Negative time value of put option 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? 
 A: You won't get any contradiction from the put call parity. On the contrary, I think the put call parity is very helpful in illustrating how puts can have negative time value.
First, we need a little confirmation on the terminology: I assume in your definitions the value of a derivative (its current price) equals its intrinsic value (what I could get if I exercised it now) plus its time value.
Then take a look at the put call parity $p_t = c_t - f_t$ in which $f_t$ is the value of a notional forward contract that has the same maturity and strike as the call and put in question. For simplicity let's assume the world follows Black Scholes assumptions (strike that, we actually don't need BL or any other model assumptions to make the following valuation of forward hold, apart from a constant risk free rate $r$), so that $f_t = S_t - Ke^{-r\tau}$ ($\tau$=time to maturity, or $T-t$). Hence
$$p_t = \color{blue}{c_t + Ke^{-r\tau} - K} + \color{red}{(K-S_t)}$$
The red part is apparently the intrinsic value, which makes the blue part the time value. Then you can see clearly that there is no guarantee that the blue part is greater than zero. Specifically, for example, it can easily drop below zero in the following two scenarios:
1). When the put is deep in the money, or equivalently when the call is out of the money. In this case $c_t\approx 0$, and $Ke^{-r\tau} - K <0$. So it's very likely they sum to a negative value.
2). When the interest rate is high or time to maturity is long. In this case $Ke^{-\tau}$ will be small and dominated by $-K$. If $c_t$ is also significantly smaller than $K$ (which happens easily when, say, $K$ is large). 
The intuition is that if you were a put holder you'd want to get paid as early as possible because early payment means larger time value of money, but the put contract stipulates that you cannot get paid prior to maturity, which entails a loss from such deferral of cash flow. This is opposite to the call, in which case you as the call holder would want to defer the payment as much as possible and the call contract actually entitles you to such deferral of payment, which means a gain in time value.
A: 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
A: Just an example for a call, but it makes it easy to understand that it could happen the same for a put: if you hold a call of an asset which follows a mean reverting proccess and not a Brownian Motion, you can, indeed, have negative time value. Being more specific, suppose you have a call on the location gas spread between the Netherlands and Belgium for the day ahead contract 7 days ahead of the current date (take into account the spread is on two almost perfect substitutive assets). This follows a mean reverting process to a historic mean of zero (this is not only consistent by a risk neutrality and non-arbitrage perspective, it is what empirically happens), then if your option is deep in the money  because there was a spyke in this spread, and it is well above the mean (lets say, the spread is now in an abnormally high value as could be 10 euros/MWh), there is an almost 100% probability that your payoff in the maturity date is below the intrinsic current value (10 euro) because it tends to revert to the mean, so, the value of the whole option must be lower than your intrinsic, and that means your time value is negative.
In the case of a put, if you hold one and the spread has a big drop to negative numbers well below the historic mean of zero, your expected value of the underlying at the maturity date is still close to the mean, so, the time value has to be negative to offset your abnormally high intrinsic value, which is not consistent from a risk neutrality perspective.
This can be easily checked running simulations of a mean reverting process and simulating the payoffs of an option on an asset with this characteristics.
Summing up: It is impossible to have negative time value if you use Black-Scholes as the valuation model, but it is indeed possible if the price of the asset you are longing or shorting trough an option follows another process, for example, a mean reverting one, and you model it like that.
A: The time value can't be negative. 
A positive time value reflects the possibility that the price of the stock might wander in such a way as to make the option in the money at expiration. Then the option gives the holder the right to exercise the option, and a make a profit.
A negative time value would reflect the possibility that the price of the stock might wander in such a way as to make the option out of the money at expiration. However, since the option gives the holder the right to exercise the option, but not the obligation, this only results in the option being worthless. So the time value can not be negative.
