These terms are just quick 'reference points' or rules of thumb to quickly simplify a complex subject to add a little bit of comparability between investments. They are the mathematical results of market price for a stock, not something that the company itself has direct control over.
First, I'll simply define a 'Valuation Multiple':
Assume you sign a contract with me: you pay me some money today, and I pay you $1,000 in 12 months.
You obviously wouldn't want to pay me $1,000 for this deal, for 2 reasons: (1) Money today can be invested in other things sooner [an opportunity cost called the 'time value of money']; and (2) if renege on our deal, you would have to pursue me legally to get your money back - and I might even go bankrupt [this is basically the 'credit risk', or the risk difference between giving me money instead of buying a bond from a strong government or corporation]. The amount you would agree to pay me, and the amount that I would accept, is based on our mutual understanding of the 'cost of money' and the risk of the project.
So would you pay me, say, $900? This would be a $100 profit, so you would have a nominal return of 11.1% [$100 profit / $900 invested]. This would be a little higher than the level of return from the stock market [on average], or put another way, about 9% higher level of risk than a simple 2% 'risk-free' government bond. If we agreed on this, then it indicates we agree that I'm pretty risky, but not much more than a portfolio of general stocks in the market. Consider that a credit card company might charge me 20%+ annual interest, so this rate implies lower risk than a general pool of credit card users.
Now what if I agree to pay you $1,000 every year, forever? How much would you pay?. That's basically what investment in a company's stock is saying. You buy a portion of the company, and have the right to future dividends [even if they have no dividend payments now, down the road you would get either future dividends or liquidated value of the company].
If you bought 0.1% of a company with a flat $1M in annual income, paid out as dividends every year, then you are buying an income stream of $1,000 per year. How much is that worth? For a very risky company, it is not worth as much [because they might go bankrupt]. For a very stable company, it is worth more [because you are more certain about future payments]. Conversely, for a company with high growth potential, it is worth a lot [because the income stream could increase], and for a company with low growth potential, it is worth less [because income could stagnate or decrease over time]. note that companies with uncertainty about the future are often the ones with possible high growth potential - so comparing the risk vs return of such companies can be difficult!
Mathematically, earning $1,000 per year if the 'required return' / 'discount rate' / 'risk rating' [nearly equivalent to an interest rate, for comparative purposes] is 11.1% like our above example, is worth today simply 1,000 / .111 = $9,009. That's right, earning $1,000 every year for 1 billion+ years is only worth $9,009 today, if the discount rate based on the risk is 11.1% annually.
The Earnings Multiple or 'Valuation Multiple' is simply the inverse calculation: If someone buy stock for $9,009 and it only gives them rights to $1,000 in income based on the company's current earnings, then the earnings multiple is 9,009 / 1,000 = 9x.**
So you can calculate Tesla's 'Valuation Multiple' based on its 900B current total market cap [market capitalization, ie: the total value of all outstanding shares based on current price of the stock being traded] and its current net income from 2021 of 17B. 900B / 17B = 52x. If Tesla was projected to have this amount of net income forever, then this implies that the proper risk-based discount rate of tesla is 17 / 900 = 1.88%. This is a lower rate than government securities, which doesn't make sense - instead, it implies that the market believes that Tesla's income in the future would be many times its current earnings.
Now to your specific questions asked:
Some of your questions are answerable and some are not, but given they are highly related I will address each individually.
What is Valuation Multiples for a stock like Amazon or Tesla?
You can quickly do that math as I have done above based on current market cap / current annual earnings.
What is P/E ratio? Is it related to Valuation Multiples?
Price / Earnings means the current share price / the earnings attributable to that share. In simplistic circumstances where there is a single share class, it is basically the same as the valuation multiple: Share price today / [company earnings / # of shares]. For Tesla using the simplistic figures above, this would still be 52:1, implying that you are paying $52 to earn the rights to $1 of annual corporate profit.
In the case of ZM, did the investors consider the Valuation is $500 per share, and is that the reason the stock went up to $500 per share?
You have it the wrong way around - 'the market' [really a collection of unrelated people each individually trying their best to make money buying stock when they think it is worthwhile and selling it when they think it isn't] bids in an open auction stock market, and what we call 'share price' is often simply the most recent trade done between buyers and sellers. So because the market last traded ZM at $500 / share, its valuation is equated to being $500 / share.
In the case of ZM, is it possible that the stock can go back to $500 per share in next 10 years if the company maintain the current Earnings Per Share?
A company's stock will go up or down based on what 'the market' bids in an open stock market auction. Hopefully this is based on relevant financial data on how the company is doing, but for companies that are difficult to value [like those with strong prospects for growth but also high risk], "hype" can be an influencing factor on share price.
What drives the P/E ratio higher or lower?
Literally every piece of financial information in the world, if that impacts what a buyer or seller is willing to trade the stock for.
