I would like to compute the evolution of the P&L of an FX plain vanilla option. Unfortunately, I am not sure about the correctness of my reasoning.
Let's imagine that I sell a 1W call option on a Monday. On that day, I will receive a premium which in my case, is computed using Garman and Kholagen pricing method. Let's say that the premium is \$10, can I say that my P&L is +\$10 right after having sold the option ? In my opinion, at the time, the P&L is \$0, not +\$10.
Let's imagine that on Day 2, if we recompute the option with the new market data, we obtain a premium of \$9. On Day 2, is it right to say that my P&L is (10-9) = \$1 ? My reasoning is that I sold at \$10 something that now only worth \$9.
If that reasoning is correct, I would continue like that until the end of life of the option.
If on top of that, I have to delta-hedge my position on a daily basis. Is it correct to do the following ?
Let's imagine that on Day 1, the ∆ is -0.5, I will buy 0.5 of spot. On that day, the P&L of the delta hedging is 0.
Let's imagine that on Day 2, the spot moved from 1.1 to 1.2. Can I compute the P&L as follows: -∆*(1.2-1.1) = 0.5*0.1 ?
I would be a great help if someone can confirm or not my reasoning.