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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.

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"Normally, $1 + r < pu + qd$" This is a very strong assumption. "Normally"? –  pincopallino Jun 16 at 7:35

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I'm not sure if I understand you perfectly, but basically, you have:

$$\mathbb{E}(S_1)= p S_0 u + (1-p) S_0 d = S_0\left[p (u-d) +d \right]$$

This is simply the expectation (i.e. there is no pricing here).

Notice that this means that:

$$\mathbb{E}\left[\frac{S_1}{S_0}\right]=p (u-d) + d$$

So the expected rate of return is $1+R_S=p (u-d) + d$.

If the investment was risk-free, it should yield $1+R_f$. So, you can say that the risk premium in there is $(1+R_S)-(1+R_f)=R_S-R_f=p (u-d) + d -1 -R_f$.

Notice that the volatility of the stock price here is determined by the choice of the $u$ and $d$ parameters.

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I think I got the quantitative explanation in Steven E. Shreve's "Stochastic calculus for finance II":

ch 5.2.2

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