Title About The Applications of Fourier Transform Methods to Option Pricing. English 12.5 MB 128
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Fourier Transform Methods for Option Pricing: An Application to Extended Heston-type Models
Contents
List of Figures
List of Tables
Acronyms
1 Overview of Fourier Transform in Finance
1.1 Introduction
1.2 The Fourier Transform
1.3 Gil-Peláez (1951) Inversion Theorem
1.4 Carr and Madan (1999) Formulation
1.4.1 The Fourier Transform of an Option Price
1.4.2 Fourier Transform of Out-of-the-Money Option Prices
2 Pricing Methods
2.1 Introduction
2.2 Direct Integration Method
2.3 Euler Monte Carlo Method
2.4 Fast Fourier Transform Method
2.5 Fractional Fast Fourier Transform Method
3 The Models
3.1 Introduction
3.2 The Heston (1993) Model
3.2.1 Characteristic Function
3.2.2 Numerical Results
3.3 The Bates (1996) Model
3.3.1 Characteristic Function
3.3.2 Numerical Results
3.4 The SVJJ (2000) Model
3.4.1 Characteristic Function
3.4.2 Numerical Results
3.5 The Double Heston (2009) Model
3.5.1 Characteristic Function
3.5.2 Numerical Results
3.6 The Mikhailov and Nögel (2004) Model
3.6.1 Characteristic Function
3.6.2 Numerical Results
4 Greeks and other Sensitivities
4.1 Introduction
4.2 The Heston (1993) Model
4.2.1 Greeks and other Sensitivities
4.2.2 Numerical Results
4.3 The Bates (1996) Model
4.3.1 Greeks and other Sensitivities
4.3.2 Numerical Results
4.4 The SVJJ (2000) Model
4.4.1 Greeks and other Sensitivities
4.4.2 Numerical Results
4.5 The Double Heston (2009) Model
4.5.1 Greeks and other Sensitivities
4.5.2 Numerical Results
4.6 The Mikhailov and Nögel (2004) Model
4.6.1 Numerical Results
5 Conclusions and Outlook
A Mean Errors and Times for the Heston Model
B Mean Errors and Times for the Bates Model
C Mean Errors and Times for the SVJJ Model
D Mean Errors and Times for the Double Heston Model
E Mean Errors and Times for the Mikhailov and Nögel Model
F Alternative Methodology for Greeks and other sensitivities
G Final Presentation
Bibliography
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##### 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

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

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

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