Replication in derivatives

I have a question about replication since I’m a little confused.

I like to use + and - signs as Long and Short respectively.

In replication: Asset + Derivative = Bond (or risk free asset)

Which from my point of view is: Long Asset + Long Derivative = Long Bond

However, it is stated that: Asset - forward = Bond Or Long Asset - Short forward = Long Bond

Why in the first case it is + sign for Derivative (forward), and in second it is - ???

In the first example, your combination of being long an asset (stock) and long a derivative (put option) creates a risk free position that will be worth the strike price at expiration. Thus, it will earn a return equal to the risk-free rate. The sign differs in the second example because it uses a forward contract rather than an option (which is a contingent claim). The long put option gives you the right to sell the underlying at expiration provided the option is in the money. However, delivery in a forward contract will take place, and is not contingent upon the spot price at expiration. To create a risk free position using a forward, you will need to be long the underlying and short the forward contract (hence the - sign). With the forward, you are entering into an agreement to sell your long position at an agreed upon price on an agreed upon future date, which eliminates price risk.