I was trying to answer to your point above : "At the end of the finite difference scheme, to obtain the value of the option at time 0 we compute

[ltr]V(S0,0)=(F(S0,0)−K)e−rTV(S0,0)=(F(S0,0)−K)e−rT which in general is not 0, in contrast with what I said above about FF being 0 at time 0. What is wrong in the reasoning?"[/ltr]

[ltr]I was just saying that the value computed is the fair price of the forward contract, not its payoff. Thus there is no reason that this quantity nullifies.[/ltr]

What's the difference between the fair price and the payoff? Moreover, you said that I'm computing the fair price (is this bad?), but so what is the payoff in this case? Thank you for help

mm...I will try to do my best to make the definition understandable (if a purist read this answer: I don't want to speak about filtration here)

I would say that a payoff is a precisely a measure [$]d P(t,x)[$], but think about it as being a function [$]P(T,x)[$], where T is called the maturity, it is easier to understand. For instance [$]P(T,x) = max(x-K,0)[$] is the payoff of an option, that is the amount of money paid at maturity T.

To define a fair price, you need first a stochastic process [$]t \mapsto x_t[$] (modeling a market, as the price of a share). The fair price of P, at time [$]t \le T[$], hypothesizing that the stochastic process is at state y, that I denote [$]\bar{P}(t,y)[$], is precisely the expectation of the payoff : [$]\bar{P}(t,y) = \mathbb{E}^{x_T}(P(T,.) | x_t = y)[$]. This is exactly what is computing the Black and Scholes PDE equation, or any Monte-Carlo based method.

Without mathematic: the payoff describes the amount of money that a contract gives, since the fair price is the price at which one should buy or sell this contract to be risk free. Another more simple phrasing: when a bank buy or sell any financial contract, they usually first compute the fair, risk free price of this contract, then add a commission depending on his client. The commission depends on obviously the bank appetite, but more subtly on the risk beard by this particular client.