Questions tagged [characteristic-function]
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21
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Heston model characteristic function
The characteristic function of $x=ln(S_T)$ in the framework of Heston model is guessed to be: $$f_j(\phi,x,v)=e^{C_j(\tau,\phi)+D_j(\tau,\phi)+i\phi x}$$
The call price is guessed to have the form: $$...
- 21
3
votes
0
answers
88
views
The little Heston Trap in DPS representation
I was wondering if the representation by Duffie, Pan, and Singleton (2000) is already accounting for the little Heston trap. DPS represent their 'general' discounted characteristic function as:
$$
\...
- 59
4
votes
2
answers
410
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Heston Riccati equation
Let
$$
\begin{align*}
dY_{t} &= \left(r - \frac{1}{2} V_{t}\right) dt + \sqrt{V_{t}}dW_{t}\\
dV_{t} &= \kappa(\theta - V_{t}) dt + \rho \sigma \sqrt{V_{t}}dW_{t} + \sigma\sqrt{1-\rho^{2}}\sqrt{...
- 107
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186
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Ito's lemma for option pricing with Levy-alpha stable drift
Consider
$$dS=\omega\left(\Lambda-S\right)dt+\sigma_S S dW_t,$$
such that such that $W_t$ is a Wiener process, $\sigma_S$ is constant, $\omega: t\rightarrow\mathbb{R}$ represents anticipated drift and ...
- 128
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0
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98
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Choice of grid for numerical integration
I have to compute an integral involving the characteristic function for pricing options in a model and it so happens that accurate approximation seems to be mostly about putting lots of points in ...
- 2,506
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1
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82
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Variation of the trading range
Example: The trading range (in points) for each of the last 5 trading days for asset A is: 5,21,2,15,32 and for asset B is: 5,6,5,5,5. Is there an indicator that ranks assets based on variation of ...
user59094
2
votes
2
answers
792
views
CIR process characteristic function
what is the characteristic function of the CIR process given by
$dv_t = \kappa (\theta - v_t)dt + \sigma \sqrt{v_t}dW_t$
Unfortunately, I could not find the answer in the literature.
I know it is in ...
- 31
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0
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265
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Computing expectation of conditional characteristic function of the Heston model and variance process $V_t$
I'm using the following Heston model:
\begin{align}
\text{d}X_t &= -\dfrac{1}{2} V_t \text{d}t +\sqrt{V_t} \text{d}B_t, \\
\text{d}V_t &= -\lambda(V_t-\kappa) \text{d}t + \sigma \sqrt{V_t} \...
- 11
4
votes
1
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383
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Ito calculus is Gaussian (using method of characteristic function)
Let $h$ be a deterministic function and define $X_{t}=\int_{0}^{t} h(s) d W_{s} .$ Show that
$$\mathbb{E} \exp \left(i u X_{t}\right)=\exp \left(-\frac{u^{2}}{2} \int_{0}^{t} h^{2}(s) d s\right),$$ ...
- 69
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Heston model with underlying BS dynamics always gives 1/2 of the right value, what am I doing wrong?
Just as an exercise I'm trying to follow this paper: https://arxiv.org/ftp/arxiv/papers/1502/1502.02963.pdf
In the section 2.2 it calculates the value of a Call using the characteristic function of ...
- 135
4
votes
1
answer
882
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Complex numbers in VBA
Hey I try to price options in VBA. To do this I need to define characteristic function and do some operations on complex numbers. For example I have this code:
...
- 433
2
votes
0
answers
205
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Characteristic function for heston model with jumps in price and variance
I need the characteristic function of the Heston model with jumps in price and variance, or in other words, the characteristic function of the Bates model (1996) adding jumps in the variance dynamics.
...
3
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0
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Characteristic function of time-changed Levy processes
Let $X_t$ be a Levy process, and $Y_t$ be a subordinator i.e. process with nondecreasing trajectories. I have to find characteristic function of $X_{Y_t}$. I know that I have to calculate:
$$E[e^{iuX_{...
- 43
11
votes
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485
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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 ...
- 2,143
3
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325
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Black-Scholes IV from Characteristic Function
I'm trying to follow Gatheral 2006 on his derivation of the BSIV from a characteristic function. The most relevant formula is (5.7) page 60.
$$\int_0^\infty\frac{du}{u^2+(1/4)}\Re[e^{-iuk}\left(\...
- 1,054
2
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1
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359
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Characteristic function of CGMY model
I have a basic question about the CGMY model which has characteristic function
$$
\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)
$$
whith $...
- 383
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1
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553
views
Options on realized volatility / variance
If I'd like to price options on variance/volatility in the Heston model. Is MC simulation and/or finite difference the only way to do it? Or is there an analytical expression for the probability ...
user34971
0
votes
1
answer
381
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Understanding FFT's complex number result on option pricing
I have been using the Carr-Madan method to price caplets using the FFT. I have followed every step closely and (i believe) successfully. I understand the procedure theoretically but I cannot interpret ...
2
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0
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Characteristic function of SDE with coefficients depending upon second coupled SDE
Say we have the following two SDEs driven by the same single Brownian:
$$ dx_t = -0.5\sigma^2g(\psi)^2dt + \sigma g(\psi)dW_t \quad\quad d\psi_t = -(H\psi_t+0.5\sigma^2)dt + \sigma dW_t$$
where $...
2
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1
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386
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Carr and Madan Fourier Transform
I am bit confused by Carr and Madan's paper. In it the authors write that the Fourier transform $ c_T(k)$ is defined by
\begin{align}
\psi_T(v) = \int_{ - \infty}^{\infty} e^{ivk} c_T(k)dk
\end{...
- 21
2
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1
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709
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Characteristic functions for options on futures
Using simple delta-probability decomposition, the price European call options a non- dividend paying asset can be computed as
\begin{equation}
C(T,K) = {S_0}{\rm{ }}{\Pi _1} - {e^{ - rT}}K{\rm{ }}{\...
- 1,471