##### Document Text Contents

Page 1

FOURIER TRANSFORM METHODS FOR

OPTION PRICING: AN APPLICATION

TO EXTENDED HESTON-TYPE MODELS

Master Thesis in Quantitative Finance and

Banking

Gorka Koldo González Sáez

Universidad del Páıs Vasco

Academic advisors:

Federico Platańıa1, Manuel Moreno F.2

1Dept. of Quantitative Economics 2Dept. of Economic Analysis

Economic Faculty Social & Legal Sciences Faculty

Univ. Complutense of Madrid Univ. of Castilla-La Mancha

Spain Spain

Submitted: 10/07/2014

Page 2

2014 by G.K. Gonz�alez S�aez ([email protected])

Page 64

C:\Users\Kuvízam\Frecuentes\Tesina FFT\LAT_LaTeX\LMT_LaTeX Master Thesis\140623_Gorka-MT_V8.0\Images_II\TDHES-STR_Price_Errors.eps

44 3. The Models

70 80 90 100 110 120 130

15

20

25

30

35

40

45

Price

Strike Price

V

a

lu

e

DI FFT TR FRFT TR

(a) Adjustment

70 80 90 100 110 120 130

10

−8

10

−7

10

−6

10

−5

10

−4

10

−3

Price

Strike Price

R

e

la

tiv

e

E

rr

o

r

(%

)

FFT TR Error FRFT TR Error

(b) Errors

70 80 90 100 110 120 130

10

−3

10

−2

10

−1

10

0

CPU Times

Strike Price

C

P

U

T

im

e

(

s)

FFT TR Times FRFT TR Times

(c) CPU Times

Figure 3.5: Adjustments, errors and CPU times for Fourier Methods in the Mikhailov

and Nögel model

Figure (3.5b) shows that the mean pricing errors in both Fourier methods are

around 10−4%, a really small value. It can also be noted that these errors are

higher than in the previous ITM and ATM options. The mean CPU times are,

respectively, around 10−2s and 103s for the FFT and FRFT alternatives.

ITM

options

Table (3.13) shows the results for ITM options.

ITM

Method Price Error (%) Time (s)

Closed Form 41.5156 0.0000 0.0170

Monte Carlo 5000 paths 41.8804 0.8786 0.1190

Monte Carlo 10000 paths 40.6857 -1.9991 0.2130

Monte Carlo 50000 paths 41.1471 -0.8878 0.8120

Monte Carlo 100000 paths 41.5893 0.1774 1.8630

FFT Trapezoidal Rule 41.5156 -0.0000 0.1410

FFT Simpson’s Rule 41.5144 -0.0029 0.0950

FRFT Trapezoidal Rule 41.5156 -0.0000 0.0140

FRFT Simpson’s Rule 41.5156 -0.0000 0.0100

S0 = 100, K = 70:46, � 1 = 2, � 1 = 0:005, � 1 = 0:2, v01 = 0:04; � 1 = 0:6

� 2 = 1:5, � 2 = 0:006, � 2 = 0:25, v02 = 0:03; � 2 = −0:6

Table 3.13: ITM results for the Mikhailov and Nögel model.

We can highlight several interesting results. First, the FRFT offers a price

that is accurate up to the fourth decimal, whereas this algorithm is the fastest

method and is even faster than the integration of semi-closed solution via the

Gauss-Laguerre quadrature. The FFT algorithm is almost as accurate than

the FRFT but it much slower, up to ten times in some cases. Monte Carlo

Page 65

3.6. The Mikhailov and Nögel (2004) Model 45

simulations provide accurate results although far from the efficiency achieved

by the Fourier algorithms.

ATM

options

The next table summarizes the results for ATM options.

