(Not an answer; rather an example.)

Here is a random walk with random $\pm 1$ $xy$-steps with equal probability (you didn't specify details), for $n=10^4$ steps, and its convex hull.
<br />&nbsp;&nbsp;&nbsp;![RandWalk10K][1]<br />
And here is the very same random walk extended to $n=10^5$ steps:
<br />&nbsp;&nbsp;&nbsp;![RandWalk100K][2]

<hr />
**Added** *15Oct15*. There is a new paper relevant to this question:

> Zakhar Kabluchko, Vladislav Vysotsky, and Dmitry Zaporozhets.
"Convex hulls of random walks, hyperplane arrangements, and Weyl chambers."
([arXiv abstract](http://arxiv.org/abs/1510.04073).)

> "We give an explicit formula for the probability that the convex hull of an
$n$-step random walk in $\mathbb{R}^d$ with centrally symmetric density of increments
does not contain the origin."

By "contain" here they mean "strictly contain in the interior of the hull."
<hr />
**Update** *1Apr2017*. The same authors have revised their paper,
establishing a formula for the expected number of $k$-dimensional faces of 
the convex hull of a random walk:

> Kabluchko, Zakhar, Vladislav Vysotsky, and Dmitry Zaporozhets. "Convex hulls of random walks: Expected number of faces and face probabilities." Feb. 2016.  ([arXiv:1612.00249 Abs](https://arxiv.org/abs/1612.00249).) 

  [1]: https://i.sstatic.net/j4LtC.jpg
  [2]: https://i.sstatic.net/6yYAp.jpg