I'm new to quant finance, and I'm confused as to whether or not European put options experience theta decay? It doesn't make sense to me that they should for a couple reasons outlined below, but everywhere I look online talks about theta decay of options (both calls and puts).
First there is a heuristic argument: the price of an underlying stock, and of the option itself, has an average growth rate of the underlying interest rate under the risk-neutral measure. If the put option is currently in the money, then each moment which goes by means that there is less time for the underlying stock to grow at that average rate, and thus there is less time for the option to eventually become out of the money. Likewise, if the option is currently out of the money, and the stock price does not move, they we haven't moved further away from the strike price. It is more probable that the option becomes in the money compared to if the stock had grown in price. This increase in probability should mean that the price of the put option increases.
The second argument relies more on computing the greeks of the option to see how one would replicate its payoff. The delta for a put option is $c_{x}(t, x) -1 = N(d_{+}) -1 < 0$. Here, $N$ is the cumulative distribution function of the normal distribution and $$ d_{\pm}:= \frac{1}{\sigma\sqrt{T-t}}\bigg[ \log\frac{x}{K} + \bigg( r\pm \frac{\sigma^2}{2}\bigg)(T-t)\bigg]. $$ Thus means that, when replicating the derivative, we always short shares of the underlying stock and go long in the money market. Thus, let's assume that at some time the underlying stock is held constant. Then the part of our portfolio which is short on the stock doesn't change at all. However, the part of our portfolio which is long in the money market has increased in value. Since this portfolio value is the price of the option, the option has increased in price over time.
Lastly, I have computed the actual theta of a European put option, but it's not clear to me if it should be positive or negative from the formula. It is equal to \begin{align*} p_{t}(t, x) &= c_{t}(t, x) + rK e^{-r(T-t)}\\ &= -rKe^{-r(T-t)}N(d_{-}) - \frac{\sigma x}{2\sqrt{T-t}}N'(d_{+}) + rKe^{-r}(T-t)\\ &= (1-N(d_{-}))rKe^{-r(T-t)} - \frac{\sigma x}{2\sqrt{T-t}}N'(d_{+}) \end{align*}
What am I getting wrong here? Does it make any sense to talk of a stock price being held constant while also passing time? Does volatility of the stock decrease and affect the put option price in that scenario? Please help me understand this better.
Thank you in advance for the help!
