Why financial derivatives has some inherent value ? Why all financial derivatives have convex payoff functions ?
Under Black Scholes option pricing method we discount the expected value of option payoff function under suitable probability measure.
Lets take an example of call option. The strike of call option is 100 and currently underlying is getting traded at 100 so at present this option is at the money (ATM) option and at present this option does not have any value. This option can have some value in future before the expiry of this option.
This option will get priced and it will definitely have some price more than 0. So why this option is getting priced more than 0 if at present the value of this option is 0. It is because in option pricing we need to use the payoff of option not payoff of underlying.
Suppose an underlying has price of 120 and 80 over some period than average price of this (50% Probability) is 100. Option on this underlying has strike of 100 than if we calculate the option value using option payoff , max( average value of underlying - strike ), than option price will be 0. So option is worth less.
If we first calculate the option values for two different underlying price and then average these option values than option will have some value. With 120 underlying price, option will have value of 20(without discounting) and with 80 underlying price, option will have no value. So average payoff will be 10.
Jenson's inequality states the same mathematically
Jason's Inequality states that for a convex function, such as the payoff of a call option, the expected value of the convex function is greater than or equal to the convex function of the expected value.
Hence all non linear financial derivative that have some value will have convex payoff function as Jason's inequality holds true only for convex functions.