ATM

Method Price Error (%) Time (s)

Closed Form 27.3676 0.0000 0.0140

Monte Carlo 5000 paths 27.7069 1.2399 0.1150

Monte Carlo 10000 paths 27.3054 -0.2271 0.2080

Monte Carlo 50000 paths 27.1933 -0.6369 0.8950

Monte Carlo 100000 paths 27.3169 -0.1852 1.6290

FFT Trapezoidal Rule 27.3676 0.0000 0.0710

FFT Simpson’s Rule 27.3664 -0.0043 0.0860

FRFT Trapezoidal Rule 27.3676 0.0000 0.0060

FRFT Simpson’s Rule 27.3676 0.0000 0.0060

S0 = 100, � 1 = 2, � 1 = 0:005, � 1 = 0:2, v01 = 0:04; � 1 = 0:6

� 2 = 1:5, � 2 = 0:006, � 2 = 0:25, v02 = 0:03; � 2 = −0:6

Table 3.14: ATM results for the Mikhailov and Nögel model.

The results are similar to the previous ones: the FRFT is the fastest algorithm,

including the Gauss-Laguerre quadrature. Once again, the Simpson’s rule for

the FFT method seems to be less accurate than the other Fourier methods.

Page 127

Bibliography

[Bat96] David S Bates. Jumps and stochastic volatility: Exchange rate

processes implicit in deutsche mark options. Review of �nancial

studies, 9(1):69–107, 1996.

[CHJ09] Peter Christoffersen, Steven Heston, and Kris Jacobs. The shape and

term structure of the index option smirk: Why multifactor stochastic

volatility models work so well. Management Science, 55(12):1914–

1932, 2009.

[Cho04] Kyriakos Chourdakis. Option pricing using the fractional fft. Journal

of Computational Finance, 8(2):1–18, 2004.

[CIJR85] John C Cox, Jonathan E Ingersoll Jr, and Stephen A Ross. A theory

of the term structure of interest rates. Econometrica: Journal of the

Econometric Society, pages 385–407, 1985.

[CM99] Peter Carr and Dilip Madan. Option valuation using the fast fourier

transform. Journal of computational �nance , 2(4):61–73, 1999.

[CT65] James W Cooley and John W Tukey. An algorithm for the machine

calculation of complex fourier series. Math. comput, 19(90):297–301,

1965.

[DPS00] Darrell Duffie, Jun Pan, and Kenneth Singleton. Transform analysis

and asset pricing for affine jump-diffusions. Econometrica, 68(6):1343–

1376, 2000.

[FO08] Fang Fang and Cornelis W Oosterlee. A novel pricing method for

european options based on fourier-cosine series expansions. SIAM

Journal on Scienti�c Computing , 31(2):826–848, 2008.

[FR08] Gianluca Fusai and Andrea Roncoroni. Implementing models in

quantitative �nance: methods and cases, volume 1. Springer Berlin,

2008.

[GP51] J Gil-Pelaez. Note on the inversion theorem. Biometrika , 38(3-4):481–

482, 1951.

[Hes93] Steven L Heston. A closed-form solution for options with stochastic

volatility with applications to bond and currency options. Review of

�nancial studies , 6(2):327–343, 1993.

[KP92] Peter E Kloeden and Eckhard Platen. Numerical solution of stochastic

di�erential equations , volume 23. Springer, 1992.

Page 128

108 Bibliography

[Lin11] Xiong Lin. The Hilbert Transform and its Applications in Com-

putational finance. PhD thesis, University of Illinois at Urbana-

Champaign, 2011.

[LK06] Roger Lord and Christian Kahl. Optimal fourier inversion in semi-

analytical option pricing. Technical report, Tinbergen Institute

Discussion Paper, 2006.

[MN04] Sergei Mikhailov and Ulrich Nogel. Heston ś stochastic volatility

model: Implementation, calibration and some extensions. Wilmott

magazine, 2004.

[Ng05] ManWo Ng. Option Pricing via the FFT. PhD thesis, Master Thesis,

Applied Institute of Mathematics, TU Delft, 2005.

[PK12] Warrick Poklewski-Koziell. Stochastic volatility models: calibration,

pricing and hedging. PhD thesis, 2012.

[Rou13] Fabrice D Rouah. The Heston Model and Its Extensions in Matlab

and C#. John Wiley & Sons, 2013.

[Zhu09] Jianwei Zhu. Applications of Fourier Transform to Smile Modeling:

Theory and Implementation. Springer, 2009.

