楼主，我也在想这个问题，从现实中看到，的确是平值期权的时间价值最大了？但为什么呢？

Intuitively , at the money option has the largest uncertainty, so that it has more time value.

Mathematically,you can consider the problem like this:

taking Call option as an example,

1. C is an nondecreasing function of S so that C(K-e)<=C(K)

2. Option's price has to satisfy some no-arbitrage inequalities like butterfly condition and bull spread condition, here we use bull spread condition:
     C(K+e)-C(K)<=e
that is C(K)>=C(K+e)-e

the two inequalities hold for any e

Now the time value (Vtime):
S>K, Vtime=C-(S-K)=C(K+e)-e
S<K, Vtime=C=C(K-e)
S=K, Vtime=C(K)

We have prove that C(K-e)<=C(K)>=C(K+e)-e.

So that Vtime(S<K)<=Vtime(S=K)
           Vtime(S>K)<=Vtime(S=K)

So time value for call is largest at S=K, and the same thing hold for put.