2
$\begingroup$

Suppose I do the following:

  • buy one lot of some underlying stock currently trading at price $S$,
  • write a call with strike price $K$, earning some premium $C$, and
  • buy a put with the same strike $K$, costing some premium $P$.

At expiration, I will sell my shares at the strike price $K$ no matter which option is in the money, so I paid $S - C + P$ for a synthetic bond of sorts with principal $K$. Ignoring dividends the implied risk-free discount factor is $D := (S - C + P) / K$.

Given the above I would expect $D$ to be more or less independent of the strike $K$. However, performing this calculation on various options expiring a year from now, I get this interesting shape that I don't understand: SPY-implied discount rate

As you can see, for near-the-money options things look fine. As of today 2023-12-22, the one-year treasury bill trades around 4.8% and SPY pays a dividend yield of 1.4% so the discount rate implied by at-the-money options (dashed line) seems very reasonable. However I do not understand what goes on to the right of the figure, where the implied risk-free rate seemingly tends to zero.

For what it's worth I observe the same phenomenon with low-dividend and dividend-less stocks:

AAPL-implied discount rate META-implied discount rate

What explains the behavior observed at higher strike prices?

$\endgroup$
9
  • 2
    $\begingroup$ This post is discussing a related issue. Namely that in a real market the put/call parity for itm/otm calls/puts (or vice versa) does not need to hold strictly. Arbitrage is always just a theoretical possibility. If the liquidity is against you there is nothing you can do to exploit it. $\endgroup$
    – Kurt G.
    Dec 22, 2023 at 18:41
  • 2
    $\begingroup$ If the objective is to see what (r-d) the market is assuming, the procedure should be: (1) from prices of puts and calls, find the forward price F, (2) From F, S and T find (r-d). $\endgroup$
    – nbbo2
    Dec 22, 2023 at 19:15
  • 1
    $\begingroup$ The main issue are borrow costs / fundings costs which you don't model. $\endgroup$
    – AKdemy
    Dec 22, 2023 at 19:52
  • 1
    $\begingroup$ I'm pretty sure this is because the options are American. High strike puts are optimal to exercise early, hence the implied discount factor should be closer to 1 on the right hand side $\endgroup$
    – dm63
    Dec 23, 2023 at 4:10
  • 1
    $\begingroup$ Ok I posted it as an answer $\endgroup$
    – dm63
    Dec 23, 2023 at 13:01

1 Answer 1

4
$\begingroup$

I'm pretty sure this is because the options are American. High strike puts are optimal to exercise early, hence the implied discount factor should be closer to 1 on the right hand side.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.