Arbitrage, Hedging, and the Law of One Price

This chapter is from the book

Understanding Arbitrage: An Intuitive Approach to Financial Analysis

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This chapter is from the book

This chapter is from the book 

Understanding Arbitrage: An Intuitive Approach to Financial Analysis

Learn More Buy

Hedging and Risk Reduction: The Tool of Arbitrage

Hedging Defined

A hedge is used to implement an arbitrage strategy. Thus, before we examine arbitrage more carefully, we must understand how a hedge works. We have all heard someone say that he or she did something just to "hedge a bet." In a strict gambling sense, this implies that an additional bet has been placed to reduce the risk of another outstanding bet. The everyday connotation is that an action is taken to gain some protection against a potentially adverse outcome. For example, you may leave early for an appointment to "hedge your bet" that you’ll find a parking place quickly. Another example is a college student’s decision to pursue a double major because he doesn’t know what jobs will be available when he graduates.

In investment analysis, a hedging transaction is intended to reduce or eliminate the risk of a primary or preexisting security or portfolio position. An investor consequently establishes a secondary position to counterbalance some or all of the risk of the primary investment position. For example, an equity mutual fund manager would not get completely out of equities if the market is expected to fall. (Just think of the signal that would send to investors.) The risk of the manager's long equity investments could be partially offset by taking short positions in selected equities, buying or selling derivatives, or some combination thereof.¹¹ This secondary position hedges the equity portfolio by gaining value when the value of the equity fund falls. The workings of such hedges are discussed next.

Often, an investor establishes a long asset position that is subsequently considered too risky. The investor consequently decides to partially or completely offset that risk exposure by taking another investment position that offsets declines in the original investment's value. A short position can be taken in the same asset that counterbalances the investor's risk exposure. The hedging transaction may be viewed as a substitute for the investor's preferred action in the absence of constraints that interfere with taking that action.¹² The constraint could be something explicit, like a portfolio policy requirement (such as in a trust) that an investor maintain a given percentage of funds invested in a stock. Alternatively, it could be a self-imposed risk-tolerance constraint where the investor wants to keep a stock with a profit but feels compelled to offset all or part of the position's risk using a hedging transaction. For instance, this could be motivated by tax treatment issues.

As noted, it is important to understand how hedging works before exploring its use in implementing arbitrage strategies. Thus, we’ll now explain how an investor constructs a hedge that holds a stock, locks in an established profit, and neutralizes risk.

Hedging Example: Protecting Profit on an Established Long Position

Investment Scenario and Expected Results of the Hedge

Consider a stock originally bought for $85 that has risen to $100. For our purposes, we'll ignore commissions associated with buying and selling securities. What should the investor do if he is happy with the $15 profit on the investment but fears that the market may fall soon? The most obvious solution is to sell the stock and take the $15 profit now. However, what if the investor is unable or unwilling to sell the stock now but still wants to lock in the profit? Perhaps the investor wants to delay realizing a taxable gain until next year or wants to stretch an existing short-term gain into a long-term gain.¹³ The investor could sell short the stock at its current price of $100, which would protect against any loss of the $15 profit. Any drop in the value of the stock would then be offset by an equal appreciation in the value of the outstanding short position. The investor has a $15 profit that could be realized by selling the stock now. However, the investor substitutes a hedging short sale transaction for the direct sale of the long position. This substitute transaction protects the profit while maintaining the original long stock position.¹⁴

The Effect of Price Changes on Hedge Profitability

What would happen if the price of the stock falls from $100 to $90? Remember that the short position locks in the proceeds from selling at $100. If the price falls to $90, the stock can be purchased at that price and returned to the lending broker, thereby generating a profit of $10. However, the profit on the long position is reduced by $10 due to the price decline. Thus, there would be no net deviation from the established profit of $15.

The hedge brings both good and bad news. The good news is that the $15 profit is locked in without risk. Yet the bad news is that the investor cannot profit further from any increase in the stock price beyond $100. This is because a price increase would raise the value of the long position but would also bring offsetting losses on the short position.

What if the price moves from $100 to $110? The profit on the long position increases from $15 to $25 a share, but the short position loses $10 a share. From a cost/benefit perspective, the "benefit" of locking in the established $15 profit comes at the "cost" of eliminating the ability to gain even greater profits. In other words, the benefit of the hedge is the floor that it places on potential losses, and its (opportunity) cost is the ceiling placed on the position’s maximum profit. This makes sense in light of the risk/return trade-off. The hedge reduces or eliminates risk and therefore reduces or eliminates subsequent expected returns. Table 1.1 summarizes the potential outcomes associated with the hedge. In this scenario, an investor buys 100 shares of stock at $85 a share, and it is now selling for $100. The investor wants to lock in the $15 profit without selling the stock. For the hedging transaction, the investor sells short 100 shares at $100 a share.

Table 1.1 The Good and Bad News of Hedging

Figure 1.1 portrays the results graphically.

Figure 1.1 Hedging to Protect Profits

The profit/loss potential of the long position originally established by buying at $85 intersects with the vertical axis at –$85 and intersects with the horizontal break-even axis at +$85. This indicates that the maximum loss is $85, which occurs if the stock price falls to zero. Furthermore, the break-even price of $85 is obtained if the price remains at its original purchase price. The positive, upward-sloping profit/loss line indicates that profits increase dollar-for-dollar as the stock’s price rises above the original purchase price of $85. Similarly, profits fall dollar-for-dollar as the stock’s price falls below the original purchase price. The maximum gain is, at least in theory, infinite.

The profit/loss potential of the short position established by selling borrowed shares at $100 intersects with the vertical axis at +$100 and intersects with the horizontal break-even axis at $100. This indicates that the maximum gain is +$100, which occurs if the stock price falls to zero. The break-even price of $100 occurs if the price remains at its original level. The negative, downward-sloping profit/loss line indicates that profits increase dollar-for-dollar as the stock’s price falls below the original short sales price of $100, and profits decline dollar-for-dollar as the stock’s price rises above the price at which the shares were sold short. The maximum loss is theoretically, but soberingly, infinite.

The most dramatic result portrayed in Figure 1.1 is the horizontal hedged profit line, which shows that profits are fixed at $15 per share regardless of where the stock’s price ends up. The horizontal line results from offsetting the upward-sloping long position profit/loss line against the downward-sloping short position profit/loss line. The opposite slopes of the two lines imply that when one position is losing money, the other is making money. Thus, the horizontal hedging profit line reflects the risk-neutralizing effect of combining the short (hedging) transaction with the investor’s original long position in the stock. Gains and losses on the two individual positions cancel each other out, thereby resulting in a fixed profit of $15 per share. This $15 profit is the difference between the original purchase price of the stock at $85 and the price at which it was sold short at $100.

The Rate of Return on Hedged Positions and Its Relationship to Arbitrage

In the preceding example, the investor locks in an ex post (after-the-fact) 17.65% return ($15/$85) through a hedge. The investor has effectively removed the position from the market and has an expected zero rate of return from that time on. Importantly, the position is riskless after the given 17.65% return is generated, and no deviation above or below that return is possible after the hedge is in place. However, insufficient data are given in the example to judge whether the ex post return of 17.65% is appropriate to the risk of the investment.

An investor cannot engage in arbitrage that profitably exploits mispriced investments without adding risk unless he can hedge. This is because the hedge is the means whereby the arbitrage strategy is rendered riskless. Hedging is an essential mechanism that allows arbitrage to structure asset prices.