FOURIER TRANSFORM METHODS FOR

OPTION PRICING: AN APPLICATION

TO EXTENDED HESTON-TYPE MODELS

Master Thesis in Quantitative Finance and

Banking

Gorka Koldo González Sáez

Universidad del Páıs Vasco

Academic advisors:

Federico Platańıa1, Manuel Moreno F.2

1Dept. of Quantitative Economics 2Dept. of Economic Analysis

Economic Faculty Social & Legal Sciences Faculty

Univ. Complutense of Madrid Univ. of Castilla-La Mancha

Spain Spain

Submitted: 10/07/2014

Page 2

2014 by G.K. Gonz�alez S�aez ([email protected])

Page 64

C:\Users\Kuvízam\Frecuentes\Tesina FFT\LAT_LaTeX\LMT_LaTeX Master Thesis\140623_Gorka-MT_V8.0\Images_II\TDHES-STR_Price_Errors.eps

44 3. The Models

70 80 90 100 110 120 130

15

20

25

30

35

40

45

Price

Strike Price

V

a

lu

e

DI FFT TR FRFT TR

(a) Adjustment

70 80 90 100 110 120 130

10

−8

10

−7

10

−6

10

−5

10

−4

10

−3

Price

Strike Price

R

e

la

tiv

e

E

rr

o

r

(%

)

FFT TR Error FRFT TR Error

(b) Errors

70 80 90 100 110 120 130

10

−3

10

−2

10

−1

10

0

CPU Times

Strike Price

C

P

U

T

im

e

(

s)

FFT TR Times FRFT TR Times

(c) CPU Times

Figure 3.5: Adjustments, errors and CPU times for Fourier Methods in the Mikhailov

and Nögel model

Figure (3.5b) shows that the mean pricing errors in both Fourier methods are

around 10−4%, a really small value. It can also be noted that these errors are

higher than in the previous ITM and ATM options. The mean CPU times are,

respectively, around 10−2s and 103s for the FFT and FRFT alternatives.

ITM

options

Table (3.13) shows the results for ITM options.

ITM

Method Price Error (%) Time (s)

Closed Form 41.5156 0.0000 0.0170

Monte Carlo 5000 paths 41.8804 0.8786 0.1190

Monte Carlo 10000 paths 40.6857 -1.9991 0.2130

Monte Carlo 50000 paths 41.1471 -0.8878 0.8120

Monte Carlo 100000 paths 41.5893 0.1774 1.8630

FFT Trapezoidal Rule 41.5156 -0.0000 0.1410

FFT Simpson’s Rule 41.5144 -0.0029 0.0950

FRFT Trapezoidal Rule 41.5156 -0.0000 0.0140

FRFT Simpson’s Rule 41.5156 -0.0000 0.0100

S0 = 100, K = 70:46, � 1 = 2, � 1 = 0:005, � 1 = 0:2, v01 = 0:04; � 1 = 0:6

� 2 = 1:5, � 2 = 0:006, � 2 = 0:25, v02 = 0:03; � 2 = −0:6

Table 3.13: ITM results for the Mikhailov and Nögel model.

We can highlight several interesting results. First, the FRFT offers a price

that is accurate up to the fourth decimal, whereas this algorithm is the fastest

method and is even faster than the integration of semi-closed solution via the

Gauss-Laguerre quadrature. The FFT algorithm is almost as accurate than

the FRFT but it much slower, up to ten times in some cases. Monte Carlo

Page 65

3.6. The Mikhailov and Nögel (2004) Model 45

simulations provide accurate results although far from the efficiency achieved

by the Fourier algorithms.

ATM

options

The next table summarizes the results for ATM options.

