19.2.1 Defining and Pricing Convertible Bonds

Convertible bonds are hybrid corporate securities, mixing fixed-income and equity characteristics into one security. In their simplest form, convertible bonds can be thought of as a combination of an unsecured corporate bond and a call option on the issuer's stock. In a bankruptcy proceeding, convertible bonds are senior to equity securities and subordinated to senior and collateralized debt issues. The yield to maturity on convertible bonds is lower than the yield on otherwise equivalent straight debt because the convertible bond's conversion feature provides an option with substantial value to the holder. Because the holder of the convertible bond owns straight debt plus an equity call option, the owner is willing to pay a higher price (and accept a lower yield) than would be acceptable for an otherwise similar straight bond. Following are formulas for the value of a convertible bond, the conversion ratio, the option strike price, the conversion value, and the conversion premium of a convertible bond:

(19.1a)

(19.1b)

(19.1c)

(19.1d)

(19.1e)

Thus, the firm in Application 19.2.1a can borrow $10 million today in a bond issue and potentially never have to repay the loan in cash, as investors may opt to be repaid with 200,000 shares of stock at some date at or before the three-year maturity of the convertible bond. Valuing the convertible bond is typically accomplished by unbundling the structure into its component parts of straight debt and the equity call option, valuing each component, and summing their values.

In practice, convertible bonds are not valued by the Black-Scholes option pricing model that is used to value short-term equity options, as assumptions (including that of constant volatility) do not apply to long-dated convertible bond issues.

19.2.2 Busted, Hybrid, and Equity-Like Convertibles

The characteristics of convertible bonds vary widely with the moneyness. Moneyness is the extent to which an option is in-the-money, at-the-money, or out-of-the-money. In the case of a convertible bond, moneyness indicates the relationship between the strike price implied by the conversion option and the price of the underlying stock. Bonds with very high conversion premiums (see Equation 19.1e) are often called busted convertibles, as the embedded stock options are far out-of-the-money. These bonds behave like straight debt because when the stock option is far out-of-the-money, the convertible bond's value is primarily derived from its coupon and principal.

Bonds with very low conversion premiums have stock options that are deep in-the-money, where the convertible bond price and the conversion value are very close. The further in-the-money that the option is, the more the convertible bond behaves like the underlying stock. An equity-like convertible is a convertible bond that is far in-the-money and therefore has a price that tracks its underlying equity very closely. Interest rates and credit spreads matter less on equity-sensitive convertibles.

Convertible bonds with moderately sized conversion ratios have stock options closer to being at-the-money and are called hybrid convertibles. Hybrids are usually the most attractive bonds for use in convertible arbitrage strategies. These hybrid convertibles are attractive for convertible arbitrage due to their asymmetric payoff profile. Exhibit 19.1 illustrates the effect of moneyness on convertible bond prices and their sensitivity to the underlying equity prices. Note the convexity in the convertible bond price for hybrid convertibles. This convexity is the essential characteristic that drives the traditional convertible arbitrage strategy. The following section on delta, gamma, and theta provides a further foundation for understanding the dynamics of convertible arbitrage.

19.2.3 Delta, Gamma, and Theta

The concepts of delta and gamma are keys to understanding the convertible arbitrage strategy. Delta is the change in the value of an option (or a security with an implicit option) with respect to a change in the value of the underlying asset (i.e., it measures the sensitivity of the option price to small changes in the price of its underlying asset). For example, if a $1 rise in the value of a stock price causes a call option to rise $0.60, then the delta of the call option is roughly 0.6.¹ Call options that are very far out-of-the-money have deltas near 0.0, whereas options very far in-the-money have deltas near 1.0. The delta of a put option is negative. Delta is the first derivative of an option's price with respect to the price of the underlying asset and is a key concept in setting the hedge ratio of a convertible arbitrage position. In a graph of an option price against the price of the underlying asset, delta is the slope of the relationship at each point along the curve.

Gamma is the second derivative of an option's price with respect to the price of the underlying asset—or, equivalently, the first derivative of delta with respect to the price of the underlying asset. That is, it measures how delta changes as the price of the underlying asset changes. Graphically, gamma is the degree of curvature in the option price versus the underlying asset price relationship. Gamma measures the rate of change in the value of delta as the price of the underlying asset changes. Gamma is near zero when an option is extremely far out-of-the-money and the delta is very small. Gamma is also near zero when an option is extremely far in-the-money and the delta is near one. Gamma tends to be largest when the option is near-the-money. As illustrated in the next section, the gamma of a position can be used to describe how hedged positions earn money during periods of high volatility in the underlying asset.

