Forward Contracts. Let $K$ be the forward price of a contract/asset agreed at time $t=0$ to be paid at $t=T$. Now, we say that the forward price $K$ is determined in such a way that the **' value of the forward contact '** at $t=0$ is zero. Can anyone explain the difference between the **current price** of the asset & the **value of the contract** at $t=0$ ? Is value of the contract the present value of the asset ( at t=0 ) ?

The value of the contract is the profit it represents. As an example, suppose I sign a contract to sell an asset in one year for $3$. I can buy the asset today for $2$ (the current price) and interest is $5\%$ for a year. If I buy the asset today, I pay $2$ for the asset plus $0.10$ interest, so I have a profit of $0.90$. That is the value of the contract. What the text is saying is that you should expect the forward price to be $2.10$ so there is no profit to be made. If the forward price were lower there would be a profit to be made by selling today and buying back in a year.