Let $\sigma_J$ be the volatility of the index $J$. Assume that $J(0)\leq B$. Consider the following 2 extreme cases:
- $\sigma_J=0 \Rightarrow \forall x\in[0,T],J(x)=J(0)\leq B$: hence you will always be paid $I(T)$ at expiry.
- $\sigma_J=\infty \Rightarrow \exists\epsilon>0, J(\epsilon)>B$: hence you will almost immediately be paid $I(0)\approx I(\epsilon)$.
Thus the payoff of your product depends on the volatility of the index $J$. More intuitively, the more volatile $J$ is, the more likely it is it will cross the barrier $B$, because the more the price varies along the life $[0,T]$.
Regarding the payment of $I$ itself, assuming a constant risk-free rate $r$, note that the value of your payoff can be written:
$$V(0)=E^\mathcal{Q}\left(D(0,T)1_{\{\max_{0\leq x\leq T}J(x)\leq B\}}I(T)+D(0,\tau)1_{\{\max_{0\leq x\leq T}J(x)> B\}}I(\tau)\right)$$
where:
$$\tau:=\min\{x:x\in[0,T],J(x)>B\}$$
If $I$ is deterministic, then $I(t)=1/D(0,t)$ and the value simplifies to:
$$\begin{align}
V(0)&=
E^\mathcal{Q}\left(1_{\{\max_{0\leq x\leq T}J(x)\leq B\}}+1_{\{\max_{0\leq x\leq T}J(x)> B\}}\right)
\end{align}$$
which is equal to $1$ because either $\max_{0\leq x\leq T}J(x)$ is above $B$ or it isn't, there are no more outcomes. On the other hand, if $I$ is risky, that is it has a stochastic term, and log-normally distributed, you would have something along:
$$V(0)=I(0)E^\mathcal{Q}\left(1_{\{\max_{0\leq x\leq T}J(x)\leq B\}}e^{-\frac{\sigma_I^2}{2}T+\sigma_IW_I(T)}+1_{\{\max_{0\leq x\leq T}J(x)> B\}}e^{-\frac{\sigma_I^2}{2}\tau+\sigma_IW_I(\tau)}\right)$$
This is a more complex product, because it depends upon the covariance structure between $I$ and $J$:
- If both are positively correlated, then if $J$ crosses $B$ (which it needs to do from below, given $J(0)\leq B$), there are more chances the value of $I$ will be high;
- On the other hand, if $I$ and $J$ have negative correlation, then if $J$ crosses the barrier it means it has gone upwards, so there will chances that the value of $I$ will have gone downwards due to negative correlation.
Note that correlation $\rho$ and volatility are related $-$ where $\sigma_{IJ}$ is covariance:
$$\rho_{IJ}=\frac{\sigma_{IJ}}{\sigma_I\sigma_J}$$
