Questions tagged [fourier-transform]

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19
votes
4answers
4k views

From Fourier Transforms to Option Values

I am trying to understand how Fourier transforms & Characteristics functions can be used to calculate option values. However, I am having difficulty following the process that is used in several ...
11
votes
1answer
261 views

From VG and NIG processes to GBM

I would like to find out if it is possible to reduce: the Madan-Seneta Variance Gamma (VG) model; the Barndorff-Nielsen Normal Inverse Gaussian (NIG) model to the standard Black-Scholes through a ...
4
votes
1answer
185 views

Most accurate Fourier transform method for extreme OTM options

I need to calculate vanilla options prices for extreme moneyness range of e.g. (0%,1000%) under the Heston model for various parameter values that satisfy Feller. Which Fourier method (or other method)...
4
votes
1answer
177 views

Heston ITM and OTM options pricing

In the Carr and Madan (1999) methodology exploiting the fast Fourier transform, the quasi-analytical price of a call is given by: $$C(t,T,K)=e^{-r(T-t)}\frac{e^{-\alpha \log (K)}}{\pi}Re\left[\int_0^\...
4
votes
2answers
128 views

Why do we need approximation in option pricing?

We know that we can get a closed form for European option price. And we can calculate directly the normal distribution accumulation. But I saw that people use many approximation methods such as ...
4
votes
0answers
62 views

Change of the stock price dynamics while pricing using the Fourier transform techniques

Right now I am trying to understand how we can use the Fourier theorem in obtaining the formula for option pricing (from Zhu J., "Modular pricing of options"). While modeling the interest ...
4
votes
0answers
155 views

Zero-rebate barrier option pricing under the Heston model

I'm trying to derive an approximation for the zero-rebate barrier option under the Heston model: $$dS_t=\mu S_tdt+\sqrt{v_t}S_tdW^S_t$$ $$dv_t=\kappa(\bar{v}-v_t)dt+\eta\sqrt{v_t}dW^v_t,\quad d\langle ...
4
votes
0answers
70 views

How can we compute copula functions by using Fast Fourier transformation?

Q1. If a copula is expressed in terms of its moment generating function then how can this copula can be computed by using Fast Fourier Transformation? Q2. Can we use copula to evaluate spread option ...
3
votes
2answers
416 views

How do you derive this Carr-Madan-like equation?

How do you derive equation (3) below? The equation is tagged as equation (11) in this paper: http://janroman.dhis.org/finance/IR/Heston%E2%80%93Hull%E2%80%93White%20Model%20Part%20I.pdf There are ...
3
votes
1answer
127 views

When do Fourier inversion methods run into problems?

So in my courses, we always priced options either with Monte Carlo methods, or some sort of PDE discretization. Then I looked up Fourier inversion methods on my own that rely on the characteristic ...
3
votes
1answer
221 views

To what extent are Lévy processes used in financial engineering?

I know that (time changed) Lévy processes are actively researched in the academic world, including tools such as minimal entropy martingale pricing measures and fast Fourier transforms. To what extent ...
3
votes
0answers
719 views

Stock market cycles with Fourier Transform - amplitude vs. phase

There is this Wikipedia article on cycles in stock market data, which describes a 5-step process of finding dominant cycles in price data where step 2 reads: Step 2: Subsequently, a cycles engine ...
2
votes
1answer
258 views

How Fourier Transform creates the filters?

I would like to know how the fourier transform creates filters to extract the constituent signals? I did learn from a book it extracts the spike information and then analyze and combine those ...
2
votes
2answers
126 views

What's some good literature for Fourier transform methods?

I am looking for literature on Fourier methods in Quantitative Finance. I've been googling and found the book "Fourier Transform Methods in Finance" (Wiley), but the book seems poorly reviewed. ...
2
votes
1answer
309 views

Fourier transform for stock price forecasting

I am trying to forecast stock prices using Fast Fourier Transform, and plot historical, "future" (i.e. real) and forecast prices on the same chart to visually compare the accuracy of the ...
2
votes
1answer
311 views

How much math is needed to understand Fourier Transform methods for option pricing?

