I am analyzing valuation of European put options in non-Gaussian environment.
I realized, that put option can easily get negative time value, when it is deep in the money (spot price close to zero). Ultimate proof, that such negative time value exists is at limit, when spot price approaches zero. Then the option value is less, than strike as the price of the underlying can move in only one direction.
This issue (negative time value of a put) has been discussed in these pages also and opinions vary: some think negative time value is not possible, but I believe my counter-example above shows, that negative time value does exist. It is not that relevant in the Gaussian world, but can become an issue with fat tails.
This negative time value leads to a paradox: when I apply put-call parity for a put with negative time value, I get negative valuation for a call. This should be impossible, as call option holder does not have, by definition, any liabilities and therefore value must be >=0.
So, what explains this contradiction: is it so, that put-call parity only applies, when underlying return is Gaussian distributed?
The so called "put-call parity" results from considering buying a European call option with a particular strike price and expiry date and selling a European put option with the same strike price and expiry date. This combination would have the same effect as buying a forward contract with the same strike price and expiry date.
It does not depend on the distribution of the distribution of changes in price of the underlying asset