I'm very much an amateur, but I do work on a derivatives trading desk and some interest in history. I'll try to answer the question but I may not be up to the normal high standards of this sub.

The simple answer to your question is that London and New York are centres of global finance because they are the host cities of two of the largest and oldest stock exchanges, and much of the other financial business that goes on (over the counter trades such as derivatives, foreign exchange deals, underwriting, etc) benefited from being in close proximity to an exchange for communications purposes. A 19th century trader benefited from having an office located near the exchange because it let him have access to market data and make trades more quickly. This is the same reason having the systems that do automated trading collocated with a contemporary exchange is beneficial.

So cities become financial centres due to the proximity of stock exchanges. The London Stock Exchange is one of the oldest stock exchanges in the world. It has operated under the name "London Stock Exchange" since 1801 (you can find a brief history on the Exchange's site ), although its roots go much further back. Activities that today take place on the exchange took place on less formal terms in coffee shops in London, the earliest example of which was John Casting's listing prices for marketable securities in 1698 at Jonathan's Coffee House. Jonathan's was eventually purchased by a group of traders in 1761.

The New York Stock Exchange also has its roots in coffee houses - in their case, the Tontine Coffee House which was situated at the corner of Wall Street and Water Street. Unlike Jonathan's in London, the Tontine was built by traders for the purpose of having a place to trade. It came about as a result of the Buttonwood Agreement of 1792, where the key traders of the day agreed to only deal with each other (thereby eliminating securities auctioneers) and not to undercut one another on commissions. The Tontine was built the following year, but was a pretty disorderly place (it was also used for social activities, banquets, balls, slave trading, impromptu political debates) and so this led to the eventual creation of the New York Stock Exchange in 1817.

Of course, these were not the only stock exchanges formed - of the city's you mentioned, Paris's exchange was completed in 1826 (construction began in 1808 and there was trading well before that), Boston's was completed in 1836, Alexandria's in 1885. Rome never had a major exchange (the Italian stock market's leading exchange, the Boursa Italiana, is based in Milan, and its antecedent was established there in 1808) - Rome was part of the Papal States until 1871, and as recently as 1836, the Pope was coming down on usury; being unable to charge interest would make it difficult to become a financial center. The Lisbon exchange actually predates London, and was established in 1769.

Why did certain exchanges flourish? Financial centres thrive in places with stable governments, and prefer ones that are friendly to the interests of the markets. This gives American and British exchanges an advantage, since both countries have had continuity of government throughout the period that the exchanges have existed, and the nature of their legal systems (with their basis in common law), also facilitates financial transactions. Changeable and codified systems of law (as exist in France, for instance), and result in the market being upset by very quick changes in the underlying legal framework. Most financial services also rely heavily on contract law, so a jurisdiction where contracts are given significant legal weight and are difficult to nullify will have an advantage in becoming a financial centre. Of course, the single largest reason that a stock market (and associated financial centre) will become dominant is the volume of deals done in that centre, which is a function of the strength of the host country's economy. A larger economy results in the capital pools in a given centre becoming larger, and so a virtuous cycle is created as greater access to capital encourages more business to be done in a given centre. The strength of the British and American economies helps explain the success of the associated stock exchanges. London is also a global center of finance due to foreign exchange (I used to cover such a London based foreign exchange desk as an analyst), and that's a historical artefact of London's role as the largest European financial centre (due to the size of the London Stock Exchange) and the need for firms that handled large FX deals due to the number of currencies in use in major economies in Europe. New York has FX firms too, but they play a lesser role because of the dominance of the US dollar in North American finance.

I can't really explain why New York's exchange eclipsed the Boston exchange, though, or for that matter, why the dominant U.S. exchange isn't the Philadelphia Exchange (which is actually the oldest US exchange, founded in 1790). I suspect this has to do with the economic history of the U.S., and possibly the historically cosy relationship the NYSE enjoyed with the powers that be in New York (particularly during the era of Tammany Hall), but anything in that regard would be pure speculation on my part and against the sub's rules. Perhaps some one with more knowledge of 19th century American history can answer that one better.

Everyone who is about to comment here: PAUSE. The "answers" here thus far have been a combination of speculation and half-assed laziness. This is not acceptable. There are actual people who study economic history, including global economics, who have produced bodies of work on the subject. Please keep that in mind, and also know answers here need to be comprehensive and informative. Please further refer to the Meta post on what it means to post a good answer in AskHistorians.

London and New York are not the only global financial centers, just the two leading ones at the moment. I'd suggest reading Banks and the world’s major banking centers, by Choi, Park, & Tschoegl for a somewhat dated overview of the world financial centers. Paris IS a financial center, just not as large as London.

There are returns to scale, which tends to create a single financial center per country/legal framework. For more see Crane's The Global Financial System: A Functional Perspective. Boston, Philadelphia, Chicago all were possible financial centers for the United States the Federal Reserve board has an very good set of financial history of specific cities that would help you to understand how New York edged out these other cities.

A good question to ask is why is Toronto a financial center when New York is so close to it. I'm sure there are a few papers on this but can't think of any off the top of my head. What you will see is that within any given country there will always be one center that is far larger than all of the others.

Financial centers come and go. Fifty years ago Tokyo per Reed's The ascent of Tokyo as an international financial center was just becoming Asia's financial center. Paris and Hamburg were important financial centers in the middle ages due to their trade fairs and the settlement of bills.

Kindleberger's The formation of financial centers: a study in comparative economic history while a it dated is a very readable overview of the entire process of the formation of financial centers if you want more information.

In college I had a class about colonialism and one of the thing the professor mentioned was that London became the a global finance center because of it being the capital of the largest empire in the world. Think of how you play a game like civilization, you take cities or areas that hold needed resources so that you can use them either to build an army, get a bonus points, or gain wealth. Now lets put that in real world examples. The British used raw goods from their colonies to create manufactured goods and then sold them back to their colonies. However, that did not stop them from exporting to other countries. The British could effectively shut off any other countries goods being sold to their colonies as well. The length of time and the wealth of their colonies eventually caused the British Empire to become very wealthy. The wealth allowed them to create financial institutions that soon became entangled in the global market. Whats more, the British had an experience with financial bubbles and had promised to pay in full (I cant remember the name of this panic, if anyone could help me out here it would be appreciated).

Now in the case of New York, it was the location of large numbers of investment banks and dominance of the US Stock Market and surpassed any other US city. The extended period of peace, lack of costly huge wars, and a period of huge economic growth allowed New York to start mirroring Londons trend. The two are very different though, London remains the leading global foreign exchange center, while New York (Wall Street really) is brilliant at financial engineering, offering market innovations, and new financial products (what crafty new things can you do with you money to make more money). Note its this kind of practice is not always safe and can create huge growth but also huge loss. This is a super dumbed down version of things like the dot com bubble and the start of the Great Recession.

Now, why didnt other countries come into power from the European continent? It was simply the fact that they didnt have the right mix of events. Spain never could create a secure empire, they simply pulled the wealth out of their colonies, France would have been a good candidate but with the Mississippi Bubble lots of people didnt have faith in the French government. That does not mean these other cities werent financial powerhouses, they just simply werent to the scale of London, and eventually New York.

Source: For the first half of the first paragraph, that was from memory from a college class about colonialism. the second half of the first paragraph and onward is from: Sassen, Saskia. "Global Financial Centers." Foreign Affairs , Vol. 78, No. 1 (Jan. - Feb., 1999) , pp. 75-87