Effect of time to expiration (option)

CFA states that:“The value of a European put option can be either directly or inversely related to the time to expiration.”
Here is its explanation :" Put option holders are awaiting the sale of the underlying, for which they will receive the exercise price. The longer they have to wait, the lower the present value of the payoff. for some puts, this negative effect can dwarf the positive effect. This situation occurs with a put the longer the time to expiration, the higher the risk free rate of interest, and the deeper it is in-the-money."
However, i don’t understand its explanation. Could anyone please explain or give me an example?

If you use the Black-Scholes model, with S the stock price, X the strike price of the option, r the interest rate, t the remaining life of the option, and \sigma the volatility, you can write down the value of a put option

V(S,t)=-\frac{S}{2}\;\textrm{erfc}\left[\frac{\log\frac{S}{X}+\left(r+\frac{\sigma^{2}}{2}\right)t}{2^{1/2}t^{1/2}\sigma}\right] +\frac{Xe^{-rt}}{2}\;\textrm{erfc}\left[\frac{\log\frac{S}{X}+\left(r-\frac{\sigma^{2}}{2}\right)t}{2^{1/2}t^{1/2}\sigma}\right]

where erfc is the complementary error function.

I’ve done some plots with r=0.05, \sigma=0.25, X=1 and S=2, S=1, and S=0.5.
The curve for S=0.5 (deep in the money) goes down as we get further from expiration.
The other 2 curves (one at the money, the other out of the money) initially go up before dipping down


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Think normally:
The more time we have the more chance we end up in the money with positive value.
Say we are out of the money
If we have only 1 day left there maybe little chnace we end up in the money
If there is a year to go there will be a much higher chance.

But there are some circumstances where time is our enemy.
You have a pur with strike 100
The underlying has gone bust asnd is worth zero
But there is 1 year to expiration.
We have to wait to be paid so time is bad for us.
This is an extreme and there are protocols if the underlying goes bust but
If we are extreme in the money, high rates we can mathematically construct scenarios where time to expiration is bad for us.

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If you look at the figure I posted,
Expiration is at the far left of the figure.
As you move away from expiration when you are in-the-money, the value of the option decreases. That’s the top curve.
As you move away from expiration when you are out-of-the-money or at-the-money, at least initially the value of the option will increase, before eventually decreasing. That’s the bottom two curves.

Eventually all the curves will approach a value of zero as times get further and further from expiration (the far right of the figure).

Time has two effects here:
(i) any payoff you actually receive has a time value so is worth less the further you are away from expiration. As you get to really long times from expiration, the value of a put option will decrease with time and will approach zero. You can see that in the figure I posted where all 3 curves are sloping downwards at t=40 (the far right of the figure).
(ii) The value of the underlying stock will change.
With the Black-Scholes model, it’s assumed that stock prices follow a lognormal random walk which means that on average (in some sense) you expect the stock price to grow exponentially with time, which is good if you’re holding a call option but bad if you’re holding a put option.

If you look at the figure I posted, at t=0 (at expiration, the far left of the figure):
Of the 3 curves, the top one, which is in-the-money, decreases as we move away from expiration, but the other two (one at-the-money, the other out-of-the-money) increase as we move away from expiration.
An out-of-the-money option is worth 0 at expiration.
If you’re out-of-the-money close to expiration, there’s a small but non-zero chance that the option will be in-the-money by the time it expires, which means that an out-of-the money option close to expiration will be worth something, so that as you move away from expiration when you are out of the money, at least initially the value of the option will increase.

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Thank you all very much!

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