Application of Moment Expansion Method to Options Square Root Model

1. Application of Moment Expansion Method to Options Square Root Model Yun Zhou Advisor: Dr. Heston

2. 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

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

4. Approach Continued • Moment Expansion terminal condition • With the SDE of volatility, formulate the backward equation • Solve the coefficients C

5. Approach Continued • To get coefficients C 1) Backward equations give a group of linear ODEs: Initial conditions:

6. Approach Continued 2) Recall matrix exponential has solution with initial value 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.

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 n(z) is probability density function

9. 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

10. 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

11. 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 $

12. European Call Option Price Same parameters on p11

13. 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 $

14. 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)

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