Finally, theta is the first derivative of an option's price with respect to the time to expiration of the option. Theta is negative for a long position in an option, since as time passes and all other values remain the same, the option declines in value. In a nutshell, theta reflects the loss in an option's time value as time passes, which can be referred to as time decay. Theta is a key concept in understanding how hedged positions lose value if there are no changes in the underlying asset or its volatility. That is, theta is a cost to the buyer of the option and a benefit to the seller of the option, as the time value decays as the option approaches expiration. The goal of many active long-option trading strategies, including convertible bond arbitrage, is to earn sufficient profits from gamma trading to overcome the predictable losses from theta.

In summary, delta is used to establish the hedge ratio in a traditional convertible arbitrage position. The positive gamma or long gamma nature of the convertible bond ensures that the hedged position will make money if the underlying asset quickly rises or falls in value. This profit is generated by the unlimited upside and limited downside nature of a long position in an option (i.e., its curvature). Finally, the theta of the long option position indicates that as time passes, the hedged position loses value in the absence of underlying asset changes. Thus, a traditional convertible arbitrage strategy's return varies directly with the level of volatility experienced in the underlying asset. The goal in convertible arbitrage is to purchase undervalued options and short sell overvalued options while hedging other risks.

19.2.4 Stylized Illustration of Convertible Arbitrage

Consider a $1,000 face value convertible bond that can be converted into one share of stock, for mathematical simplicity. The stock currently sells for about $1,000, so the implicit option is at-the-money. Exhibit 19.2 shows the five prices that the convertible bond can currently have for five possible stock prices. Notice that the convertible bond's price moves nonlinearly with respect to large changes in the underlying asset price, just like a call option does, with smaller losses to the downside and larger gains to the upside. This behavior is due to convexity and is a key to the profit potential.

Assume that the current price of the stock is $1,000 and the price of the convertible bond is $1,100. If the stock rises or falls $20, the convertible bond moves in the same direction but with half the magnitude (i.e., $10).² The delta of the convertible bond is therefore 0.50, and the hedged position would be a long position of one convertible bond and a short position of 0.5 shares of stock. The hedged position is said to be delta-neutral. A delta-neutral position is a position in which the value-weighted sum of all deltas of all positions equals zero. In this example, the sensitivity of the 0.5 short-sold shares to the equity price equals the sensitivity of one convertible bond to the equity price, offsetting each other and leaving the combined positions insensitive to small changes (i.e., a change of $20) in the stock price.

The last line of Exhibit 19.2 illustrates that the hedged position breaks even for very small changes in the stock price; the combined positions retain a constant value of $600. But the combined positions are profitable for either a $40 up or a $40 down movement in the underlying asset. This illustrates that even though the positions are delta-neutral, the hedge benefits from large movements in either direction. The profit is generated by the positive gamma of the convertible bond, wherein losses of the bond slow down when the stock declines, and profits accelerate when the stock rises. If a large price change in the underlying asset takes place, the hedged position makes a profit, and the positions are adjusted to being delta-neutral based on a new hedge ratio at the new price levels. If the underlying stock price does not move, the convertible bond will slowly decline to its par value at maturity, and the hedged position will fall to $500, illustrating the negative theta.

In a convertible arbitrage strategy, when the underlying stock price has changed and the positions (i.e., the hedge ratio) have been adjusted to bring the exposure back to being delta-neutral, it does not matter whether the stock price moves back to its original value or continues moving in the same direction. The reason it does not matter is that once the stock price has changed and the arbitrageur has reset the hedge to reflect the new hedge ratio by expanding or contracting the short position in the stock, the positions are returned to being delta-neutral. Once the positions are returned to delta neutrality, the positions return to the profit and loss exposures illustrated in Exhibit 19.2, and the arbitrageur returns to being able to profit whether the next move in the stock is up or down.

