- Discrete Time Market Models 81
Exercise 4.6
Given the bond and stock prices in Exercise 4.5, is there an arbitrage
strategy if short selling of stock is allowed, but the number of units of
each asset in a portfolio must be an integer?Exercise 4.7
Given the bond and stock prices in Exercise 4.5, is there an arbitrage
strategy if short selling of stock is allowed, but transaction costs of 5%
of the transaction volume apply whenever stock is traded.4.1.3 Application to the Binomial Tree Model
We shall see that in the binomial tree model with several time steps Condi-
tion 3.2 is equivalent to the lack of arbitrage.
Proposition 4.2
The binomial tree model admits no arbitrage if and only ifd<r<u.
Proof
We shall begin with a one-step binomial tree. This will then be used as a
building block in the case of several time steps.
One step.Suppose thatr≤d.If so, then:
- Borrow 1 dollar at the risk-free rate.
- Buy 1/S(0) shares.
That is to say, construct a portfolio withx=1/S(0) andy=−1, the value
of which isV(0) = 0. After one step, eitherS(1) =S(0)(1 +d)andV(1) =
−r+d≥0, orS(1) =S(0)(1 +u)andV(1) =−r+u>0, leading to arbitrage.
Suppose thatu≤r. In this case:
- Buy one bond.
- Sell short 1/S(0) shares.
The resulting portfolio withx=− 1 /S(0) andy= 1 will once again have initial
valueV(0) = 0. After one step this portfolio will be worthV(1) =r−u≥0if
the stock price goes up, orV(1) =r−d>0 if it goes down, also realising an
arbitrage opportunity.
Finally, suppose thatd<r<u. Every portfolio withV(0) = 0 must be
of the formx=a/S(0) andy=−afor some real numbera. Consider the