no code implementations • 21 Jun 2019 • Martin Forde, Stefan Gerhold, Benjamin Smith
Finally, using L\'{e}vy's convergence theorem, we show that the log stock price $X_t$ tends weakly to a non-symmetric random variable $X^{(1/2)}_t$ as $\alpha \to 1/2$ (i. e. $H\to 0$) whose mgf is also the solution to the Rough Heston VIE with $\alpha=1/2$, and we show that $X^{(1/2)}_t/\sqrt{t}$ tends weakly to a non-symmetric random variable as $t\to 0$, which leads to a non-flat non-symmetric asymptotic smile in the Edgeworth regime.
no code implementations • 27 Oct 2016 • Martin Forde, Hongzhong Zhang
Using the large deviation principle (LDP) for a re-scaled fractional Brownian motion $B^H_t$ where the rate function is defined via the reproducing kernel Hilbert space, we compute small-time asymptotics for a correlated fractional stochastic volatility model of the form $dS_t=S_t\sigma(Y_t) (\bar{\rho} dW_t +\rho dB_t), \, dY_t=dB^H_t$ where $\sigma$ is $\alpha$-H\"{o}lder continuous for some $\alpha\in(0, 1]$; in particular, we show that $t^{H-\frac{1}{2}} \log S_t $ satisfies the LDP as $t\to0$ and the model has a well-defined implied volatility smile as $t \to 0$, when the log-moneyness $k(t)=x t^{\frac{1}{2}-H}$.
