March 5th, 2013, 11:59 am
Quote1.a) In any arbitrage free model, the level zero has to be absorbing for the stock price. Thus, if the stock price is ever zero it has to stay there forever with probability one. Consequently, all call options on it expire worthless with probability one.1.b) You might want to replace this with . As the stock price becomes large, the probability of exercising the option at maturity goes to one. Thus the proportion of the time value that comes from the optionality component (and not the financing) goes to zero. The call option is then like a forward contract where the forward price is paid immediately and this is exactly the above value.These BCs can also be 'deduced' mathematically from Fichera theory and that the solution is continuous at S = 0 and at S = SMax.Better still, transform S to (0,1) and apply Fichera again. You could use Fichera for puts as well. If financial motivation breaks down, use Fichera and vice versa.BTW S = SMax is a numerical trick and not part of PDE formulation. QuoteBtw, is it possible for you to clarify why is it enough to solve PDE with three boundary conditions (for call option price)? Mb, it is enough 2 boundary conditions? This jargon is wrong. There are 2 BC and 1 initial (or terminal) condition, although QF literature does use this incorrect terminology.t = T is NOT a boundary.//Integate once in t to give an unknown constantIntegate twice in S in t to give 2 unknown constants3 unknowns ! Quote2) What this boundary condition (basically, it says that after time of expiration the slope of the price of european option should be equal to 1)This BC cannot be motivated by Fichera AFAIK, maybe it is pathological case?(?)
Last edited by
Cuchulainn on March 4th, 2013, 11:00 pm, edited 1 time in total.