Consider BS-model with parameters: Stock = 100, Strike = 100, Texp = 1 year, Vol = 13%, Rf Rate = 3%. For these parameters the BS put price is 3.76. Then consider the same parameters but with Texp = 20 years. The new BS put price is 3.29. Consider now Texp = 100 years. The put price is even lower still at 0.09.

What gives? Does it make sense that longer expiry options can be valued less?

  • 1
    $\begingroup$ Note that the upper bound of the put price is $K e^{-r T}$ and thus decreases with time-to-expiry. The forward on the other hand grows at $e^{r T}$. $\endgroup$ Commented Sep 8, 2020 at 17:26
  • $\begingroup$ how about 5 years? $\endgroup$ Commented Sep 8, 2020 at 18:04
  • $\begingroup$ Is 5 years the turning point? $\endgroup$
    – roz
    Commented Sep 8, 2020 at 18:22
  • $\begingroup$ turning point is slightly to the right of 5 in this case. $\endgroup$ Commented Sep 8, 2020 at 19:03
  • $\begingroup$ I've been messing around with it. I guess this result is because at that long time span, the rf drift makes it so that all the paths almost surely are OTM for those puts regardless of the vol. Is this about right? Is there a better, more precise explanation? $\endgroup$
    – roz
    Commented Sep 8, 2020 at 19:05

1 Answer 1


It's the interest rate component. That is more meaningful in the formula. Note that the call becomes more expensive.

Think about it this way. You could buy the call and sell the put instead of being long the stock. This gives you a synthetic long position. You need to pay the market the cost of borrow (r). That makes the calls more expensive and the puts cheaper.

In a twisted way you could say that the market expects the price of the stock to rise by the risk free rate.


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