Question

5. Consider the single period binomial model as in Section 1.2.2. Suppose that d <1+r <u. Show that if there exists an arbitrage opportunity (as in Definition 1.5), then one can find an arbitrage opportunity with V = 0. This means that there is no net cash flow at time 0. (Note: This is a step in the proof of Proposition 1.7 which you should go through carefully.)

Answer 1

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Explanation:

Proof. Let's start with d21+ r then the following strategy is an arbitrage: Begin with zero initial endowment and at time t= 0 borrow the amount So from the money market in order to buy one share of the stock. Even in the worst case of a tail in the coin toss (i.e., when S = Sod) the stock at time t = 1 will be worth Soda So (1 + r), which is enough to pay off the money market debt. Also, the stock has a positive probability of being worth strictly more then the debt's value, since y >d21+ r, and thus Sou > So(1 + r). If u S1 + r then the following strategy is an arbitrage: • Begin with zero initial endowment and at time t = 0 sell short one share Of the stock and invest the proceeds So in the money market account. Even in the worst case of a head in the coin toss (.e., when S = Sou) the cost Si of repurchasing the stock at time t = 1 will be less than or equal to the value So (1 + r) of the money market investment at time t = 1. Since d <u51+ r, there is also a positive probability that the cost of buying back the stock will be strictly less than the value of the money market investment. We have therefore established the following implication: No arbitrage = d <1 + r <u. and x = V0(x, 0) = 0 (that is, a trading strategy needs no initial investment) because, an arbitrage as a definition (or a free lunch) is a trading strategy that begins with no money, has zero probability of losing money, and has a positive probability of making money.