# Questions tagged [stochastic-volatility]

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### 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 ...
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### In what cases characteristic function of (log-)price process is known?

Hey I know that we can use characteristic function of log-price process to price different options. But when we know the characteristic function? I know that we can take Levy processes and constant ...
314 views

### Characteristic function of the Bates model

I have a misunderstanding concerning the derivation of the SVJ model : Firsty,I understand how to reach the final differential equation from : \begin{gather} dS_t = (r - q - \lambda t (e^{m-\frac{\nu}{...
106 views

### Is $C(K,S_t)$ a (local) martingale if PCS is broken?

When put-call symmetry holds $$P(S_t,K) = C(K,S_t) = \frac{K}{S_t} C \left( S_t, \frac{S_t^2}{K} \right)$$ where $P$ is the market price of a put option and $C$ is the market price of a call option. ... 484 views

### Bergomi Volatility Model

I was studying on the Bergomi volatility model(using forward variance represented as $\xi_{t}^{T}$).However I don't understand how the author passes from the sde to the first step by only integrating ...
305 views

### Non-constant Volatility of the Volatility in Stochastic Volatility Models

In pricing financial derivatives, we often first assume that the volatility of the stock price is constant. $$\mathrm{d}S(t) = \alpha S(t) \mathrm{d}t + \sigma S(t) \mathrm{d}W(t)\text{.}$$ The ...
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### Heston Nandi Garch Implementation Problem for Python

I have a coded my own Garch class in order to implement the Heston-Nandi Garch model. ...
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### Fractional Brownian Motion's Covariance Proof

Let's have the non independent Brownian motion such : $B_{H}(r)=\frac{1}{A(H)} \int_{R}\left[\left\{(r-s)_{+}\right\}^{H-1 / 2}-\left\{(-s)_{+}\right\}^{H-1 / 2}\right] \mathrm{d} B(s), \quad r \in R$ ...
I was looking at the paper of Raval and Jaquier The Log Moment Formula For Implied Volatility available here : https://arxiv.org/pdf/2101.08145.pdf On the page 4 they wrote(with $<logS>_T$ and \$&...