Note, however, that for the arbitrageur to make more money on gamma than is being lost on theta, which is known as time decay, the stock must keep experiencing substantial price changes. These price changes dictate the relationship between realized volatility and implied volatility. Realized volatility is the actual observed volatility (i.e., the standard deviation of returns) experienced by an asset—in this case, the underlying stock. The implied volatility of an option or an option-like position—in this case, the implied volatility of a convertible bond—is the standard deviation of returns that is viewed as being consistent with an observed market price for the option. A traditional convertible arbitrage strategy is a play on whether the realized volatility is equal to, less than, or greater than the implied volatility of the convertible bond price when the position was established. The keys to convertible arbitrage success are to buy convertible bonds with underpriced conversion options (i.e., implied volatility that is too low), short sell convertible bonds with overpriced conversion options (i.e., implied volatility that is too high), and maintain hedges by taking offsetting positions in the underlying equity to control for risk. By far the most common strategy is to take a long position in the convertible bond and hedge the market risk of the position by taking a short position in the underlying equity. Fund managers who follow this strategy believe that the implied volatilities of convertibles are too low when compared to expected realized volatility.

19.2.5 Background on Short Selling

Convertible arbitrage provides an excellent context to discuss details of short selling. The most common convertible arbitrage strategy involves short selling large quantities of the common stock underlying the convertible bond's embedded option. Short selling is common in a variety of alternative investments and provides a major distinction in the managerial expertise required in alternative investments relative to traditional investments.

The steps in selling assets short include the following:

1. Borrowing the assets from an entity that currently owns them. There is an active market between entities that borrow assets and entities that lend assets, known as securities lending. Securities lending is generally facilitated by an intermediary, usually an investment bank or a brokerage firm.
2. Selling the borrowed assets into the market.
3. Eventually closing the position by purchasing the assets from the market and delivering them to the entity from which they were borrowed.

The borrower of the short position posts collateral equal to the price of the assets plus margin, also known as a haircut, usually of 2%. Thus, if Fund A borrows $100,000 of stock from ABC Brokerage Firm and short sells that stock into the market, Fund A must place the proceeds of the sale (i.e., $100,000) and 2% more (i.e., $2,000) as collateral to provide protection to the lender against the risk that the borrowed stock will rise in price at the same time that Fund A becomes unable to fulfill its obligation to return the stock.

The lender of the securities earns interest on the collateral but typically offers the borrower of the securities a rebate. A rebate is a payment of interest to the securities' borrower on the collateral posted. A typical rebate is a little less than current short-term market interest rates (e.g., the general collateral rate less 0.25%). Thus, the goal of the securities lender is to receive a spread between the interest rate the lender is able to earn on the collateral and the rebate paid to the securities borrower. Also, the borrower must pay any dividends due on the short stock position so that the securities lender can effectively receive dividends on the lent shares. Note that the securities lender takes the risk that the borrower will default and be unable to return the shares at the same time that the collateral will be insufficient to repurchase the shares in the marketplace.

Most securities lending is performed on an overnight basis, wherein securities lenders may demand return of the shares at any time and may require regular adjustment of the collateral amount to reflect the current market price of the borrowed securities. However, some short sales can be performed as term loans of perhaps six months, wherein the lender agrees not to demand return of the securities until the term has ended.

There are special risks to having short positions in equity securities, especially for stocks that are popular targets of short sellers. As the quantity of a stock's outstanding shares being lent to short sellers increases, the competition to find new stock to borrow increases. Entities that hold the stock put that stock “on special.” In this context, a special stock is a stock for which higher net fees are demanded when it is borrowed. To the short seller, this means receiving a smaller rebate. For example, general collateral stocks, which are stocks not facing heavy borrowing demand, may earn a 2% rebate when Treasury bill rates are at 2%, whereas stocks on special may earn zero rebates or even negative rebates, wherein borrowers must pay the lenders money in addition to the interest that the lender is earning on the collateral. In general, investment banks and brokers that see the supply and demand for shorting stocks determine which stocks are on special.

When numerous speculators establish highly similar large positions, it is often referred to as a crowded trade. In the case of traders establishing large short positions, the trade is often termed a crowded short. The security being shorted can become a special stock, and in extreme cases, the security can only be made available for short sales at extraordinarily high borrowing rates as high as 20% or more.

When the inventory of stock available to borrowers becomes extremely tight, short sellers may find their position bought in, meaning the broker revokes the borrowing privilege for that specific stock and requires the trader to cover the short position. If shares cannot be borrowed through another lender on affordable terms, this leaves a convertible arbitrage manager without a hedge to the convertible bond position, which is likely to lead the trader to sell the bond to reduce the stock market risk of the portfolio.

