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 1 st part: present value of the spot asset before optimal exercise 2 nd 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 1-3 rd Moments E(x^3)=6. mu3-9. mu2 v+k2 x (v ( x)+theta ( x))+v (4.5 rho sigma v-0.75 v2+sigma2 ( rho2+1.5 x))+k (v (v ( x)+theta ( x))+rho sigma (theta ( x)+v ( x)))+mu (v (-9. rho sigma+4.5 v)+k (theta ( x)+v ( x))) Due to computation limit, only 3 rd 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, 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,