American Put Early Exercise and Strike-Dependent Parity Rates

Summary

The document examines why a discount factor inferred from put-call parity appears to vary with strike. Its proposed synthetic position combines a long share, a short call, and a long put with the same strike, producing a terminal payoff tied to that strike. Under a simple European-style parity argument that ignores dividends, the initial net cost divided by the strike would imply a discount factor that should not depend strongly on strike.

The accepted explanation is that the options are American. High-strike puts can be optimal to exercise before expiration, so their market prices reflect early-exercise value. Applying a European parity calculation to those prices therefore does not produce a strike-independent estimate of the risk-free discount factor; for higher strikes, the inferred factor may move closer to one. The discussion offers a qualitative explanation rather than a derivation or empirical test. Dividend effects, financing assumptions, and option-market frictions are not developed, so the observation should be interpreted within the document’s simplified setup.

Key ideas

• A parity-based discount factor from a synthetic bond should be approximately strike-independent under the simplified European setup.
• American options can be exercised before expiration, affecting their prices relative to European options.
• High-strike puts may have value from optimal early exercise.
• Using American option prices in a European-style parity calculation can make the inferred discount factor vary with strike.
• The document gives a qualitative explanation without quantifying dividends or market frictions.

Tags

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


# 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?

## Answer by dm63 (score 5, accepted)

https://quant.stackexchange.com/a/77800

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.