New answers tagged variance
6
Studying zero-coupon bond prices in the CIR (1985) short rate model, $\text{d}r_t=\kappa(\theta-r_t)\text{d}t+\xi\sqrt{r_t}\text{d}W_t$, Hirsa (2013, Section 1.2.6.2) states that the characteristic function of the realised interest rate $R_t=\int_0^t r_s\text{d}s$ is
\begin{align*}
\varphi_{R_t}(u)=\mathbb{E}\left[e^{iuR_t}\right] = A_t(u)e^{B_t(u)r_0},
\end{...
4
A partial but general answer:
Let $\mathcal{F}_t^W$ be the filtration generated by $W$. Since $X_T = \int_t^T v_u du$ is $\mathcal{F}_T^W$ measurable, the Clark-Ocone-Haussman formula states
$$
X_T = E_t[X_T] + \int_t^T E_u \left[ D_u^W X_T \right] dW_u
$$
with $D_u^W X_T$ denoting the Malliavin derivative of $X_T$ with respect to $W_u$.
Hence,
$$
Var(X_T) = ...
5
The probablility of a jump of $J = \phi$. ( in either direction so I'll assume $\frac{\phi}{2} = $ probability of J and $\frac{\phi}{2} = $ probability of -J ). The probability of a jump of $0 = (1-\phi)$.
So, the expectation of the of jump amount, MM,
$ = E(MM) = \frac{\phi}{2} \times J + \frac{\phi}{2} \times -J + (1-\phi) \times 0 = 0$
The variance, $\...
Top 50 recent answers are included
