# Is this the present value of a short position on an option?

Consider a European put option, whose price at time $$0$$ is $$\Pi_0$$. Set: $$\mathcal{L}_0=\Pi_0 - P(0,t_M)\Pi_{t_M}$$ where 0 < $$t_M$$ and $$P(0, t_M)$$ is the discount factor from time $$0$$ to time $$t_M$$. Is that correct to state that $$\mathcal{L}_0$$ represents the present value of a short position on the put option? That is, at time $$0$$ you have a positive value equal to $$\Pi_0$$ and at time $$t_M$$ you will have to rebuy the put option at its current price (i.e. $$\Pi_{t_M}$$), which enters negatively (and discounted, since you have to evaluate everything at time 0) in the value of your position.

Close to that. Your formula represents the profit (or loss) of that position as seen from $$t_0$$, which is so far a stochastic term observable at expiry.