Short sellers should monitor the availability of shares trading in the market to ensure that they can be purchased without substantially increasing the market price when they are needed to cover a short position. Short sellers need to be aware of the possibility of a short squeeze. A short squeeze occurs when holders of short positions are compelled to purchase shares at increasing prices to cover their positions due to limited liquidity. As the ratio of shares being sold short increases relative to the total number of freely floating shares, it becomes increasingly difficult to borrow additional shares, and the potential for a short squeeze increases. Several hedge fund managers being forced to buy in and cover their short positions simultaneously can put upward pressure on the price of the shorted security. The upward movement of the stock price may cause other hedge fund managers to cover their short positions, putting even more upward pressure on the stock price. As more and more hedge fund managers scramble to cover their short positions, the price of the underlying stock can rise rapidly, leaving the last few hedge fund managers squeezed out of their positions at especially elevated prices.

19.2.6 Convertible Bond Arbitrage Background

Convertible bond arbitrage offers the potential to earn alpha when the options implicit in the bonds are mispriced. Why might convertible bond prices be attractive? First, convertible bonds have a relatively small issuance base, with a global convertible bond market size estimated by Invesco in March 2013 at less than $200 billion, down from $300 billion in 2007. As a small and complex asset class, convertible bonds may offer liquidity or complexity premiums to skilled hedge fund managers who are able to evaluate them and identify the potential mispricing that results from their complexity. A complexity premium is a higher expected return offered by a security to an investor to compensate for analyzing and managing a position that requires added time and expertise. Convertible bonds, already made complex by the conversion options, become especially complex when the bonds stray from the plain-vanilla package of corporate debt plus a conversion option to having the additional complexities of callable or putable convertibles, dual currencies, and/or forced conversions.

Convertible bond arbitrage funds develop computerized systems to scan the universe of convertible bonds and compare convertible bond prices to the price of the straight debt and equity call option package. Each hedge fund creates customized assumptions for the straight bond yield and the volatility of the underlying equities. The analysis of the underlying straight bond focuses on the firm's credit risk, whereas the analysis of the equity volatility focuses on historical return volatilities and current option prices. When the convertible bond is undervalued relative to the sum of its parts, the hedge fund purchases the convertible bond and shorts the underlying equity. Less often, the convertible bond is viewed as overvalued and sold short with a long position in the underlying equity. Also, the convertible bond position is sometimes hedged with positions in equity options in addition to or in place of positions in equities. Further, to hedge the interest rate risk and credit risk of the convertible bonds, the manager sometimes establishes positions in interest rate derivatives or credit derivatives.

19.2.7 Four Sources of Returns to Convertible Bond Arbitrage

Fund managers who are able to develop accurate predictions of equity volatility relative to the volatility implied by convertible bond prices can earn superior returns by buying undervalued convertible bonds and shorting the underlying equity. In the past two decades, convertible bond arbitrage trading tended to focus on long positions in convertible bonds and to generate superior returns, especially in the mid- to late 1990s, indicating that convertible bonds themselves offered consistently superior returns.

Note, however, that if investors in convertible bonds consistently earned superior returns, the bonds might offer higher than necessary yields, which make convertible bonds an expensive source of corporate financing (i.e., the return earned by the bond investor is the cost of capital to the firm). In perfect capital markets, the risk-adjusted costs of all sources of financing would be forced toward equality, since investors would avoid buying securities with returns too low and corporations would avoid issuing securities with returns too high. There are two elements necessary to support the argument that convertible bonds should consistently offer superior risk-adjusted returns. First, demand to buy convertible bonds must be restricted such that it prevents convertible bond prices from increasing to the point of offering normal risk-adjusted returns. Second, suppliers of convertible bonds (corporations) must be of sufficient size to suppress convertible bond prices to the point of allowing superior returns.

The argument that there is limited demand from convertible bond investors appears plausible. The complexity of convertible bond analysis and hedging, combined with restrictions on the ability of traditional investment managers such as mutual fund managers to short equity, may limit the number of investors willing and able to perform convertible bond arbitrage. But why would corporations issue convertible bonds if they were consistently underpriced? More broadly, are there solid reasons to believe that convertible bonds will continue to be issued at prices that offer consistently high risk-adjusted returns to investors and therefore higher costs to issuers? There are four especially persuasive reasons to believe that issuers may, at least periodically, continue to offer convertible bonds at attractive prices:

