Martingale Measures Depend on the Chosen Numeraire

Summary

The answer introduces the relationship between a chosen numeraire and the probability measure used for pricing. Its central idea is that, under the measure associated with a numeraire, the price of another asset divided by that numeraire has no expected drift. This gives a practical way to formulate risk-neutral probability questions in a one-period up-and-down model.

For the two cases raised by the question, the answer suggests setting the stock-to-bond ratio or the bond-to-stock ratio to have zero expected drift under the corresponding measure, then solving for the implied probability of each move. It is a brief hint rather than a worked solution: no equations, numerical example, or explanation of measure-change assumptions are provided. The note is most useful as a conceptual pointer for studying numeraire-based pricing, and readers need further material to carry out the derivation rigorously.

Key ideas

• A numeraire determines the probability measure under which normalized asset prices are martingales.
• The ratio of an asset price to the chosen numeraire has no expected drift under the associated measure.
• In a discrete up-and-down model, the martingale condition can be used to solve for implied move probabilities.
• Changing the numeraire changes which price ratio is treated as driftless.

Tags

# What is martingle measure with risk free asset in numeraire or stock price in numeraire


# What is martingle measure with risk free asset in numeraire or stock price in numeraire












What is martingle measure with risk free asset in numeraire or stock price in numeraire

## Answer by dm63 (score 2, accepted)

https://quant.stackexchange.com/a/39278

First of all it is martingale not martangale. Secondly it is numeraire not numerator. It sounds like you need to study the basics of risk- neutral pricing.

A hint would be that the ratio of two asset prices has no expected drift in the appropriate measure. You are supposed to find the implied probability of the up- move and the down move assuming in (b) that the ratio of stock price to bond price has no expected drift and in (c) that the ratio of bond price to stock price has no expected drift.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.