HiI am trying to get my head around the use of numeraires in option pricing and would appreciate if you could comment on my chain of thought (or lack of it!) Thanks a ton in advance!Q. Why do we need to express the asset 'S' in terms of the numeraire 'P' and then find the measure under which 'S/P' is a martingale? Why not use the measure under which 'S' is a martingale?A. Any asset 'S' needs to be expressed in terms of a numeraire 'P' because two assets ('S' and 'P') are required to create the (dynamic) hedging portfolio which forms the basis of the 'no-arbitrage' argument for option pricing. Q. Why not find a common measure under which both 'S' and 'P' are martingales?A. It is(?) not possible to find a common measure under which both S and P are martingalesQ. When we express 'S' in terms of 'P', then are we using 'P' to create the dynamic hedging portfolio? A. ?Q. If yes, then are there any restrictions on what we can choose as 'P' (apart from the restriction that P should be an asset)? i.e. Can the hedging portfolio be created using any two assets as long as one of them is S?A. ?Q. When we choose P as the numeraire, instead of the risk-free asset, we still assume that we are in a risk-neutral world. Does this mean that risk-neutrality has nothing to do with choosing the risk-free asset as the numeraire?A. ?

Can someone pls help!! Thanks in advance

Ok, let me give a shot, some of the questions can be most easily understood thinking in terms of binomail trees. In binomial tree, you are using risk free cash as numeriaire:Q1: in this case, the risk-neutral probability is chosen so that both (stock/cash) and (option/cash) are martingale. And by the construction of binomial tree, you have an no-arbitrage relationship automatically.Q2: It is not possible, in any measure, return of cash is fixed in all state of the future world, so it is not an martingale no matter how you assign your probability.Q3: Yes, you can use P as a numeriare, you automatically get a hedging portfolio in binomial tree.Q4: P has to be chosen so that the market is complete, ie, payoff matrix is invertable, ie, you can get valid probability from the payoff of P and S. For example, if P is perfectly correlated with S that you are trying to price, you cannot get meaningful probability in binomial tree, in another words, the market is not complete if S and P are perfectly correlated -> state price does not exist.Q5: risk-neutral probability means people's risk aversion preference has been included in the assigned probability. For this to work, the market has to be complete, and by no-arbitrage argument, under these condition, any risk aversion preference will give same seucirty price as if people are risk-neutral, which is the easest to price by just taking expectation. That is why we like risk-neutrality. So if market is complete, ie, state price exist and is unique, it does not matter which numeraire you use, they have to be all consistent with same set of state price. But the assigned probability under different numeraire is of course different. Thinking in binomial tree, you could use option and stock to create cash, the probability is certainly different from the normal probability that is using cash and stock to replicate option. However, the state price are the same under those two different approaches.