# Tag Info

## Hot answers tagged fourier-transform

14

Intuition Yes it is possible. Both, the NIG and the VG process are exponential Lévy processes, i.e. they model the stock price via $S_t=S_0e^{X_t}$, where $X_t$ is a Lévy process. Here's a recent answer to the topic. Your question boils down to the following: if $X_t$ is a general Lévy process (VG, NIG, etc.), can we find parameters of $X_t$ such that $X_t$ ...

7

I'll give it a start and stick with Fourier methods. The approaches from Carr and Madan (1999) and Fang and Oosterlee (2009) are indeed known to be inaccurate for highly OTM options. I'd suggest to try out one the following three alternatives. The first one seems to be the most relevant one. Saddlepoint Method I begin to cite Hirsa (2013): The saddlepoint ...

5

The fourier transform is $$\hat{w}_c= \int_{-\infty}^\infty (\sqrt{y}-K)^+ e^{-i\phi y}dy = \int_{K^2}^\infty (\sqrt{y}-K) e^{-i\phi y}dy$$ Now do a change of variable with $t=\sqrt{i\phi y}$ and solve the resulting integral.

3

A filter is a mathematical operator that serves to convert an original time-series into another time-series or function. The purpose is to remove some particular features (e.g. trends, business cycle, seasonalities and noise) that are associated with specific frequency components. To get the basic idea think of a moving average which is nothing but a filter ...

3

There are a number of tricks. My favourite is to use the Black--Scholes price as a control. The integrals become much better behaved. You compute the difference of the Heston price from the BS price which is the same for calls and puts so there are no ITM vs OTM issues. In my paper with Chao Yang, we investigate the problem of what implied vol to use in ...

3

You do not need to know too much Fourier theory. Of course it helps, but it is not necessary. There are many different applications of Fourier methods e.g. Carr Madan (1999): you damp the option price as a function of the log-strike price and compute the fourier transform of the entire option price Bakshi and Madan (2000), Duffie, Pan and Singleton (2000): ...

3

Im going to hazard a guess that your problem is u**(N-i). Large exponents are notoriously poor performers, I would first look to restructure that aspect of the code and then isolate other poorly performant sections afterwards. For example you might observe that: S_T[i] = S0 * u**(N-i) * d**(i) is equivalent to: S_T[i] = S0 * u**N * (d/u)**i then u**N ...

3

For Fourier methods, you always need the characteristic function of the log-asset price $\ln(S_t)$. In the Black-Scholes model, $\ln(S_t)\sim N\left(\ln(S_0)+\left(r-\delta-\frac{1}{2}\sigma^2\right)t,\sigma^2t\right)$. It is well-known that the characteristic function of $X\sim N(m,s^2)$ is given by $$\phi_X(u)=\exp\left(imu-\frac{1}{2}s^2u^2\right).$$ You ...

2

Pricing path dependent options (Asians, lookbacks, barriers, Americans) is much harder with Fourier. MC simulations are easier in these cases. Recall the characteristic function only contains information about the terminal stock price $S_T$. For European style options, Fourier methods are extremely popular; in particular because they apply to a very wide ...

2

No need to buy a book for a first introduction of applications to finance. Here is a good review to start with. After reading the review you can then move on to more specialized texts. https://pfadintegral.com/docs/Schmelzle2010%20Fourier%20Pricing.pdf

1

Y in the CGMY model is not defined for negative integer values due to divergence of the gamma function at those values, and implicitly the characteristic function. However, in the case of negative non-integer $x$ we extend the gamma function in the sense that whenever $x \in (-\infty,0) \setminus \mathbb{Z_{-}}$, we define the value of $\Gamma(x)$ via the ...

1

Levy processes are not used for pricing derivatives and are useless in practice. When the task at hand is to price a derivative, i.e., working in the risk neutral measure, then using Levy processes is worse than useless, it is dangerous and should actively be avoided. You can add entropy risk measures, FFTs and other (practically) useless concepts from ...

1

We can only get closed-form solutions under certain assumptions about the market dynamics, e.g. in the Black-Scholes framework (share prices follows a GBM), the European option can be valued with the well-known Black-Scholes formula. For other assumptions where no closed-form solution exists or is known (e.g. share price is a Levy process), FFT methods are ...

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