Complete Market with 2 Stocks

[stochan]

Here is a question for you guys...

Suppose I have a one-period world, with 2 stocks that can either go up or down (so at time 1 there are 4 states of the world). I also have a risk free asset. Can such a market be complete ? How can I price an option that gives me 1 if both stocks go up and 0 otherwise ?

Assume that the no arbitrage conditions are met

 

[euroazn]

Here is a question for you guys...

Suppose I have a one-period world, with 2 stocks that can either go up or down (so at time 1 there are 4 states of the world). I also have a risk free asset. Can such a market be complete ? How can I price an option that gives me 1 if both stocks go up and 0 otherwise ?

Assume that the no arbitrage conditions are met

What kind of information is available about the stocks i.e correlation, probability of rise, etc.
I do think the market is complete though, I can't see why not.

 

[stochan]

What you know is, for each stock, the price up and the price down. You also know the yield of the bond, and that there is no arbitrage. You will have proven that the market is complete once you are able to replicate every option. Can you find the price of a digital that gives 1 if they both rise and 0 otherwise ?

In general I claim that such a model cannot be complete.

 

[euroazn]

Well generally you can't synthesize the digital with just the two stocks and the risk-free, if that's what you mean.

Suppose our portfolio is (a,b,c), or a of stock 1, b of stock 2, and c of the risk free.
There are 4 state equations (for 3 unknowns!) that need to be satisfied if our portfolio replicates the digital:
Equation 1: a(stock 1 up price)+b(stock 2 up price)+c(risk free up)=1
Equation 2: a(stock 1 up price)+b(stock 2 down price)+c(risk free up)=0
Equation 3: a(stock 1 down price)+b(stock 2 up price)+c(risk free up)=0
Equation 4: a(stock 1 down price)+b(stock 2 down price)+c(risk free up)=0

Equations 2&4 imply a=0; Equations 3&4 imply b=0. Then c=0 and we have a problem.

There's no dynamic replication since we only have one period.

So in that sense the market isn't "complete".

 

[stochan]

Great, it is my same conclusion.

Now, the major question is: do you agree that we have proved that any discrete-time market with more than one stock (plus the risk-free asset) is incomplete?

 

[ChrisN]

Sorry to be so naive, but you have proved the above on the condition that both stocks go up. How valuable is that proof?

It is a very specific example isn't it? I might be wrong but I can't really get what you are trying to prove.

If you could explain to me (so I can learn as well) that would be great!

 

[stochan]

Where did I say that they are both going up?
What I say is that, if:

- we have at least 2 stocks
- we have a discrete time market (so it exists time t1)

then we can write an option on 2 stocks, expirying at time t1, that pays off 1 if they both go up and 0 otherwise. Such an option cannot be replicated, and so the market is incomplete (completeness means I can replicate everything - for details please give a look to wikipedia)

 

[ChrisN]

Well the option pays off only if both go up, that's what I am saying.

 

[stochan]

OK. And why don't you like it ? To prove incompleteness it is enought to find one option that can't be replicated...

 

[ChrisN]

Don't get me wrong... it's not that I don't like it. I am just trying to understand!

 

[stochan]

 

[euroazn]

Don't get me wrong... it's not that I don't like it. I am just trying to understand!

He's using the definition of an incomplete market that states that a market is incomplete if there exists a gamble which you cannot buy; stochan provided such a gamble.

stochan
Yes I suspect you are right; it shouldn't be too hard to prove with simple linear algebra.
That doesn't prove the real market is incomplete though
That's because that option can be one of the assets provided!

 

[stochan]

Yes I suspect you are right; it shouldn't be too hard to prove with simple linear algebra.

Well you have already proved it! The system you wrote down doesn't have solutions

That doesn't prove the real market is incomplete though.

Sure. Real markets are incomplete for other reasons...

That's because that option can be one of the assets provided!

No one was claiming that no option can be replicated!

 

[euroazn]

Well you have already proved it! The system you wrote down doesn't have solutions

Yeah, but strictly speaking we haven't proved the general case for more than 2 stocks.