I am trying to back out the put call parity price of an American call option for a 10 min period with tick data (using CME ES Futures Options in this example, see plot below), using the standard PCP formula for European options, where $q=0$.

$$C =Se^{-qT}+P-Ke^{-rT}$$

Some background info:

Call strike: 4700

Underlying: ESM2 (ES Futures Jun22)

Underlying futures spot (in the 10 minutes): ~4458.375

Time to maturity: 0.19452054 (71 calendar days)

interest rate: Implied using the ATM option bid and ask prices, and then averaged

$$r_{implied} = \frac{1}{2}(r_{bid} + r_{ask})$$

$$r_{bid/ask} = -\frac{1}{T}[ln(S+P_{bid/ask}-C_{bid/ask}) - ln(K_{ATM})]$$


My goal is to use put call parity to derive the call's bid/ask prices in the event of a single sided TOB, as a proxy of its 'fair price'. Orange line is the call option's bid/ask prices, green line is the equivalent put option's bid/ask prices, and blue line is the put call parity derived call bid/ask prices.

E.g. if the orange bid was missing due to illiquidity, the supposedly 'fair price' should be the upper blue line, which is quite bad.


enter image description here

The 4700 Put bid/asks used to model the 4700 call prices:

enter image description here


a) I understand that put call parity cannot be directly used for American options, a change is to add inequalities like this post mentions: What changes to put-call parity are necessary when evaluating american options on non-dividend paying assets? I am not quite sure how to (and if it really makes a difference at all) make use of this fact in my current methodology.

b) The ITM 4700 put (green line) has a much larger spread than the OTM 4700 call (orange line), and as a result, the computed put call parity call bids and asks (blue) have a similar spread to the put, which isn't good at all as it is quite far off the actual call bids and asks. Am I doing something wrong here? My goal is to use put call parity to derive the call's bid/ask prices in the event of a single sided TOB, as a proxy of its 'fair price' but as per the plots it just seems like a hopeless endeavor, as the spread of the put call parity bid/ask (blue) is entirely dependent on the spread of the opposite option (green). How do I make this work so the put call parity estimate is better (i.e. blue line closer to orange)?

c) Is the implied rate method a sensible way to go about getting a suitable rate to plug in?



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