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Martingales and Numeraires

Kerry Back’s excellent Springer book entitled “A Course in Derivative Securities” makes a remark about the numeraire/martingale way of looking at options pricing:

“It seems worthwhile here to step back a bit from the calculations and try to offer some perspectives on the methods developed in this chapter. The change of numeraire technique probably seems mysterious. Even though one may agree that it works after following the steps in the chapter, there is probably a lingering question about why it works. The author’s opinion is that it may be best simply to regard it as a ‘computational trick’. Fundamentally it works because valuation is linear. … The linearity is manifested in the statement that the value of a cash flow is the sum across states of the world of the state prices multiplied by the size of the cash flow in each state. The change of numeraire technique exploits the linearity to further simplify the valuation exercise … After enough practice with it, it will seem as natural as other computational tricks on might have learned.”

Thinking of it as a computational trick indicates how unintuitive the result is. Anyone have a better introductory proof?

Published in Models