Postulates on Finance - YouTube

Hao (Well), Postulates on Asset Prices

Asset price is the key variable in financial market.

So, First and foremost

the marketable primitive security must have a single price for trading.

Thus, the first postulate:

Each primitive security has a unique price.

And here, We let X be the collection of the (payoffs of the) primitive securities

thus we have a function

this function (maps) from the payoff space to a real number

and we know

Portfolio is an essential concept in finance

and (where) a collection of securities are held and treated as a whole.

which (is) called a portfolio

The payoff and price of a portfolio is formed by linear combination

Thus, the second postulate: Law of portfolio

here, the payoff of a portfolio, (equals to) the linear combination of payoffs

payoffs of its constituent asset

similarly, the price of a portfolio

equals the linear combination of prices of its

again, constituent assets

Thus, we have this

relationship

this is a linear function

and, immediately, we have this result, through origion

which means, if a(n) asset of a portfolio

has no value in the future, it costs nothing now

and from this postulate, or this assumption

we have the following understandings

first, the payoff space is a vector space

because it's a, say, eh---, linear combination

(thus) forms a vector space

for convenience, we still use the symbol X

and the pricing function, (Weierstrass) P, now,

(is) extended to the whole payoff space

and, we here can reach this conclusion

the pricing function is linear and zero-axial

here, pass the origon

n we've introduced the concept of weak-positive

weak-positive is, first, non-negative

non-negative , and second, cannot always be zero

or can not be zero almost surely

and we know, all primitive assets are weak-positive, or called

in financial words, limited liability

thus, from the market practice, we have this potulate

for any limited liability portfolio, the price is positive

this is because

A claim right entails obligations

If you have some rights, you must have some obligations

thus, if the payoff is weak-positive

then the price must be positive

this is a restriction on the pricing function

otherwise, you have the (weak-)positive payoff

and the price is zero, then there is an arbitrage opportunity

because you get something out of nothing

right?

and here, this postulate

this postulate reveals some market convensions

first, all of the primitive securities are limited liability

thus, their payoffs are weak-positive

and their price must be positive

and there (it) is very important (that)

this one, one, the payoff of one, is a risk-free asset

and the price of this one must be positive

and if this is positive

the risk-free interest rate is well defined