Suppose we are in a one-period "CAPM" world with 5 risky securities, whose returns are normally distributed.There is also a risk-free security.Are markets complete in this example? If not, what other assumptions must be made to make them complete?More generally, are markets complete in the simple one-period CAPM?Please -- don't give me fancy talk -- just the straight answers to these 2 questions.

Markets are complete if all possible future state situations/assets - including shorting assets and creating derivatives on the risky assets - can be constructed using the existing assets. It is best considered a theortical construct against which to measure a real market.Moreover, completeness is not dependent on the number of risky assets in the market. Consider put-call parity. A put can be synthesised by a combination of the risk-free asset, the asset and a call (long the call, short the stock and invest the strike at risk free rate). If calls on the asset are not available in the market, then it can't be considered complete.

No, how you will be able to replicate, say call at the money of the first asset? (there are too many possibilities)The one period market is complete iff you have n possible future states and n-1 differentlinearly independent risky assets. If You have a continuum...(well, a countable number of assets will suffice)HTHP.S.And pray, what does it _precisely_ mean return? (Simple but not trivial question)

QuoteOriginally posted by: rogermcMarkets are complete if all possible future state situations/assets - including shorting assets and creating derivatives on the risky assets - can be constructed using the existing assets. It is best considered a theortical construct against which to measure a real market.This is a generic general statement in fancy language. It does not answer my question about CAPM and those 5 assets.

QuoteOriginally posted by: pjakubenashow you will be able to replicate, say call at the money of the first asset? (there are too many possibilities) The one period market is complete iff you have n possible future states and n-1 differentlinearly independent risky assets. If You have a continuum...(well, a countable number of assets will suffice). You have touched on the reasons why I am confused about the 5 asset CAPM example. First, it is often said that markets ARE complete in CAPM because there is no ambiguity in the price of those 5 assets. This makes sense. But then, my question is, why are markets complete when you have 5 assets and a continuum number of states (e.g., number of states is much greater than number of assets)?

The basic definition of complete markets is if you have the same number of linearly independent securities and states of nature (or future states). If this does not hold, then you do not have complete markets. Thus, if your 5 risky assets are "pure" securities, that is, they pay a value of $1 in one state and zero in another state, and you do not have assets that payoff the same in the same state, then you need to have five states of nature to have a complete market. The risk-free security's return can be replicated by a linear combination of the five risky assets, so this security is redundant. For example, if you have asset1=[1 0 0 0 0], asset2=[0 1 0 0 0], ..., asset5=[0 0 0 0 1], then a risk-free security that paysoff [x x x x x] can be replicated by a linear combination of assets 1 through 5. However, one of the assumtions of the CAPM, if I remember correctly, is that investors have the same beliefs or expectations (or something like that). Hence, in order for this to happen, you must have (at least sufficiently) complete markets. To see why, if you do not have complete markets, then the introduction of a new security will change the portfolio mix of some investors (because some would prefer this new security and sell other securities that they own in order to attain their mean-variance efficient portfolio) and, consequently, the supply and demand of assets in the market will also change. This would then lead to a change in the price of the securities, and homogenous expectations will not be attained.

One should note that there are two definitions that are not the same1) linearly independent securities = states of nature (or future states)2) all possible future state situations/assets - including shorting assets and creating derivatives on the risky assets - can be constructed using the existing assetsYou can have a market like Shanghai which has a no-short rule, in which case you could have a situation in which the number of linearly independent securities = states of nature, but which because a regulatory rule, you cannot construct an asset out of existing assets.The terminology I use to distinguish the two is underdetermined if condition 1) is violated, and externally constrained if condition 1) holds and condition 2) is violated. It's important to distinguish the two different definitions of incomplete, because I have read that the no-short rule on Shanghai makes Black-Scholes pricing impossible because it makes the market incomplete. This confuses violating condition 2) with condition 1), and there is an argument that if the law of one-price holds that a no-short condition gives you Black-Scholes prices if GBM holes.

can't you argue that for large banks the "no short" rule does not matter - since they can internally "borrow" shares they hold on behalf of customers when they need to short on another transaction?hence the market is still complete...since banks will be trading most actively anyway and tend to influence the market more than individual traders.

QuoteOriginally posted by: twofish 1) linearly independent securities = states of nature (or future states)It is true that IF #state = #indep securities, thenmarkets are complete. However, I believe this is not a REQUIREMENT for complete markets. There are some textbooks that will say that markets are complete whenever there are no trading constraints that forbid traders from constructing their own derivative securites to complete the market (if the market originally just one asset). In other words, markets can be effectively completed by trading even if one starts with just one or two assets and a continuum of states.

QuoteOriginally posted by: csaOne of the assumtions of the CAPM, if I remember correctly, is that investors have the same beliefs or expectations (or something like that). Hence, in order for this to happen, you must have (at least sufficiently) complete markets. To see why, if you do not have complete markets, then the introduction of a new security will change the portfolio mix of some investors (because some would prefer this new security and sell other securities that they own in order to attain their mean-variance efficient portfolio) and, consequently, the supply and demand of assets in the market will also change. This would then lead to a change in the price of the securities.What you say makes sense. The introduction of a new security changes the market portfolio and, hence, changes all beta values.Therefore, CAPM with a finite number of assets and a continuum of states is never complete. However, this makes me queasy: I have never found any authoritative reference that states that "markets are incomplete in a CAPM world with a finite number of assets."