1. Agents (corporate managers) may underestimate the true costs of issuing convertible bonds. Convertible bonds offer yields that substantially underestimate expected returns when those yields are based on coupons and principal amounts. Issuers may find the lower yields to be attractive, as the coupon interest rate on convertible bonds is lower than the interest rate paid on the straight bonds issued by the firm. The issuers may not fully appreciate the potential harm to share prices from dilution when the implicit options are exercised. Dilution takes place when additional equity is issued at below-market values, and the per-share value of the holdings of existing shareholders is diminished.
2. Agents of small firms may have no choice but to issue convertible bonds at attractive prices. Convertible bonds are rarely registered in a public offering. In the United States, most convertible bonds are sold as 144A exempt securities, meaning they are exempt from the registration requirements of the SEC (Securities and Exchange Commission). As a result, most convertible bonds cannot be sold to retail investors, and trade only among institutional investors. The lack of a public market for these convertible bonds makes them less liquid than stocks or regular bonds. Consequently, their prices may be lower and their returns higher as a premium for bearing liquidity risk.

3. There is a potentially substantial conflict of interest between straight bond investors and shareholders with regard to preferred corporate asset volatility. Straight bondholders prefer low asset volatility to decrease the probability of bankruptcy. Equity holders have a risk exposure that can be viewed as a call option on the firm's assets, and therefore they may prefer high asset volatility. Shareholders have an incentive to increase the volatility of the firm's assets after the issuance of debt, in order to transfer wealth from bondholders to themselves. Since bondholders are aware of this potential risk, they demand a higher yield for compensation, and suboptimal corporate investment decisions may result. The incentive to take on excessive risk is reduced if convertible bonds are issued, as any increase in volatility benefits the convertible bondholders as well as the equity holders. In short, convertible bonds reduce agency costs and lead to a lower cost of capital for the firm.

   Asymmetric information between corporate managers and investors regarding asset volatility can exacerbate the problems with issuing straight debt. Since convertible bonds are hybrid investments, their prices are less sensitive to the credit risk of the issuing firm. This insensitivity makes it easier for the firm and potential bondholders to agree on the value of the bond when convertible debt is used and there is substantial uncertainty about the riskiness of corporate assets.

4. Indirect equity issuance costs are a factor. Corporations use convertible bonds as an indirect way to issue equity because their cost of directly issuing new equity may be high. For instance, when managers opt to issue new stock at current price levels, potential buyers of the new shares may conclude that managers and current shareholders view the current price as being above its fair value, making them willing to bring in new investors. The inadvertent information signal caused by issuing equity could depress share prices as market participants react to the concern that the firm is in worse financial position than originally believed and as reflected in the share price. Since most convertibles are converted into equity only if the stock price increases, the signal conveyed to the market is not viewed as negatively as when equity is issued.

19.2.8 Components of Convertible Arbitrage Returns

The components of convertible arbitrage returns include interest, dividends, rebates, and capital gains and losses. Exhibit 19.3 depicts these components for the case of a traditional convertible bond strategy of being long the convertible bond and short the underlying stock. The first component of the return of a traditional convertible arbitrage strategy is the income component. Assuming a long position in the convertible bond and a short position in the stock, the investor earns the coupon interest paid on the bond, pays any dividends due on the short stock position, and earns a rebate on the cash proceeds from the short sale of the stock. If there are any costs to financing the position, such as the cost to borrow the stock or the interest paid on leveraged positions, those costs are deducted from the arbitrage income.

The second source of the return to the convertible bond arbitrageur is the gain on stock trading (and, to a lesser extent, the possible gain or loss on the eventual sale of the convertible bond), as illustrated in Exhibits 19.2 and 19.3. In the traditional convertible arbitrage trade of being long the convertible bond, the larger and more frequent the stock price moves, the greater the profits from gamma trading. Profits from gamma trading, though, are offset through theta, or time decay. The goal of gamma trading is to earn more in profits from gamma than the option value loses in time decay. This goal is met when the realized volatility of the stock exceeds the implied volatility priced into the option on the day the convertible bond is purchased.

This simplified discussion of convertible arbitrage has held constant other sources of risk and return, such as interest rates, credit spreads, and implied volatility. Although some convertible bond managers are content to maintain a simple hedge of the convertible bond against the underlying stock, other managers may seek to hedge other risks or add further value through derivative strategies related to interest rates, credit spreads, volatility, or stock price anticipation. For example, more sophisticated hedging strategies use interest rate and credit derivatives to hedge interest rate and credit spread risks such that the arbitrage is more of a pure play on realized volatility relative to implied volatility.

