If stock price is determined by what people are willing to pay then why is changing a stock price never an option for an average investor?

Mutual funds trade differently than stocks. A mutual fund is a collection of small pieces of a lot of different things. The price of a share of a mutual fund is the price of all that stuff.

For things like stocks and ETFs you can put in whatever orders you want. If nobody else wants to make up the other side of that transaction then your orders wait until someone does (or until they expire) As an example, see how regular individual traders were recently able to substantially influence the price of a few stocks such as gamestop.

If you want to trade now, you get whatever price someone is willing to buy or sell at right now. If you are okay with waiting you might be able to set the price yourself.

Too obvious to really be an answer, but... When you kick a mountain, why doesn't it move?

Unless you're someone like Warren Buffet, you simply don't have enough money to significantly affect the price of a stock all by yourself. But if stock ABC is currently trading at $100, you can certainly place an order for shares at $95. If enough other people do that, and the people who hold shares of ABC want the $95, that becomes the new price.

stock price is determined by what the public is willing to pay for a given security

yes, the market is an auction

the public never seems to have the option to raise or lower the price of a stock

debatable. the public can influence the share price, but a discontinuous jump in price doesn't happen immediately without major amounts of buying/selling pressure underpinning the move.

there is no option to change a bid price

false. anyone can place a limit order on securities such as stocks and options. The market decides whether or not the transaction is fulfilled, though.

retail traders can bid a price that's lower than the MVPS (market value per share)

likewise, sellers can ask for a price that's higher than the MVPS.

their orders won't necessarily ever be filled.

e.g., if you place a limit order on a share of a stock at say $100, and the stock is trading at $105, it's unlikely to execute because no one wants to sell their share to you at $100 when everyone's able to sell their shares at $105. If someone in error set a limit sell price at $100, though, it's possible that share could go to you for $100, but this would be a rare event.

there is only the option to buy or sell at the existing price

that is known as a market order. that might be true of some of the mutual funds through your broker, in which case you can call your broker and ask them if there's a way for you to trade ETFs, which should allow you to place limit orders. when it comes to individual stocks and options using my two brokers (Robinhood and ETrade), I have complete freedom to set limit orders, or even schedule them for a later date. but that doesn't mean that they'll ever get filled.

who is making the price changes

the invisible hand

Is it the large hedge funds

large financial institutions can have the power to move stocks, and in many cases they own the majority of outstanding shares, but ultimately everything comes down to "how desirable is this stock right now? how much am I willing to pay for one share? what will be my price target before I sell? is the company well positioned to succeed and meet my alpha requirements?". that's the type of questions going through investors' minds when trading in the market. that includes both small investors as well as large investment firms.

sudden, intense desires among institutions or the public to buy long or short a specific security are what propel that stock's price upward or pressure it downward. in the event of an upward breakout, eventually the price will be bid up to and settle onto a price point at which momentum is halted because no one is willing to purchase shares above that price. in the event of a downward breakout, the price will fall to possibly even comically low levels, at which point buyers will see opportunities to open a position, and the price will stop falling at the pricepoint where the demand meets the supply. these individual decisions cumulatively represent the supply and demand dynamics seen in the market.

I'll offer up a different perspective to the other answers which hopefully will make clear why a small investor can't ever make a noticeable difference. This will use a fictional example with made up numbers, but the principle is the same for the real markets.

Imagine we have Foobar Ltd, a company listed on a stock exchange. At any given time there is an order book which contains a list of all buy/sell (technical term: bid/offer) orders. The order book is essentially a list which summarises for each given price, how many shares are being offered for sale or purchase at that price. For example, at a given moment of time the order book might look like this:

Bids:   [1,000,000 @ $100.00] [2,000,000 @ $ 99.99] [1,500,000 @ $ 99.98]
Offers: [1,300,000 @ $100.01] [2,100,000 @ $100.02] [1,800,000 @ $100.03]

Now, you, as an ordinary investor, might try do one of the following to lower the price of Foobar Ltd:

1. Sell some shares to use up the existing orders and force the price to the next set of orders.
2. Place a bid order at a new, lower, price.

Let's try scenario 1. You, being a small investor, decide to sell 100 shares using the available orders. The best price you can currently get (from the bid offers on the order book above) is $100. So you sell your shares and now the order book looks like this (changed entry tagged with < >):

Bids:   <  999,900 @ $100.00> [2,000,000 @ $ 99.99] [1,500,000 @ $ 99.98]
Offers: [1,300,000 @ $100.01] [2,100,000 @ $100.02] [1,800,000 @ $100.03]

There are simply too many other bids at $100.00 for your sale to have made any noticeable difference. You'd have had to sell 1,000,000 shares for $100,000,000 to reduce the bid price to $99.99.

Now let's try scenario 2. You decide you're not happy about the fact that you have to pay $100.01 to buy 100 shares of Foobar Ltd (using available orders) or $100.00 (placing an order of your own at the current price), so you decide to place a bid at $99.99. Now the order book looks like this (changed entry tagged wtih < >):

Bids:   [1,000,000 @ $100.00] <2,000,100 @ $ 99.99> [1,500,000 @ $ 99.98]
Offers: [1,300,000 @ $100.01] [2,100,000 @ $100.02] [1,800,000 @ $100.03]

Your tiny order has marginally increased the orders available at $99.99, but there are still the 1,000,000 bids at $100.00. Anyone selling their shares will sell the first 1,000,000 at $100.00 before they even look at your order. So again, your action has made no noticeable difference.

But, throw in a few 10,000s of other small investors all making the same decision as you (or a few large institutional investors) and now the price will change because collectively the action is large enough to make a difference.

In summary:

"They say the stock price is determined by what the public is willing to pay for a given security."

This means that the price is determined by what the "public" is willing to pay collectively, not what you, by yourself, are willing to pay. Note that "public" here includes what you are thinking of as public (ordinary investors) but also institutional investors who buy/sell in bulk, and market makers (those that profit from placing simultaneous bid/offer orders with a price spread, and help to create liquidity by making sure there are always orders available).

Bob and Matt wrote excellent answers about Net Asset Value, which I think is the at the core of your question. I just wanted to address as simply as possible the auction aspect of the market. You asked:

If stock price is determined by what people are willing to pay then why is changing a stock price never an option for an average investor?

If you bought it, that's (at least) what you were willing to pay; you may have been willing to pay more which makes the current price a good deal. That's you participating in the market auction. You could also have not bought. Not buying is also market information that the price may be too high.