1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ "Normally, $1 + r < pu + qd$" This is a very strong assumption. "Normally"? $\endgroup$ – pincopallino Jun 16 '14 at 7:35
0
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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

ch 5.2.2

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.