It's a question that gets brought up a lot, so I figured I'd work on a breakdown of the components of stock prices. A lot of the details aren't particularly important in the sense that you'd probably never actively consider them in a game (and for good reason), but I've included them for my future reference and for completeness's sake.

So, when you hover over your name in the player list, you get a popup that looks something like this (shamelessly stolen from one of Zultar's videos because I didn't have a valid replay):

Let's look at each compentent individually.

HQ value
The base value of your headquarters. This goes up whenever you upgrade and is not tied to faction type, but is tied to the number of players, as follows:

8 players - 600k +120k per level, for a total of 1,080k at HQ5
7 players - 700k +140k per level, for a total of 1,260k at HQ5
6 players - 800k +160k per level, for a total of 1,440k at HQ5
5 players - 900k +180k per level, for a total of 1,620k at HQ5
4 players - 1,000k +200k per level, for a total of 1,800k at HQ5
3 players - 1,100k +220k per level, for a total of 1,980k at HQ5
2 players - 1,200k +240k per level, for a total of 2,160k at HQ5

That's a rather large difference! The base HQ values have significant implications for debt and bond ratings, which will be discussed later.

Cash
Fairly self-explanatory, this is your cash on hand. Do note the implication that when you spend cash, you're losing the stock value that it was contributing (though that's often just getting shuffled to another category)

Resources
Also rather self-explanatory, this is the total value of your resources if you sold everything at that moment. This includes any money/resources tied up in researching patents or upgrades, initiating hacks, and launching things offworld.

Structures
A bit of an odder category, this comprises the value of resources invested into your buildings and HQ upgrades, though not using market prices. While structures do form a significant portion of your stock value, the difference in value between buildings isn't great enough that this should be prioritized over anything else when choosing buildings.

From the moment a building lands on the claimed tile, it adds the value of the resources used to create it to your structures total.

When it finishes, it instead values the building as follows:
$120 per steel, $40 per iron/aluminum/silicon/carbon, $160 per glass, $200 per electronics

The resource value of HQ upgrades is also included in the structures total, using the base prices of steel = $60, aluminum = $20, glass = $80, electronics = $100, and carbon = $20 to calculate.
3200, 9800, 19600, 32800, 49200 - robotic
4000, 12000, 24000, 40000, 60000 - expansive/scientific
2800, 8400, 16800, 28000, 42000 - scavenger

One important takeaway is that scavengers will have noticeably lower structures values than any other faction, with a penalty of $60 of structures value per steel required to make the building ($24k, or $0.24 in stock value, less for an offworld!) and a penalty of $0.18 in stock value from upgrade value at HQ5, relative to scientifics/expansives. Robots will also lose a full $0.108 in stock value relative to scientifics/expansives at HQ5. As of beta 10, the expansive half-steel-cost bonus is also reflected in the structures value, meaning an expansive offworld, for example, is worth $24k less than a scientific or robotic offworld, and the same as a scavenger offworld.

I should also point out that a structure completing can actually lead to a decrease in stock price as the resource value is converted from market prices to the fixed prices used to calculate structure value. 

Assets - just the sum of HQ value, cash, resources, and structure value
Debt - also self-explanatory, the debt you've accumulated over the course of the game through auto-purchasing, auctions, and interest

Bond rating
These have no direct effects on your stock price, but can over time, due to debt interest. The debt thresholds vary depending on the number of players and a table can be found here: http://offworldtradingcompany.gamepedia.com/Debt#Rating_Formula

Total value (TV) = assets - (debt*multiplier)
Your share price trends to your Target Value per Share, which is generally equal to TV/100,000. This isn't always true; past some value, there are diminishing returns on total value, such that your target stock value will be less than TV/100,000. If anyone knows more about where these values lie, I'd be interested to know.

Other modifiers:

• Stocks bought - every stock bought in you above the minimum starting stock gives an additive 5% boost to your stock value, until all 10 stocks are owned, which is when the multiplier jumps to 50%. This means the last share has a disproportionate effect on your stock price, increasing this multiplier by 10% in 2-3 player games, by 15% in 4-5 player games, and by 20% in 6-8 player games.
• Stocks owned - the value of stock you own in other players. This value is based on the target value per share of the stock owned times the fraction of stock owned, so it does not include the effects of any of the modifiers in this section. 
• For example: You buy one stock in someone with a stock price of $10.00. This is the first stock anyone has bought in him, so it raises the stocks bought modifier to +5%, bringing his stock price to $10.50. This stock only adds $10.00/10 = $1.00 to your stock price, despite his current stock price being $10.50. Any further stocks you buy will also only add $1.00 to your stock price, pretending everything else is constant. 
• Note that stock in a player with a negative stock value is valued at nothing for the purposes of this modifier, despite the negative real value and the $1 minimum stock price.
• Difficulty - if there's a stock price modifier for your chosen difficulty, it's applied as a multiplicative boost after everything else is considered

What this all means
Here are some examples of various approaches to increasing your stock price and how they compare. Please let me know if my math or thinking is wrong anywhere.

