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Application of Moment Expansion Method to Options Square Root Model

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Approach

- Heston (1993) Square Root Model

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

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

Approach Continued

- Moment Expansion

terminal condition

- With the SDE of volatility, formulate the backward equation
- Solve the coefficients C

Approach Continued

- To get coefficients C

1) Backward equations give a group of linear ODEs:

Initial conditions:

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.

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

n(z) is probability density function

Black-Scholes Solution with Adjustment

Base 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

Implement the Moments

- 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

Heston Closed Form Solution Test

Parameters

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 $

European Call Option Price

Same parameters on p11

Test on Volatility of Volatility

Parameters

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 $

Test on Corrado’s Results

Parameters

Initial Stock Price = $100

Volatility = 0.15

Option Maturity = 0.25 y

Interest rate = 0.04

Strike Price = $(75~125)

Red line represent -Q3

Blue line represent Q4

In equation (1) (p9)

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

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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