# No-Arbitrage condition

No Arbitrage states the inadmissibility of Free Lunch in applying a structure (metric?) to the prices.

Considering a title with price
and cash flow
, or every time
, where
it is the discretized time
.

Then we have the two No-Arbitrage condition:

- Weak No-Arbitrage : if
- Strong No-Arbitrage: if
- and

Let’s suppose a world where I can borrow money with fixed interest rate
, which will be the price
of a product with interest
over one year?

I can borrow money until I expect to be payed back by the revenue A, so the cash flow at and of year would be 0, which is
. The price
of the portfolio is given by the price of the product minus the borrowed money at time 0:
.

Weak No-Arbitrage fixes then the price
giving
.

Same can be applied for selling, giving
, thus the meeting of buying and selling is, unsurprisingly
.

*But, if we use the strong arbitrage condition -satisfied, since we assumed the interest of the product
- we get that
for the buyer, and
for the seller (?). So buying and selling cannot be met. And why should they? You gain nothing by selling or buying if you can lend and borrow money indefinitely for a fixed interest rate, thus is better not to do anything at all…*