Two players coin flipping 
Two players are flipping a coin. If head, player A wins 1 point. If
  tail, player B wins 1 point. The person who first wins 2 points wins
  the game. The loser must pay the winner $\$1$. 
However, if player A has an option he can use to increase the stake of
  the game to $\$2$-game, what is the value of such option?

I received this question during an interview but I am unsure of the answer I provided.
The original game has an EV of $0$ and using the option will yield an EV of $0.25$ according to my calculations. The final answer will then be that the option has a value of $0.25$ USD.
Is this way of reasoning correct? Can anybody provide a thorough explanation of the process and show me the right way of approaching this problem?
 A: Yes, your reasoning looks correct to me. There is a 0.5 chance of a H on the first throw. In that event A exercises the option. If the next throw is H he wins 2. If the next throw is T, then his expected win is 0.
On the other hand, there is a 0.5 chance of a T on the first throw. In that event, A does not exercise the option. If the next throw is T he loses 1. If the next throw is H, then his expected win is 0.
So adding that up, with the option his expected win is $0.25\times2-0,25\times1=0.25$, compared with an expected win of 0 without the option. So the option is worth 0.25.
Note that A the only time A benefits from exercising the option is immediately after a H on the first throw. If there is a head and he delays, then either there is a H on the next throw and it is too late - the game is over - or there is a T on the next throw and he has no advantage. Similarly, if there is a T on the first throw his position is worse, so it hurts him to exercise the option.
A: It actually depends on whether A pays for the right to have the option (whether he uses it or not) or if A pays only for using the option. In the first case you are right. In the second (it seems more likely, at least to me) the value is a bit higher.
To clarify this in the first scenario a game could go as follows:
Game starts. Player A pays 0.25$ \$ $ for the option.
Coin flipped - B wins.
Coin flipped - B wins.
A pays 1$\$ $.
In total A loses 1.25$\$ $
In the second scenario:
Game starts.
Coin flipped - B wins.
Coin flipped - B wins.
A pays 1$\$ $.
In total A loses 1$\$ $,  
He doesn't pay the 0.25$\$ $ since he didn't use the option to raise the bet.