Can you tell me what you mean by a "CAPM" world? In your first post, you mentioned:"Suppose we are in a one-period "CAPM" world with 5 risky securities, whose returns are normally distributed.There is also a risk-free security."Aside from the "5 risky securities", you have basically identified several of the assumptions that were used to derive the theoretical CAPM as originally proposed by Sharpe. You can also add to the list of assumptions homogenous expectations, perfectly divisible assets, and frictionless markets (I may have left out one or two additional assumptions). There is nothing in the assumptions used to derive the CAPM that states a continuum of future states. If there were a continuum of states and you have a finite number of assets (e.g. 6) then, by definition, the markets are not complete.So, I think in your initial post, to make the markets complete you would need to assume x number of linearly independent securities among the 6 that you have (5 risky + 1 risk-free) and x number of future states. On a general sense, I think that the homogenous expectations argument that I made in my previous post would be a likely answer to the market completeness of the CAPM. The reason is that the other assumptions are needed to show different things. For example, normally distributed assets (or quadratic utility function) is used so that we can use mean and variance. The risk-free security with unlimited borrowing or lending assumption is for two-fund separation. CAPM is a one-period model according to the theory.As an aside, I want to offer my opinion regarding your post on derivative securities and complete markets. Often derivative securities are used to "complete the market". For example, a call option has a nonnegative payoff only when certain states occur and zero in other states. Hence, you can construct call options that payoff only in one state. Therefore, with a series of call options, you can create the required number of linearly independent securities to complete the market. However, the definition of complete markets still holds, which is the same number of linearly independent securities to future states.

QuoteOriginally posted by: tibbarcan't you argue that for large banks the "no short" rule does not matter - since they can internally "borrow" shares they hold on behalf of customers when they need to short on another transaction?Heh. Heh. Heh. Yes you can argue that and the derivation that convinced me that no short had no effect on option prices, involves a securities company just doing that.Later I found that's thats exactly what security company traders in Shanghai do. :-)Since these transactions are internal, hard to police, and don't have public policy consequences (i.e. you can do this to allow your traders to short, but not your clients and if things go bad, then its your problem). This sort of thing is not generally forbidden. However there was a situation in which different entities acting under the same QFII allocation were trading shares among themselves, and the regulators put a stop to that, because there was a public policy issue involved (i.e. bad risk management in one firm would infect another one).Shanghai has some interesting rules regarding cancellation and creation of warrants that have some QF implications that I'm trying to figure out (unless someone can find a paper that has already done it). Basically they require superhedging of their warrant issues, and I'm trying to figure out what the optimal strategy is for a securities company to issue or cancel a warrant. Right now, I'm stuck because I don't have the visualization tools to look at my data, so I'll be spending the next few months writing them.Quotehence the market is still complete...since banks will be trading most actively anyway and tend to influence the market more than individual traders.It doesn't matter what terminology people use as long as it is unambigious, but unfortunately "in/complete market" isn't. I got into a screaming flame war a few months back with people who used the argument "shanghai markets don't follow Black-Scholes because the market is incomplete because of the no-short rule." This argument is wrong because "complete market" is being used in two different senses.If the market is "externally constrained" you can calculate the Black-Scholes price without the constraints and then put the constraints back in. If the market is "underdetermined" you can't get that far since there is no unique risk-neutral measure.

QuoteIf the market is "underdetermined" you can't get that far since there is no unique risk-neutral measure.Perhaps you should look at determining a unique risk-neutral measure according to a suitable utility function?

QuoteOriginally posted by: csaCan you tell me what you mean by a "CAPM" world? So, I think in your initial post, to make the markets complete you would need to assume x number of linearly independent securities among the 6 that you have (5 risky + 1 risk-free) and x number of future states. On a general sense, I think that the homogenous expectations argument that I made in my previous post would be a likely answer to the market completeness of the CAPM. The consensus I am getting from people here is that with a finite number of (primitive) assets and an infinite number of states, 1-period CAPM is incomplete.QuoteThe reason is that the other assumptions are needed to show different things. For example, normally distributed assets (or quadratic utility function) is used so that we can use mean and variance. The risk-free security with unlimited borrowing or lending assumption is for two-fund separation. hmmm. I thought 2-fund separation refers to the fact that the efficient frontier is spanned by any 2 risky portfolios on the frontier. This is differentfrom what you're saying about the CML being spanned by the risk-free asset and the market portfolio.QuoteAs an aside, I want to offer my opinion regarding your post on derivative securities and complete markets. This would seem to be irrelevant in 1-period capm with an infinite number of states since you can never span an infinite number of states with a finite number of securities in one trading round.

QuoteOriginally posted by: KackToodlesQuoteOriginally posted by: csaCan you tell me what you mean by a "CAPM" world? So, I think in your initial post, to make the markets complete you would need to assume x number of linearly independent securities among the 6 that you have (5 risky + 1 risk-free) and x number of future states. On a general sense, I think that the homogenous expectations argument that I made in my previous post would be a likely answer to the market completeness of the CAPM. The consensus I am getting from people here is that with a finite number of (primitive) assets and an infinite number of states, 1-period CAPM is incomplete. The issue of homogeneity seems quite irrelevant. QuoteThe reason is that the other assumptions are needed to show different things. For example, normally distributed assets (or quadratic utility function) is used so that we can use mean and variance. The risk-free security with unlimited borrowing or lending assumption is for two-fund separation. hmmm. I thought 2-fund separation refers to the fact that the efficient frontier is spanned by any 2 risky portfolios on the frontier. This is differentfrom what you're saying about the CML being spanned by the risk-free asset and the market portfolio.QuoteAs an aside, I want to offer my opinion regarding your post on derivative securities and complete markets. This would seem to be irrelevant in 1-period capm with an infinite number of states since you can never span an infinite number of states with a finite number of securities in one trading round.