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Application of Moment Expansion Method to Options Square Root Model

This paper explores the application of the Moment Expansion Method to the Heston Square Root Model to price European call options. Utilizing Heston's closed-form solution, the study investigates the impact of different volatility parameters and integrates skewness and kurtosis adjustments for improved pricing accuracy. We analyze the backward equations derived from stochastic differential equations and solve for coefficients to develop efficient pricing models. The results are compared with traditional methods to highlight enhancements in option valuation, providing valuable insights for financial modeling and derivative trading strategies.

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Application of Moment Expansion Method to Options Square Root Model

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  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

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