i
• Moment Expansion M ( x, v, , n)   Cij n ( ) xi v j
i 0 j 0
terminal condition M ( x, v, 0, n)  xn
With the SDE of volatility, formulate the backward equation
•
1
2
•
vM xx   vM xv  12  2vMvv  (  12 v)M x  k (  v)Mv  M
Solve the coefficients C
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Approach Continued
• To get coefficients C
1) Backward equations give a group of linear
ODEs:
Ci', j  kjCi , j  12 (i  2)( j  1)Ci  2, j 1  (  (i  1) j   (i  1))Ci 1, j 
( 12  2 ( j  1) j  k ( j  1))Ci , j 1  12 (i  1)Ci 1, j 1
Initial conditions:
C (n,0) | 0  1
C (i, j ) | 0  0
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Approach Continued
2) Recall matrix exponential
y'(t )  A y(t )
has solution with initial value b
y (t )  e b
At
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Moments
• 1-3rd Moments
E( x)  x(t )  (  12 v)(T  t )
E( x2 )  v(t )  (2 2  2v(t )   v(t )  0.5v(t )2  k (1  x(t ))(  v(t )))(T  t )
• 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.
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Option Price from Moments
• European call option payoff, K is the strike
price
C(T , S (T ))  (S (T )  K )
• Corrado and Su (1996) used Skewness and
Kurtosis to adjust Black-Scholes option price
on a Gram-Charlier density expansion

g ( z )  n( z ) 1  3!3 ( z 3  3z )  44!3 ( z 4  6 z 2  3) 
ln( St / S0 )  (r   2 / 2)t
z
 t
n(z) is probability density function
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Black-Scholes Solution with
Adjustment
Base on Gram-Charlier density expansion, the option price is
denoted as C
GC
CGC  CBS  3Q3  ( 4  3)Q4
(1)
Q3  3!1 S0 t ((2 t  d )n(d )   2tN (d ))
Q4  4!1 S0 t ((d 2  1  3 t (d   t ))n(d )   3t 3/ 2 N (d ))
Where
CBS  S0 N (d )  Kert N (d   t ) is the Black-Scholes option
Pricing formula, N(d) is cumulative distribution function
Q3 , Q4
represent the marginal effect of nonnormal skewness and
Kurtosis
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Implement the Moments
• Get moments from the Moment Function by
E( xn )  M ( x, v, , n) |   0
•After get
3  E( x3 )  M ( x, v, ,3) |   0
4  E( x4 )  M ( x, v, , 4) |   0
(2)
(3)
•Implement (2) and (3) into (1), will get Option price based on
Gram-Charlier Expansion
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Heston Closed Form Solution Test
European Call Option Price difference between Heston and BS
0.15
rho=0.5
rho=-0.5
0.1
Parameters
Option price difference
0.05
Mean reversion k=2
Long run variance theta =0.01
Initial variance v(0) = 0.01
Correlation rho = +0.5/-0.5
Volatility of Volatility = 0.1
Option Maturity = 0.5 year
Interest rate mu = 0
Strike price K = 100
All scales in $
0
-0.05
-0.1
-0.15
-0.2
70
80
90
100
110
Asset Price
120
130
140
11
European Call Option Price
European Call Option Price between Heston and BS
45
40
35
Option price
30
25
20
15
10
5
0
70
80
90
100
110
Asset Price
120
Same parameters on p11
130
140
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Test on Volatility of Volatility
European Call Option Price difference between Heston and BS
0.06
sigma=0.1
sigma=0.2
0.04
Parameters
Option price difference
0.02
Mean reversion k=2
Long run variance theta =0.01
Initial variance v(0) = 0.01
Correlation rho = 0
Option Maturity = 0.5 year
Interest rate mu = 0
Strike price K = 100
All scales in $
0
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
70
80
90
100
110
Asset Price
120
130
140
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Test on Corrado’s Results
Price Adjustment by Kurtosis and Skewness
0.3
Parameters
-Skewness
Kurtosis
0.2
Initial Stock Price = $100
Volatility = 0.15
Option Maturity = 0.25 y
Interest rate = 0.04
Strike Price = $(75~125)
Price Adjustment
0.1
0
-0.1
-0.2
Red line represent -Q3
Blue line represent Q4
In equation (1) (p9)
-0.3
-0.4
-40
-30
-20
-10
0
Option Moneyness (%)
10
20
Ke rt  S0
Moneyness(%) 
100
 rt
Ke
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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
15
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
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