In very simple terms:
Buying the first stock in someone will cost you an amount equal to their stock value*1000, lowering your stock value from cash by their stock value/100. At the same time, your stock value increases by their current stock value/10, which means money spent on the first stock is 10x more valuable for your stock price than the cash used for it. Even in the worst case, with 10 stocks bought in someone (bringing the +50% multiplier into play) and needing to pay double price for owned stock, the stock is 3.33x as valuable for stock value as the cash equivalent (i.e. you'll gain $2.33 in stock value for every $100,000 you spend on stock in this scenario).
On Manager difficulty, paying debt will add $5 to your total value for every $1 you pay, meaning that every $25,000 of debt paid adds $1 to your stock value (paying off $25,000 in debt raises your stock value by $125,000, but it costs you $25,000 cash to do that, leaving a net gain of $100,000 in stock value, or $1 in stock price). With 10 stocks bought, that becomes $6.00 per $100,000 of debt paid, which means paying debt is six times as valuable for your stock price as holding cash.
Therefore, buying unowned stock (the worst ratio will be in a 3-player game with 9 stocks owned, giving a net gain of 10/1.4 - 1 = $6.14 per $100,000 spent) will always be better than paying off debt if you only care about raising your stock price in the moment.

Adding another layer:
With all stocks owned, paying off $25,000 in debt increases your stock price by $1.50, as mentioned above. This means that your opponent will need an extra $3,000 for every share he needs to buy, which will be a minimum of 5. If you're making equal amounts of money, you must own at least nine stock in yourself for this to pay off.

Accounting for interest:
For simplicity's sake, let's say you're paying debt down immediately before an interest tick and this one day is all you're considering.
If you're still stuck in D debt after paying down debt, every $100,000 of debt paid will keep your stock price $7.80 ($6.00*1.3) above where it would've been with the interest applied. You must own at least seven stocks in yourself.
If you manage to pay down exactly enough debt to get to a C rating, assuming a 2.5m asset value (which would be roughly accurate for the late game in a 4-player match), you'd be saving $1.875 in stock value from avoiding the higher interest rate alone (difference of 10% interest on 250k debt, with no offsetting cash cost). The breakeven point will depend on how much debt you had to pay to get back to the C rating. For example, if you paid $100,000, you would gain $9.675 in stock value. Assuming again that your income is equal to your opponent's, you'd need at least six stock in yourself to take an advantage from paying this amount of debt.
Any further debt payment would have smaller returns, as the avoided interest is lower and moving up bond ratings both has smaller effects on interest rates and affects a smaller amount of debt.

So based on this analysis alone, my go-to panic move of paying down all my debt when I'm in trouble may be costing me. That said, the common rule of thumb to only pay down debt if you own all your stock seems too conservative.

Other considerations:

• Paying down enough debt to get out of D levels of debt also gives you access to the black market again. While black market is generally very expensive late-game and thus of questionable value, it is still a powerful and possibly pivotal tool, especially with how strong offworld markets get in the late-game.
• The game may last for more than one interest tick, in which case paying down debt sooner reduces your principal and prevents the extra interest from compounding.
• In situations with more than one opponent, paying down debt not only makes you more expensive to buy, but also makes you less of an immediate threat since you lose the purchasing power of the cash. This may not save you if there are still compelling reasons to target you first, but you might combine this with stock sells or something similar to get someone to switch targets. Surviving longer doesn't necessarily give you a significantly greater chance of winning, though.
• All those estimates were done assuming equal future income. If you have reason to believe you're earning more than your (single) opponent, there's no reason to treat a dollar spent by you as equal to an extra dollar he'd need.
• All that said, the cash used to pay debt (or buy stock to increase your stock price) obviously comes with an opportunity cost. Funneling all that cash into hacks or using some means to invest that cash into making greater future income can be more effective.

I was originally going to deeper into the implications here, but trying to weight these other considerations properly is hard and I've sort of run out of ideas (and laziness struck). If anyone has anything to add or wants me to expand on something, feel free. Hopefully someone finds this useful.