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Questions tagged [characteristic-function]

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0answers
46 views

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(\...
2
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1answer
48 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
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1answer
87 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 ...
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0answers
30 views

Deriving the set where the moment generating function is well-defined from the structure of the assets

So lets assume I have the following structure \begin{align} dS_1 = & \ rS_1 dt + S_1 \left( \sqrt{V_1} dW_{11} + \sigma _{1m} \sqrt{V_m} dW_{12} \right) \nonumber \\ dS_2 = & \ rS_2 dt + S_2 \...
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0answers
135 views

Heston 1997 Characteristic Function

Can anybody provide me with the characteristic function of the log price for the 3/2 model of Heston (1997) ?
0
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1answer
202 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 ...
1
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0answers
126 views

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 $...
3
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1answer
118 views

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{...
2
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1answer
136 views

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{ }}{\...