I know of FDM and MC methods for option pricing, but have little to no experience with Fourier Methods. I am intending to dive into the literature on using Fourier methods for option pricing, ...
2
votes
1answer
233 views

Characteristic function of CGMY model

I have a basic question about the CGMY model which has characteristic function $$ \Gamma(-Y_p)\left((M-iu)^{Y_p}-M^{Y_p}\right)+\frac{C_n}{C_p}\Gamma(-Y_n)\left((G+iu)^{Y_n}-G^{Y_n}\right) $$ whith $...
1
vote
1answer
82 views

Transform of payoff function $w_c=(\sqrt{y}-K)^+$ [closed]

I am working on a project where I price EU call options written on the VIX index. The payoff function of interest looks like $w_c=(\sqrt{y}-K)^+$ where K is the strike price and y is the value of $...
1
vote
1answer
355 views

Fourier transform of a European put

In book The concepts and practice of mathematical finance, in the context of illustrating the stochastic volatility model, the Fourier transform $\hat{P}(\xi, V, T)$ of a European put $P(x, V, T)$ is ...
1
vote
1answer
233 views

How to find characteristic function in Fourier Cosine method (COS method) by Fang and Oosterlee

Fang and Oosterlee (2009) introduced Fourier-Cosine method (COS method) in their paper. The formula to price an option is approximately $$e^{-r\Delta t} \sum_{k=0}^{N-1}' Re\left\{ \phi\left( \frac{k\...
1
vote
1answer
284 views

Python Numpy FFT array size limit?

I am trying to find the price of an Option based on the fft technique within the binomial model and it works fine until N>40000 where I start getting negative values and weird convergene and I am not ...
1
vote
1answer
202 views

Option pricing using discrete fourier transform (python)

I am trying to implement the pricing formula for a European (call) option given in Ales Cerny's paper "Introduction to Fast Fourier Transform in Finance" (paper can be found here), as ...
1
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0answers
40 views

Fourier transform of price function

If the expiry value is given by $f(x,T) = e^{-c x}$ for $x \ge a$ and 0 otherwise and c is a +ve constant, prove that in the Fourier domain: $$ (c + j \omega) F(\omega, 0) = e^{-rT} e^{-a(c+j\omega)}...
1
vote
0answers
64 views

Fourier transform Carr-Madan method on an arbitrar initial $S_0$ values

As mentioned in Carr-Madan's paper, here, the European call option is: $$ C_T(k)=\frac{e^{\alpha k}}{\pi}\int_0^\infty\mathcal{Re}\left(e^{-iuk}\psi(u)\right)du $$ where $$ \psi(u)=e^{-rT}\frac{\phi_T(...
0
votes
1answer
55 views

Heston model with underlying BS dynamics always gives 1/2 of the right value, what am I doing wrong?

Just as an exercise I'm trying to follow this paper: https://arxiv.org/ftp/arxiv/papers/1502/1502.02963.pdf In the section 2.2 it calculates the value of a Call using the characteristic function of ...
0
votes
0answers
50 views

Cyclic analysis for trading signal generation

I would like to build trading signals using cyclic analysis in order to obtain a forecast afterwards. I had a look in literature and Hurst analysis, Fourier, etc, are used However, I am struggling to ...
0
votes
0answers
79 views

Pricing options by IFT under the Heston and Nandi (2000) model: odd behavior

I am working on option pricing using GARCH models and, currently, I am coding the pricing of options under the Heston and Nandi (2000) model. This model admits a quasi analytical formula for pricing ...
0
votes
0answers
405 views

Beginner FFT (Fourier) transforms on closing prices for Apple

I don't know math very well, but I have been programming for many years. I would like to use FFT as a parameter to a ML model. The FFT is diving down sharply. I tried many stocks and its the same. ...
0
votes
1answer
272 views

Understanding FFT's complex number result on option pricing

I have been using the Carr-Madan method to price caplets using the FFT. I have followed every step closely and (i believe) successfully. I understand the procedure theoretically but I cannot interpret ...