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

Download Presentation

Application of Moment Expansion Method to Options Square Root Model

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

Application of Moment Expansion Method to Options Square Root Model

Yun Zhou

Advisor: Dr. Heston

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:

Approach Continued

2) Recall matrix exponential

has solution with initial value b

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