derivatives value at the beginning of a contract

The value is determined by the underlying (or derived) asset or security. 

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Generally exchange traded derivatives do have a zero value at the point of purchase/sale, however OTC derivatives can be traded “off market” and require a transfer of funds from/to a counterparty at the time of initiation. 

An “off market” FRA for example could set the fixed rate slightly higher than the theoretical “fair” fixed rate and therefore have an initial negative value to the fixed rate payer and a positive value to the fixed rate receiver. In this case, the fixed rate receiver would pay an initial outflow to the fixed rate payer to compensate for this positive value.