Introduction to No-arbitrage

Contracts, prices, and no-arbitrage

Consider the following contract

• Pay price $p$ at time $t=0$
• Receive $c_k$ at time $t=k,k=1,...,T$ Note that the cash flow c could be negative!

The no-arbitrage condition bounds the price $p$ for this contract

• Weak No-Arbitrage: $c_k\ge 0$ for all $k\ge 1\to p\ge 0$

• Strong No-Arbitrage: $c_k\ge 0$ for all $k\ge 1$ and $c_l>0$

  for some $l\to p>0$

Essentially eliminate the possibility of a free-lunch

The rationale for the weak no-arbitrage condition: Suppose $p<0$:

• Since $c_k\ge 0$ for all $k\ge 1$, the buyer receives $-p>0$ at time $0$, and then does not lose money thereafter. Free lunch!
• The seller can increase the price as long as $p\le 0$, and still have buyers available.
• Buyers will be willing to pay a higher price in order to compete.

Assumptions underlying no-arbitrage

The rationale for the strong no-arbitrage condition

• Suppose $p\le 0$.
• Recall that for some $c_l>0$ for some $l\ge 1$. Therefore, free lunch as long as $p\le 0$.
• We can only guarantee that $p>0$ but not the precise value! Implicit assumptions underlying the no-arbitrage condition
• Markets are liquid: a sufficient number of buyers and sellers
• The price information is available to all buyers and sellers
• Competition in supply and demand will correct any deviations from no-arbitrage prices

Pricing a simple bond

What is the price of a contract $p$ that pays A dollars in 1 year? Suppose one is able to borrow and lend unlimited amounts at an interest rate of $r$ per year.

Construct the following portfolio:

• Buy the contract at price $p$
• Borrow $\frac{A}{1+r}$ at interest rate $r$ Cash flows associated with this portfolio:
• Price of portfolio: $z=p-\frac{A}{1+r}$
• Cashflow in 1 year: $A-A=0$ Weak No-arbitrage: $c_1\ge 0$ implices price $z\ge 0$, i.e., $p\ge \frac{A}{1+r}$

Next, construct the following portfolio:

• Sell the contract at price $p$
• Lend $\frac{A}{1+r}$ at interest rate $r$ Cash flows associated with this portfolio:
• Price of portfolio: $z=p-\frac{A}{1+r}$
• Cashflow in 1 year: $-A+A=0$ Weak No-arbitrage: $c_1\ge 0$ implices price $z\ge 0$, i.e., $p\le \frac{A}{1+r}$ Two results together imply $p=\frac{A}{1+r}$. Surprise? The result relied on the ability to borrow and lend at rate r.
• What if borrowing and lending rates are different?
• What if borrowing and lending markets are elastic?