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Hot answers tagged stochastic-volatility

3

I write down the solution for the Heston model. You can directly generalise the result. Let $f=f(t,s,v)\in C^{1,2,2}(\mathbb{R}_+^3)$ be a real-valued function (portfolio value) and consider the two-dimensional stochastic process $(S_t,v_t)$ with \begin{align*} \mathrm{d}S_t&=(r-q) S_t \mathrm{d}t+\sqrt{v_t} S_t \mathrm{d}W_{1,t}, \\ \mathrm{d}v_t&=\...

3

Intuitively, in a (log)-space homogenous diffusion model $$S_t \propto S_0, \forall t \geq 0$$ such that implied volatilities will only depend on the moneyness level and not on the absolute spot level, which is precisely the definition of sticky delta. Mathematically, consider a (log)-space homogeneous diffusion model (be it stochastic or not)  \frac{...

2

I use Gatheral's notations. The SVI-Jump-Wings (SVI-JW) parameterization of the implied variance v (rather than the implied total variance The raw and natural parametrizations describe the total implied variance for one slice (fixed tenor). The SVI-JW describes the implied variance for one slice (fixed tenor). The total implied variance slice for a fixed ...

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