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## Monday, January 13, 2014

### present value of 33 shares, using share as numeraire

We all know that the present value of \$1 to be received in 3Y is (almost always) below \$1, basically equal to exp(-r*3) where r:= continuous compound risk-free interest rate. This is like an informal, working definition of PV.

Q: What about a contract where the (no-dividend) IBM stock is used as currency or "numeraire"? Suppose contract pays 33 shares in 3Y... what's the PV?

%%A: I feel the PV of that cash flow is 33*S_0 i.e current IBM stock price.

I feel this "numeraire" has nothing to do with probability measure. We don't worry about the uncertainty (or probability distribution) of future dollar price of some security. The currency is the IBM stock, so the future value of 1 share is exactly 1, without /uncertainty/randomness/ i.e. it's /deterministic/.

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Similarly, given a zero bond will mature (i.e. cash flow of \$1) in 3Y, PV of that cash flow is Z_0 i.e. the current market value of that bond.