Some convertible arbitrage hedge funds attempt to identify and hedge the underpriced embedded options of a convertible bond by buying the convertible bond and selling short an exchange-traded call option on the underlying stock. This technique can be effective when there is a large spread in implied volatility between exchange-traded and embedded call options. However, there is rarely a clean match between listed and embedded options in terms of exercise periods, and many convertible bond issuers do not have options listed on their stock. Interest rate hedges are less common in convertible arbitrage funds. Given that much of the convertible bond universe is below investment grade, credit spread changes can be significantly more important than changes in risk-free or investment-grade interest rates. Credit derivatives, detailed in Chapter 26, can be useful vehicles with which to hedge credit risk.

Rather than hedging various risk exposures, managers may speculate on them. Profits from a convertible arbitrage position can be substantially enhanced when managers have the ability to consistently predict the future path of interest rates, credit spreads, stock prices, or volatility. For example, to add value through credit spread anticipation, the manager may perform fundamental credit analysis on each issuer, seeking to purchase bonds with improving credit quality and tightening credit spreads, while avoiding bonds whose credit quality is deteriorating, which can lead to widening credit spreads. For stock price anticipation, the manager deviates from delta neutrality in an attempt to profit from stock price moves in a particular direction. The manager applies heavy delta hedges (a net short position) to stocks expected to underperform, and light delta hedges (a net long position) to stocks for which higher prices are anticipated. Fund managers are also likely to diversify their portfolios across issuer, sector, maturity, and so forth to reduce both idiosyncratic risk and exposures to industries and sectors.

19.2.9 Details Regarding Convertible Bond Arbitrage

Let's take a close look at how a traditional convertible arbitrage strategy tries to enhance returns. Specifically, this section details how a position that is delta hedged can earn gains from gamma (convexity) that more than offset the losses from theta (time decay).

Delta hedging is shown in this example to reduce risk; however, delta hedging does not eliminate the potential for net capital gains. Due to the nonlinear nature of their payoff, most at-the-money convertible bonds exhibit a desirable property known as positive convexity, or high gamma. That is, they appreciate in value from an immediate upward stock price change more than they depreciate from the same sized downward change in the underlying stock price. This section shows that a delta-hedged position will actually benefit from any movement in the underlying stock due to this convexity. The traditional arbitrage strategy is a speculation that there will be enough movements in the stock price (i.e., volatility) to generate gains from gamma that more than offset the losses from theta (the time decay of the position).

Consider an example of convertible bond XYZ with a conversion ratio of 8. A convertible bond arbitrageur believes that the implicit option in the bond is undervalued and that therefore the entire bond is underpriced based on the arbitrageur's estimate of the future volatility of the underlying asset (stock). How can the arbitrageur exploit such a mispricing? Buying the cheap convertible is clearly part of the solution, but it is not sufficient. Simply waiting for market prices to adjust is not an arbitrage because the long convertible position comes with a variety of risks that could easily wipe out the expected gains. To arbitrage, it is necessary to both buy the cheap convertible bond and hedge its risks, a dynamic process that is very similar to what arbitrageurs of listed options do on a regular basis.

The primary risk of holding a long convertible position comes from the potential variations in the underlying stock price. This equity risk can be easily eliminated by selling short an appropriate quantity of the underlying stock. This quantity corresponds to the convertible's delta multiplied by the number of shares into which the bond may be converted. Let's assume that the delta of the XYZ convertible bond is 0.625. To hedge the equity risk, an arbitrageur would need to sell short delta times the conversion ratio (0.625 × 8 = 5.0) shares of stock per $1,000 face value of the convertible bond bought. If the stock price gains $1, the convertible bond will gain approximately delta (times 8) dollars and the short stock position will lose delta (times 8) dollars, so that the overall variation will be near zero. Conversely, if the stock price drops by $1, the convertible bond will lose approximately the same number of dollars that the short stock position will gain. As illustrated in Exhibit 19.4, for a small change in the price of the stock, the arbitrageur's position will be hedged.

But this approximation ignores a key aspect to the hedge: Although delta hedging reduces the risk from changes in the underlying stock price, it does not eliminate return. Return of the strategy can be enhanced because, ignoring theta, the hedged position generates a small gain whether the underlying stock moves up or down, due to the position's gamma. This important concept is detailed later.

First let's focus on the need to rebalance the original delta-hedged position. In our example, when the stock price changes, the delta of the convertible bond will no longer be 0.625, and therefore the net delta of the position will no longer be equal to zero. The net delta of a position is the delta of long positions minus the delta of short positions.

