Consider a one step binomial tree model for stock price. The classical setup is as below:
At time $t=0$, the stock price is $S_0$.
At time $t=1$, the stock has probability $p$ to jump up to price $S_1^u = uS_0$ (define $u:=S_1^u/S_0$); and probability $q=1-p$ to jump down to price $S_1^d = dS_0$ (define $d:=S_1^d/S_0$).
Let the risk free interest rate as $r$. Normally, $1+r < pu + qd$, so $$\frac{1}{1+r} \mathbb{E}[S_1] = \frac{pu+qd}{1+r}S_0 > S_0$$ , or, at time $0$ the stock is priced as $S_0$, lower than the discounted value of stock price expectation at time $1$.
Now, my question is , can we interpret the underbid of the stock price, $\frac{1}{1+r} \mathbb{E}[S_1] - S_0$, as the risk premium, or the cost because of uncertainties?
If so, is there any related formula? For example, an equation linking the stock price and its volatility $\sigma$?
Ps. in the risk neutral world, the probabilities are $\tilde p=\frac{1+r-d}{u-d}$ and $\tilde q = 1-\tilde p=\frac{u-1-r}{u-d}$, and we have $S_0 = \frac{1}{1+r} (\tilde p S_1^u + \tilde q S_1^d) = \mathbb{\tilde E}\left[\frac{S_1}{1+r}\right]$ , in other words, the stock price $S_t$ is a martingale, under numeraire $(1+r)$ and risk neutral measure. These are all smart and classic, but not related to my question.
