Let c(k) be the function that gives the price of a call option for every value of the strike price: as Breeden and Litzenberg first showed, it is closely related with the "risk neutral" probability. This relation is here investigated starting from the hypothesis that c(k) is actually defined by the market. It is shown that the properties c(k) must satisfy for no arbitrage to be possible exactly allow to define a probability density function with a finite mean; the derivative of c(k) in zero gives the risk neutral probability of the event "the value of the stock vanishes at maturity", and this probability can, in the continuous case, be zero or not zero regardless of what the corresponding "real" probability does. Finally, a very easy geometric interpretation is given for the "risk neutral variance" of the variable that represents the maturity value of the call.
Cacciafesta, F. (2006). Risk neutral probabilites, and options curves. In Atti del trentesimo convegno AMASES.
Risk neutral probabilites, and options curves
CACCIAFESTA, FABRIZIO
2006-09-14
Abstract
Let c(k) be the function that gives the price of a call option for every value of the strike price: as Breeden and Litzenberg first showed, it is closely related with the "risk neutral" probability. This relation is here investigated starting from the hypothesis that c(k) is actually defined by the market. It is shown that the properties c(k) must satisfy for no arbitrage to be possible exactly allow to define a probability density function with a finite mean; the derivative of c(k) in zero gives the risk neutral probability of the event "the value of the stock vanishes at maturity", and this probability can, in the continuous case, be zero or not zero regardless of what the corresponding "real" probability does. Finally, a very easy geometric interpretation is given for the "risk neutral variance" of the variable that represents the maturity value of the call.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.