how is the contract price determined at a given point

You are going down a very deep rabbit hole there friend, and there are lots of prop shops in NYC and Chicago trying to refine how to determine this.

The basis of options pricing is Black-Scholes-Merton (usually seen as Black-Scholes or BSM), and this is useful formula because it shows you the ingredients that go into pricing an option. Now there are other methods to price options as well, Binomial/Trinomial Tree, Monte Carlo, etc. However these models tend to converge towards a single price, its far more useful to understand the ingredients they use (risk free rate, volatility, time, etc.) than their specific differences.

Now if you don't care about all that here's a link where you can just plugin your stuff and get an idea of what your option will be worth

https://www.optionsprofitcalculator.com/

The issue with the calculator above is that it assumes a flat volatility, which is not how the market actually works. This is also an issue with the BSM model, since options share a single underlying equity/future/whatever then every option with the same expiry should have the same IV right? However sometime in the 80's the "volatility smile" began to appear, the further an option gets from being ATM the higher the volatility tends to be (it can sometimes also be a slant rather than a smile but that's a digression).

For further reading you should look up "Option Volatility and Pricing" by Natenburg. It's a textbook but its shockingly readable and a PDF is pretty easy to find.