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# Mathematics Achieves Command of Financial Risks

## Financial Derivatives

Consider two examples of financial derivatives: first, an introductory example from sports betting; and second, a more traditional one from the stock market.

In both examples, suppose you wish to receive a particular risky payoff.   Suppose you are willing to pay, today, for a contract that promises you that payoff in the future.  Unfortunately, nobody is offering to sell you such a contract.  Can you synthesize the desired contract from the underlying assets that are tradeable?

Example 1:  The Yankees are about to play the Dodgers in the World Series.  You want to receive \$100 if the Yankees win the World Series, and to receive -\$100 if they lose the World Series.   Unfortunately, nobody is offering to make such a wager with you.  But suppose that a casino allows you to place wagers on who will win each individual game of the World Series. Immediately before each individual game (1,2,3,4 and, if necessary 5,6 and 7), you are free to decide how much to bet on that game. Your goal is to win exactly \$100 if the Yankees win the series. Note \$99 is not good enough! If the Yankees lose the series, your loss should be exactly \$100. How much should you wager on each individual game, to create the desired World Series contract? (INSERT LINK TO APPLET HERE). For an explanation of the why this example works out, see the solution.

Example 2: You want to receive the dollar amount by which Amazon’s stock price exceeds \$120 at year end (and to receive nothing if it doesn’t exceed \$120).  Unfortunately, nobody is offering to sell you such a contract, known as a call option.  But suppose that the stock market allows you to buy and sell shares of Amazon stock.  How many shares should you buy/sell, in order to create the desired call option payoff max(Amazon-120, 0) at year end?  (INSERT LINK TO ANIMATION HERE?)

Example 2: A stock price has infinitely many possible paths.  We don’t know which one will actually occur.  Let F(x) be some given mathematical function.  Challenge: trade stock in such a way that your portfolio has final value F(final stock price), no matter what path occurs.  In other words, replicate the financial derivative which pays F(final stock price),

Recent developments in mathematics and computation have revealed how to manage various forms of risk, by trading in financial markets.  In particular, we now understand how to create/replicate/synthesize various financial derivatives (such as the \$100 World Series contract, and the Amazon call option) by holding carefully-chosen quantities of underlying assets (namely, the indivdual game wagers, and the Amazon stock, respectively).

Three of the major categories of risk traded in the derivatives markets are equity risk, interest rate risk, and credit risk

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