14 problems

Regularity-reduction conjecture. The regularity conditions required for this central limit theorem can be reduced to those of Proposition 1 of Mykland and Zhang.

Pareto-tail conjecture. At the cost of increasing the model's complexity, replacing the exponential jump law by a law with Pareto tails should yield even better results.

Let $h(t,y;s)$ solve the Heston-model partial differential equation specified in the surrounding analysis, with terminal condition $h(s,y;s)=1$. Exponential-affine solution conject…

In the fractional stochastic volatility model, let the parameter vector be $\theta=(\xi,\lambda,\nu,H)$, and let $g^{(2)}_{0}(\theta)$ and $g_{\ell}(\theta)$, for…

Let $f_j(x,v,\tau;\phi)$, for $j\in\{1,2\}$, denote the characteristic function solving the stated partial differential equation with initial condition…

Single-discretization conjecture. In practice, single discretization gives a good approximation for the expectation $\mathrm{E}f(S_T)$, better than double discretization or direct…

Stationary-variance stochastic-volatility conjecture. All stochastic-volatility models in which the variance process has a stationary distribution should exhibit similar large-matu…

Practitioners' conjecture. The large-maturity forward smile should be the same as the large-maturity spot smile.

Let $\rho$ denote the correlation parameter, $r$ the interest-rate parameter, and let $\mu$ and the remaining model and discretisation parameters take their admissible values. The…

The scheme uses a correlation parameter $\rho$ in the stochastic-volatility model, and its von Neumann stability condition is denoted by … is satisfied also for non-vanishing corre…

Let $R$ and $N$ be the time-homogeneous processes under $mathbb Q$, and for $(t,u,v)\in\mathcal Q$ define … Here $S_d$ denotes the space of real symmetric $d\times d$ matrices, and…

At-the-money limit conjecture.

Stochastic-volatility conjecture. We conjecture that in a stochastic volatility model these amounts will be different.

Let $S$ follow a continuous stochastic-volatility model and let $V_T=\langle\log S\rangle_T$ be its realized variance. Let $\tilde S$ be the corresponding local-volatility model ca…