MATH EXPLORERS' CLUB Cornell Department of Mathematics 


 10. Short-selling prohibition  
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Short-selling prohibition

Massive short-selling is a practice that is often observed after the burst of a price bubble. Examples are the Dutch Tulip-mania in the seventeenth century, the U.S. stock price crash in 1929, the NASDAQ price bubble of 1998-2000 and more recently the housing price bubble. Since the practice of short-selling is alleged to magnify the decline of asset prices, it has been banned and restricted many times during history. As such, short-selling bans have been commonly used as a regulatory measure to stabilize prices during downturns in the economy. The most recent example was in September of 2008 with the prohibition of short-selling by the U.S. Securities and Exchange Commission (SEC) for 799 financial companies in an effort to stabilize those companies. At the same time the U.K. Financial Services Authority (FSA) prohibited short selling for 32 financial companies. On September 22, Australia enacted even more extensive measures with a total ban of short selling.

However, short-selling prohibition is not only seen after the burst of a price bubble. In certain cases, the inability to short-sell is inherent to the specific market. In the housing market, for example, primary securities such as mortgages cannot be sold short. Additionally in some of the developed markets and most of the emerging markets around the world short-selling securities is not feasible. In the table below we highlighted, as of 2005, the developed markets were short-selling is not feasible or illegal and listed the emerging markets where short-selling is feasible. Also, in most of the examples presented so far, the ability to short-sell the risky assets is crucial in our arguments. In this section we will state the fundamental theorem of asset pricing when the risky asset in the model cannot be sold short and explore some of its consequences. In particular, we will see that in this case the market is no longer complete.

Example: Dollar Vs Euro with short-selling prohibition

The arbitrage opportunity presented in example 2 of lesson 1 involves short-selling the Euro. Recall that in this example the dollar's interest rate is 10% and the possible rates of return of the risky asset are d=-0.2 and u=0.1. Suppose now that the Euro cannot be sold short. One might ask then, does there exist an arbitrage opportunity in this market that does not sell the Euro short? In order to answer this question we argue as follows. Suppose that (x1,y1) is a trading strategy (see lesson 2) with nonpositive initial value, i.e. x1+1.5y1≤0, and 0≤y1 (this accounts for the fact that the Euro cannot be sold short). The set of these strategies can be represented by a cone in the plane (see figure below). On the other hand the set of strategies that yield nonnegative outcomes under both states of the world corresponds to strategies (x1,y1), such that 1.1x1+1.65y1≥0 and 1.1x1+1.2y1≥0. These strategies can be represented by a cone in the plane as well (see figure below).

The intersection of these cones is the strategy (0,0) which is not an arbitrage opportunity (no strictly positive profit with positive probability). We conclude then that prohibiting short-selling the Euro makes this market arbitrage free.
Activity 1
  1. Consider example 3 of lesson 2 with time horizon equal to T=1. Arguing as above, prove that if short-selling the GM stock is not allowed then the market is arbitrage free.
  2. Can you find conditions on the dollar's interest rate r under which the market of part a) remains arbitrage free?


The Fundamental Theorem under short-selling prohibition

When the conditional expectation under a probability Q of a process at a future time is less than or equal its present value, the probability Q is called a supermartingale measure. If the conditional expectation of the process at a future time is greater than or equal its present value, the probability Q is called a submartingale measure. A probability that is a supermartingale and a submartingale measure simultaneously is a martingale measure.
In the previous three lessons, when studying extensions of the model exposed in module 1, we have used the fundamental theorem of asset pricing as presented in lesson 3. The fundamental theorem of asset pricing under short-selling prohibition is somehow independent of this formulation. Suppose that we have the binomial model of lesson 1 with 0 < p <1, but short-selling the risky asset is not allowed whatsoever. This model is arbitrage free if and only if there exists an equivalent probability measure, Q*, such that at any time t

which holds if and only if d < r.

Intuitively, the condition above says that the expectation of the discounted price process St+1/(1+r)t+1 under Q* is less than or equal St/(1+r)t, hence the discounted price process tends to decrease according to Q* (see tangent), but since short-selling is prohibited the investors cannot "arbitrage" this price decline and the market is arbitrage free. Observe that in the example above d=-0.2< r=u=0.1 .
Activity 2

Verify the answer obtained in part b) of activity 1 by using the Fundamental Theorem under short-selling prohibition. Does anything change if we assume that GM's stock pays dividends?


Market incompleteness

Recall from lesson 5 that the binomial model exposed in module 1 is complete, i.e. any contingent claim can be perfectly replicated by trading on the riskless and risky assets. Under no arbitrage, this property still holds in the extensions presented in lessons 7 through 9. However when short-selling is not allowed the model is no longer complete. The payoffs (under the two possible states of the world) that are obtained with strategies that do not sell the risky asset short can be represented by a semi-plane as shown in the figure below.

As we saw in lesson 5 the price of a contingent claim in a complete market is equal the cost of replication which is in turn equal to the discounted expectation under the unique risk neutral probability. Under short-selling prohibition there is not a unique risk neutral probability, because there are infinitely many supermartingale measures (see theorem and tangent above), and some of the claims cannot be perfectly replicated, one might ask then what is the fair price of a contingent claim when short-selling is restricted? There is not a unique answer to this question but usually the contingent claims are priced by the minimum cost of a strategy with payoff greater than or equal the payoff of the claim under any state of the world (this is commonly known as the minimum super-replication price). It turns out that if the payoff of the claim is C this price at time 0 is equal to

For instance, in the model exhibited in the beginning of the lesson, the price of a call option with maturity of one day and strike price $1.5 has payoffs of $0.15 and $0 if the price increases and decreases respectively. Any positive probability q below (r-d)/(u-d)=1 is a supermartingale probability and hence

Activity 3

For the model presented in the beginning of the section find the prices of a put option with strike price $1.5 and a straddle.


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