Computing confidence or probabilities requires choosing a model for your data.
There are many ways you could do this, each involving different levels of difficulty in estimation and assumptions; for instance, whether you want to use a time series method to model the underlying "up vs down process" versus trying to fit a probability distribution to try to model the streaks and their size itself. (The former case makes it easy to compute the chance of streaks continuing; the latter is what would be necessary to compute confidences.)
Anyway, here is a simple idea: use a Markov chain.
A common assumption here is time-homogeneity, which says that the distributions governing the process do not change over time.
An order $k$ Markov chain means that the value of the process at time $t$ depends only on the last $k$ values.
Specifically, for example:
\begin{align*}
P(X_t=x_t|X_{t-1} = x_{t-1},\ldots,X_0=x_0)
&= P(X_t=x_t|X_{t-1}=x_{t-1},\ldots,X_{t-k}=x_{t-k})
\end{align*}
where $x_i=U$ means it moved up and $x_j=D$ means it moved down.
You then have to fit your Markov chain to the data (i.e. estimate the various transition probabilities).
For instance, for $k=1$, the transition probabilities can be written as a matrix:
$$
P =
\begin{bmatrix}
P(X_t=U|X_{t-1}=U) & P(X_t=D|X_{t-1}=U)\\
P(X_t=U|X_{t-1}=D) & P(X_t=D|X_{t-1}=D)
\end{bmatrix}
$$
In this highly simplified case, one can compute the probability of an "up" streak continuing as just $P(X_t=U|X_{t-1}=U)$.
You can then compute the probability of a streak occurring directly as well, e.g. a streak of size 3, starting from $ X_t=D $, as
$$P(X_{t+3}=U|X_{t+2}=U)\,P(X_{t+2}=U|X_{t+1}=U)\,P(X_{t+1}=U|X_{t}=D)$$
Also, since you are using financial data, you might want to consider continuous time stochastic process models instead. E.g. for stock prices a simple model is geometric Brownian motion, and there are more sophisticated ones (e.g. see here).
