= Mathematical finance
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<Mathematical finance> uses <probability theory>, <linear algebra> and <mathematical optimization> to study <financial markets>, <portfolio> decisions and <contingent claim> prices. One question asks which <portfolio> best serves a specified objective; another asks what price a <contingent claim> must have to exclude <arbitrage>. The latter can have an exact answer even when future <stock> prices are uncertain.
The central idea is <claim replication>: construct a <self-financing portfolio> delivering the same <financial payoff> as a contract in every possible state. The <law of one price> then identifies the contract's price with the <portfolio>'s initial cost. The <binomial market> makes this beautiful connection between uncertainty and <linear algebra> visible using just two equations. <Risk-neutral pricing> expresses the same cost as a discounted <expectation>.
The one-step example below fixes today's <stock> price at 100, terminal prices at 120 or 80, and the <bank account>'s gross growth factor at 1.05. A <European call option> with <strike price> 100 pays 20 or zero. Its <replicating portfolio in a binomial market> costs $250/21$, independently of the physical up <probability>, provided both states remain possible. These are conclusions within the specified <financial market> model.
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