ATM

Method Price Error (%) Time (s)

Closed Form 27.3676 0.0000 0.0140

Monte Carlo 5000 paths 27.7069 1.2399 0.1150

Monte Carlo 10000 paths 27.3054 -0.2271 0.2080

Monte Carlo 50000 paths 27.1933 -0.6369 0.8950

Monte Carlo 100000 paths 27.3169 -0.1852 1.6290

FFT Trapezoidal Rule 27.3676 0.0000 0.0710

FFT Simpson’s Rule 27.3664 -0.0043 0.0860

FRFT Trapezoidal Rule 27.3676 0.0000 0.0060

FRFT Simpson’s Rule 27.3676 0.0000 0.0060

S0 = 100, � 1 = 2, � 1 = 0:005, � 1 = 0:2, v01 = 0:04; � 1 = 0:6

� 2 = 1:5, � 2 = 0:006, � 2 = 0:25, v02 = 0:03; � 2 = −0:6

Table 3.14: ATM results for the Mikhailov and Nögel model.

The results are similar to the previous ones: the FRFT is the fastest algorithm,

including the Gauss-Laguerre quadrature. Once again, the Simpson’s rule for

the FFT method seems to be less accurate than the other Fourier methods.

Page 127

Bibliography

[Bat96] David S Bates. Jumps and stochastic volatility: Exchange rate

processes implicit in deutsche mark options. Review of �nancial

studies, 9(1):69–107, 1996.

[CHJ09] Peter Christoffersen, Steven Heston, and Kris Jacobs. The shape and

term structure of the index option smirk: Why multifactor stochastic

volatility models work so well. Management Science, 55(12):1914–

1932, 2009.

[Cho04] Kyriakos Chourdakis. Option pricing using the fractional fft. Journal

of Computational Finance, 8(2):1–18, 2004.

[CIJR85] John C Cox, Jonathan E Ingersoll Jr, and Stephen A Ross. A theory

of the term structure of interest rates. Econometrica: Journal of the

Econometric Society, pages 385–407, 1985.

[CM99] Peter Carr and Dilip Madan. Option valuation using the fast fourier

transform. Journal of computational �nance , 2(4):61–73, 1999.

[CT65] James W Cooley and John W Tukey. An algorithm for the machine

calculation of complex fourier series. Math. comput, 19(90):297–301,

1965.

[DPS00] Darrell Duffie, Jun Pan, and Kenneth Singleton. Transform analysis

and asset pricing for affine jump-diffusions. Econometrica, 68(6):1343–

1376, 2000.

[FO08] Fang Fang and Cornelis W Oosterlee. A novel pricing method for

european options based on fourier-cosine series expansions. SIAM

Journal on Scienti�c Computing , 31(2):826–848, 2008.

[FR08] Gianluca Fusai and Andrea Roncoroni. Implementing models in

quantitative �nance: methods and cases, volume 1. Springer Berlin,

2008.

[GP51] J Gil-Pelaez. Note on the inversion theorem. Biometrika , 38(3-4):481–

482, 1951.

[Hes93] Steven L Heston. A closed-form solution for options with stochastic

volatility with applications to bond and currency options. Review of

�nancial studies , 6(2):327–343, 1993.

[KP92] Peter E Kloeden and Eckhard Platen. Numerical solution of stochastic

di�erential equations , volume 23. Springer, 1992.

Page 128

108 Bibliography

[Lin11] Xiong Lin. The Hilbert Transform and its Applications in Com-

putational finance. PhD thesis, University of Illinois at Urbana-

Champaign, 2011.

[LK06] Roger Lord and Christian Kahl. Optimal fourier inversion in semi-

analytical option pricing. Technical report, Tinbergen Institute

Discussion Paper, 2006.

[MN04] Sergei Mikhailov and Ulrich Nogel. Heston ś stochastic volatility

model: Implementation, calibration and some extensions. Wilmott

magazine, 2004.

[Ng05] ManWo Ng. Option Pricing via the FFT. PhD thesis, Master Thesis,

Applied Institute of Mathematics, TU Delft, 2005.

[PK12] Warrick Poklewski-Koziell. Stochastic volatility models: calibration,

pricing and hedging. PhD thesis, 2012.

[Rou13] Fabrice D Rouah. The Heston Model and Its Extensions in Matlab

and C#. John Wiley & Sons, 2013.

[Zhu09] Jianwei Zhu. Applications of Fourier Transform to Smile Modeling:

Theory and Implementation. Springer, 2009.