UMD AMSC 663 - Application of Moment Expansion Method to Options Square Root Model

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Application of Moment Expansion Method to Options Square Root ModelApproachApproach ContinuedSlide 4Slide 5Slide 6MomentsOption Price from MomentsBlack-Scholes Solution with AdjustmentImplement the MomentsHeston Closed Form Solution TestEuropean Call Option PriceTest on Volatility of VolatilityTest on Corrado’s ResultsNext SemesterReferences1Application of Moment Expansion Method to Options Square Root ModelYun ZhouAdvisor: Dr. Heston2Approach•Heston (1993) Square Root Models vt tdW dW dtr=st t t t tdS S dt v S dWm= +( )vt t t tdv k v dt v dWq s= - +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 varianceκ is the rate at which νt reverts to θ. is the vol of vol, or volatility of the volatilitys3Approach 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 use1 2( , , ) ( , )C s v t SP KP t T P= -( )( , )r T tP t T e- -=4Approach Continued•Moment Expansion terminal condition• With the SDE of volatility, formulate the backward equation• Solve the coefficients C ( , ,0, )nM x v n x=12( )stdx v dt vdWm= - +ln ( )x S t=21 1 12 2 2( ) ( )xx xv vv x vvM vM vM v M k v M Mtr x x m q+ + + - + - =0 0( , , , ) ( )n n in i jiji jM x v n C x vt t-= ==��5Approach Continued•To get coefficients C 1) Backward equations give a group of linear ODEs: Initial conditions:'1, , 2, 1 1,221 1, 1 1, 12 2( 2)( 1) ( ( 1) ( 1))( ( 1) ( 1)) ( 1)i j i j i j i ji j i jC kjC i j C i j i Cj j k j C i Cr s ms q+ - ++ + -= + + + + + + + ++ + + - +00( ,0) | 1( , ) | 0C nC i jtt====6Approach Continued 2) Recall matrix exponential has solution with initial value b '( ) ( )y t A y t=( )Aty t e b=7Moments•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.2 2 2( ) ( ) (2 2 ( ) ( ) 0.5 ( ) (1 ( ))( ( )))( )E x v t v t v t v t k x t v t T tm m r s q= + - - + + - - -12( ) ( ) ( )( )E x x t v T tm= + - -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 3433 4 23! 4!20( ) ( ) 1 ( 3 ) ( 6 3)ln( / ) ( / 2)tg z n z z z z zS S r tztmmss-� �= + - + - +� �- -=( , ( )) ( ( ) )C T S T S T K+= -n(z) is probability density function9Black-Scholes Solution with Adjustment3 3 4 4213 03!2 3 3/ 214 04!( 3)((2 ) ( ) ( ))(( 1 3 ( )) ( ) ( ))GC BSC C Q QQ S t t d n d tN dQ S t d t d t n d t N dm ms s ss s s s= + + -= - -= - - - +0( ) ( )rtBSC S N d Ke N d ts-= - -GCCBase 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 Kurtosis3 4,Q Q10Implement the Moments( ) ( , , , ) | 0nE x M x v nt t= =33( ) ( , , ,3) | 0E x M x vm t t= = =• 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 Expansion44( ) ( , , , 4) | 0E x M x vm t t= = =11Heston Closed Form Solution Test70 80 90 100 110 120 130 140-0.2-0.15-0.1-0.0500.050.10.15Asset PriceOption price diferenceEuropean Call Option Price diference between Heston and BS rho=0.5rho=-0.5ParametersMean reversion k=2Long run variance theta =0.01Initial variance v(0) = 0.01Correlation rho = +0.5/-0.5Volatility of Volatility = 0.1Option Maturity = 0.5 yearInterest rate mu = 0Strike price K = 100All scales in $12European Call Option PriceSame parameters on p1170 80 90 100 110 120 130 140051015202530354045Asset PriceOption price European Call Option Price between Heston and BS13Test on Volatility of Volatility70 80 90 100 110 120 130 140-0.12-0.1-0.08-0.06-0.04-0.0200.020.040.06Asset PriceOption price diferenceEuropean Call Option Price diference between Heston and BS sigma=0.1sigma=0.2ParametersMean reversion k=2Long run variance theta =0.01Initial variance v(0) = 0.01Correlation rho = 0Option Maturity = 0.5 yearInterest rate mu = 0Strike price K = 100All scales in $14Test on Corrado’s Results0(%) 100rtrtKe SMoneynessKe---= �ParametersInitial Stock Price = $100Volatility = 0.15Option Maturity = 0.25 yInterest rate = 0.04Strike Price = $(75~125) Red line represent -Q3Blue line represent Q4In equation (1) (p9)-40 -30 -20 -10 0 10 20-0.4-0.3-0.2-0.100.10.20.3Option Moneyness (%)Price AdjustmentPrice Adjustment by Kurtosis and Skewness -SkewnessKurtosis15Next 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 solution16References•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,


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