# Why does the risk-free rate implied by put-call parity vary with strike prices?

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:

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:

What explains the behavior observed at higher strike prices?

• 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. Dec 22, 2023 at 18:41
• 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). Dec 22, 2023 at 19:15
• The main issue are borrow costs / fundings costs which you don't model. Dec 22, 2023 at 19:52
• 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
– dm63
Dec 23, 2023 at 4:10
• Ok I posted it as an answer
– dm63
Dec 23, 2023 at 13:01