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## Change of Numeraire

There is another way to view risk-neutral probabilities. Recall in the derivation of the option price, we converted the option on 100 shares of stock into ownership of 40 shares of stock minus \$2,000. In effect, we expressed the option price in terms of shares of stock instead of dollars. The technical term for this is a change of numeraire. By stating prices in shares of stock rather than dollars—in other words, by using the stock instead of dollars as money—we make the option riskless. A share of stock is always worth one share of stock, whatever the price is in dollars. Since the option price can be stated as a fixed number of shares, at least in theory and for an instant, the option is also riskless in the stock numeraire.

This is a general technique in derivative pricing, and financial engineering in general. Picking the right numeraire can make a hard problem easy (and we saw in Chapter 1 that it can reverse the answer). It can make a problem that required estimation of probabilities into one that can be answered directly from observation. The numeraire is what connects frequency to degree of belief. And it's fundamentally a concept of money. Looking at it another way, the way to make degree of belief correspond to frequency is to measure things in the right units. Reversing that logic, the thing you choose for your numeraire fundamentally alters your interpretation of statistical evidence. That's why the nature of money is so crucial to an economy. Of ...

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