As the stock price increases, the option component moves further in-the-money and the convertible bond becomes more equity sensitive (see Exhibit 19.4). The delta of the convertible bond increases, so the arbitrageur must adjust the hedge by shorting more shares. Conversely, as the stock price declines, the option moves out-of-the-money, the delta of the convertible bond declines, and the arbitrageur must reduce the hedge by buying back some shares.

For example, if the delta rises to 0.70 due to a stock price increase, the short position must be expanded from 5.0 shares to 5.6 shares (8 × 0.70). If the delta falls to 0.50 due to a stock price decrease, the short position must be contracted to 4.0 shares (8 × 0.50). The hedge needs to be rebalanced repeatedly as the stock price moves, in a strategy known as dynamic delta hedging. Dynamic delta hedging is the process of frequently adjusting positions in order to maintain a target exposure to delta, often delta neutrality.

A key question for most arbitrageurs is how often they should rebalance their hedges. Arbitrageurs usually rehedge based on a time or price formula. In the former case, rehedging takes place at prespecified time intervals, such as every day or every hour. In the latter case, rehedging takes place whenever the stock price changes by a certain amount (e.g., every $1 move or every 1% move in the stock price) or when the size of the necessary adjustment reaches a certain threshold.

Let's look more closely at the convexity, or gamma, that drives the traditional convertible bond arbitrage strategy. Gamma refers to the asymmetric valuation profile generated by movements in the underlying stock price. In other words, gamma is illustrated by the curvature in Exhibit 19.5.

Exhibit 19.5 illustrates why the gamma of the convertible bond generates a gain to the hedged position when the underlying stock moves up or down. But Exhibit 19.5 does not illustrate the downside risk. The worst outcome for the traditional convertible bond arbitrageur is when the stock price remains unchanged. When the stock price does not change, the hedged position loses value due to the theta (time decay) of the long position in the implicit option. When the underlying stock price experiences less volatility than is implied by the bond price, the losses from the theta of the option more than offset the gains from the gamma, and the strategy underperforms.

Saying that a convertible bond is cheap is equivalent to saying that the corresponding implied volatility is too low. If realized volatility is higher than implied volatility, then the profits illustrated in Exhibit 19.5 should dominate the theta, resulting in net profits for the strategy. Conversely, if the realized volatility is below the implied volatility, the loss due to theta will outweigh the profit made from the realized volatility, and the position will underperform a risk-free investment, perhaps even incurring a loss.

19.2.10 Return Drivers of Convertible Bond Arbitrage

The mispricing of convertible bonds can be relatively large or small. Minor differences in the volatility used to price the embedded stock option in a convertible bond can generate substantial price differences. For example, if a three-year convertible bond is mispriced by two volatility points (e.g., 25% volatility is used to price the bond rather than 27%), the convertible bond may be underpriced by 1%, a mispricing that may take three years to fully correct. In cases of small degrees of mispricing, convertible bond arbitrage hedge funds may apply leverage to increase the expected returns. Before the 2008 financial crisis, it was not uncommon to see convertible bond hedge funds trade at leverage of over eight times investor capital. Since 2008, it has become more difficult to leverage positions, with the result that some convertible bond funds may now forgo leverage, while others may be able to reach a maximum leverage of only four times investor capital.

It is easy to see why hedge fund managers are tempted to use leverage, as they earn incentive fees on each additional dollar of returns they earn. But leverage is a two-edged sword to investors, as it magnifies both gains and losses. However, incentive fee–based hedge fund managers disproportionately participate in the gains but not the losses; thus, as detailed in Chapter 16, the managers may increase the value of their incentive fee option by taking larger risks.

The market crisis of 2008 created unprecedented risks and opportunities for convertible bond arbitrage. The Credit Suisse Convertible Bond Arbitrage Index declined by more than 25% during the last four months of 2008. This decline may have been caused by illiquidity and large amounts of forced selling of convertible bonds, as prime brokers forcibly reduced the availability of leverage, and the large portfolio of the now-defunct Lehman Brothers was quickly sold into the market. Once this selling subsided, the opportunities in the convertible bond market were unprecedented, as mispricing reached record levels. The yield on U.S. investment-grade convertible bonds reached 14.9% in March 2009, wider than the straight bond yield of 11.0% of the same issuers. A convertible bond arbitrage fund could apparently buy the convertible bond and sell short the straight bond of the same issuer, receiving a free option and an extra yield of 3.9%. Exhibit 19.6 summarizes the risks of convertible bond arbitrage.

