Bergische Universität Wuppertal
Angewandte Mathematik & Numerische Analysis
Prof. Dr. Matthias Ehrhardt
Prof. Dr. Michael Günther
PD Dr. Jörg Kienitz Seminar in Summer term 2019 :

Stochastic Volatility and Fourier Transformation
(2 SWS)

In this seminar we consider Fourier Transform Methods applied to Option Pricing problems. We consider a real valued random variable X and consider the expectation (1)

where i denotes the imaginary unit, i.e. i = . If the distribution X of X has a density with respect to Lebesgue measure, denoted by fX(x), this expectation can be written as In this seminar we consider the Heston model that is specified by the following system of Stochastic Differential equations:

 dS(t) = μ(t)dt + σ(t) dW(t), (2) dV (t) = κ(t)(θ(t) - V (t))dt + ν(t)dZ(t), (3) ⟨W(t),Z(t)⟩ = ρ(t)dt, S(0) = S0, V (0) = V 0,

where W and Z are correlated Brownian motions and the quantity ρ is called the correlation. Furthermore, κ the mean reversion, θ the long term mean, ν the volatility of variance. The spot asset value is S0 and the spot variance value is V 0.

Given a strike price K and a maturity T for option pricing problems we would be interested in quantities such as (S(T) K) or (S(T) K). This would enable us to assign a probability to the event that at time T is asset is above or below the strike price and, thus, a Call, resp. Put option is exercised or not.

The bad news here is that the probability distribution, nor the density, of S is not available in closed form and, thus, pricing an option using the above mentioned probabilities and using a Black-Scholes-Merton type formula is not possible. However, for a lot of models including the Heston model the characteristic function which is nothing but the Fourier transform of the probability density is available in closed form. Furthermore, it comes in handy that a mathematical theorem is available that relates the probability density/distribution to the characteristic function – called the Gil-Pelaez inversion. It holds that In this seminar we will understand the notion of the Fourier transform, study the basic properties and prove the Gil-Pelaez theorem. Equipped with the results we wish to start digging into the practical aspects of using Fourier transforms for option pricing. After understanding the different methods proposed in the literature we wish to implement the methods in Matlab/VBA and apply the methods for pricing and calibration. There is already code available, e.g. Kienitz, Wetterau (2012).

#### The following presentation titles are available:

1. Fourier Transforms; Theory
2. The Heston Model; Theory
3. The Characterstic Function of the Heston Model; Theory
4. The Carr-Madan Method and its implementation
5. The Lewis Method and its implementation
6. The COS Method and its implementation
7. The Joshi Control Method and its implementation
8. The Lord-Kahl Method I; Theoretical Results
9. The Lord-Kahl Method II; Implementing the Full Method for Heston

We aim to have all presentations on a single day. The presentations will be held at our industry partner locations. In this case it is Talanx Asset Management. Talanx is a Cologne based asset manager and is located near the station Cologne Deutz.

### General Literature and Code (Matlab/VBA):

We base the practical part on the following books, papers, code repositories.

• Cristomo, R., (2014) An Analysis of the Heston Stochastic Volatility Model: Implementation and Calibration using Matlab, SSRN
• Kienitz, J. and Wetterau, (2012) D., Financial Modelling, Wiley
• Kienitz, J. and Wetterau, D., (2012) Financial Modelling, Wiley and Mathworks (https://de.mathworks.com/academia/books/financial-modeling-kienitz.html)
• Rouah, F., (2015) The Heston Model and Its Extensions in VBA, Wiley
• Saez, G.K.G. (2014) Fourier Transform Methods for Option pricing: An application to extended Heston-type models - Masters Thesis
• Schmelzle, M., (2010) Option Pricing Formulae using Fourier Transform:
Theory and Application

It is determined at the beginning of the seminar how the practical part is structured an what is expected from the presenter.

Note: It is essential that you not only copy and paste the code but fulfill a specific task that is agreed on before you prepare your topic and your presentation.

### Literature for presentations:

We base the presentations on the following papers:

• Carr, P. P. and D. B. Madan (1999) Option valuation using the fast Fourier transform, Journal of Computational Finance 2(4), 61-73.
• Duffie, D., Pan, J., and Singleton K., (2000) Transform analysis and asset pricing for affine jump-diffusion, Econometrica 68, 1343-1376
• Fang, F. and C. Oosterlee (2008) A novel pricing method for European options based on Fourier-cosine series expansions, SIAM Journal on Scientific Computing 31(2), 826 84.
• Gil-Pelaez, J. (1951) Note on the inversion theorem, Biometrika 38(3-4), 481 482.
• Lewis, A. (2001) A Simple Option Formula for General Jump-Di7usion and other Exponential Lvy Processes, Envision Financial Systems and OptionCity.net, California. Available at http://optioncity.net/pubs/ExpLevy.pdf
• LeFloch, F. (2014) Fourier Integration and Stochastic Volatility Calibration, SSRN
• LeFloch, F. (2016) An adaptive Filon quadrature for stochastic volatility models, SSRN
• Joshi, M. and Yang, C., (2014) Fourier Transforms, option pricing and controls, SSRN
• Lord, R. and C. Kahl (2007) Optimal Fourier Inversion in Semi-Analytical Option Pricing, Discussion Paper TI No.2006-066/2, Tinbergen Institute. Available at SSRN: http://ssrn.com/abstract=921336
• Wang, C., (2017) Pricing European Options by Stable Fourier-Cosine Series Expansions, SSRN