Hi

@[email protected], I can try with a quick illustration, but admittedly I think this is a counterintuive idea because, as you suggest, this is an

exception to the general rule that an option will experience "time decay;" i.e., its value will decrease as maturity approaches. To generate the exception, all I need to do is price the put option when it is deeply in-the-money. So, in this example, the prices are genuine Black-Scholes-Merton (here is the XLS at

https://www.dropbox.com/s/ppu83yebn6a7tq2/0821-theta-put.xlsx?dl=0) and the assumptions are:

- S = 10, K = 50 so this is deeply in the money
- Volatility, σ = 40%; although somewhat (also) counterintutively this is an exceptional case where volatility doesn't much matter; put another way, vega is nearly zero for deeply in- or out-of-money options
- Rf = 3.0%
- The Term varies: {1.0 year, 0.75 years, 0.50 years, 0.25 and 1/250 is the last value which represents only one day until maturity}

Note that, per this

**exception**, put option value is actually increasing as maturity decreases. Mathematically, we can explain this: these values are effectively the minimum option values, given by K*exp(-rT) - S0. For example, with a one year maturity, the put value is effectively the minimum value of $50.00*exp(-3%*1.0) - $10.00 = $38.522. In this way, notice the value is really just increasing because as a function of the discounted strike price!

At the same time, the intuition that you cite, the way that I look at this is: Imagine you hold the one-year European put option when S = 10.00 and K = 50.00. This is a volatile stock (40%). The intrinsic value is $40.00.

*If it were American style, don't you want to exercise now and collect the $40.00?* Compare this to the scenario where the stock is steady at $10.00 for the entire year: true, you will get the same $40.00, but it will be one year later, so your PV is lower. On the other hand, volatility represents the possibility of "random diffusion" in either direction.

**Unlike the call** option (which has theoretically unlimited upside), this put can never be exercised for more than $50.00 because the stock can never fall below zero. Don't you want to exercise now also because the volatility has "more time to reduce your intrinsic value" than it "has room" to increase your value (it only has + 10 more that is even possible). I hope that's helpful!

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