# Put call parity with american options

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

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

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

Goal:

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.

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

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?