19.2.11 Historical Return of Convertible Bond Arbitrage Funds

Exhibit 19.7 summarizes the monthly returns of the HFRI Relative Value Fixed-Income Convertible Arbitrage Index from January 2000 to December 2014, along with the overall HFRI Relative Value Index and several major market indices, following the standard format used throughout this book and detailed in the appendix. Exhibit 19.7a indicates that the cross-sectionally averaged returns of convertible arbitrage funds exhibited returns and risk between those of global bonds and U.S. high-yield bonds. All indices except global bonds experienced negative skews and leptokurtosis. However, the two relative value indices exhibited much larger negative skews and fat tails, consistent with the view of relative value strategies being similar to writing out-of-the-money options. Note also the high autocorrelation coefficients of the relative value indices.

Exhibit 19.7b indicates similarity in cumulative wealth indices between the two relative value indices and the global bond index, except that the convertible arbitrage funds experienced a much greater decline and recovery related to the financial crisis that began in 2007. Exhibit 19.7c indicates high correlations of both the convertible bond index returns and the overall relative value index returns to the returns of world equities and U.S. high-yield bonds.

Finally, Exhibit 19.7d indicates an important point. During months of small movements in the global equity market, the returns of the convertible bond index are modestly related. However, the most extreme convertible bond index returns, especially to the downside, are associated with similar moves in the global equity index. In summary, past returns may or may not indicate future return behaviors. However, when a strategy can be reasoned to behave like short positions in out-of-the-money options, and when 15 years of historical return data confirm that behavior, an analyst can have reasonable confidence that future returns are likely to behave similarly. However, this confidence should not extend to predictions regarding average returns or risk-adjusted performance. The likelihood of relatively strong risk-adjusted performance of convertible arbitrage beyond 2014 may depend on capacity and other conditions that differ from the conditions that generated the results depicted in Exhibit 19.7.

19.3.1 Volatility and Vega Overview

Any security that contains a nontrivial option feature may be viewed as having a direct relationship between its price and the volatility of the underlying asset, holding all other values constant. Often multiple security prices depend on the same underlying asset volatility (or related asset volatilities). Examples include options on the same asset that differ with regard to strike price, expiration date, type of option (e.g., European, American, Bermuda, range, and knockout), and being calls or puts. This permits traders to speculate on the relative performance of the multiple securities with option characteristics and with the same underlying asset.

A key concept in volatility arbitrage and options in general is vega. Vega is a measure of the risk of a position or an asset due to changes in the volatility of a price or rate that helps determine the value of that position or asset. For example, in the case of an option, vega is the first derivative of the option price with respect to the implied volatility of the returns of the asset underlying the option. Vega risk is the economic dispersion caused by changes in the volatility of a price, return, or rate.

A key distinction in volatility involves differences between implied volatility, anticipated volatility, and realized volatility. In all three cases, volatility is defined as the standard deviation of returns. Implied volatility, as discussed earlier, is the level of volatility in an option's underlying asset inferred by the current price of the option based on a particular option pricing model. Implied volatility is a mathematical computation performed by searching for the level of asset volatility that when inserted into a specified option pricing model generates a model price that equals the current market price of the option. Anticipated volatility is the future level of volatility expected by a market participant. Realized volatility, as discussed previously, is a statistically based estimate of the actual historical volatility experienced in the marketplace. Market participants often develop anticipations of volatility based on observations of realized volatility. They then compare the anticipated volatility with the implied volatility of options, taking long option positions when their anticipations of volatility exceed the implied volatility and short option positions when their anticipations of volatility are lower than the implied volatility.

It is especially important in discussing volatility arbitrage funds to be careful regarding the use and meaning of the term volatility. Outside investments, volatility is interpreted as simply indicating dispersion. Within investments, volatility is typically used specifically as and synonymously with standard deviation. However, in the area of volatility derivatives and variance derivatives, the terminology is evolving, and it is not always clear that volatility refers solely to standard deviation or that variance derivatives reference only variance.

Sinclair presents some stylized observations regarding volatility, many of which are key assumptions behind some volatility arbitrage portfolio strategies and risk management techniques:

1. Volatility is not constant, but it mean-reverts, clusters, and has long memory. As such, many traders will model volatility using a regime-switching model.
2. Volatility tends to stay low for some extended period of time until a market shock occurs and volatility transitions to a higher level for some period of time.
3. The volatility of volatility can be high, but in the long run, volatility tends to revert toward some long-term average level.
4. In equity markets, volatility tends to increase as price levels decline.
5. Volatility tends to rise more quickly in response to stock prices falling than it falls in response to stock prices rising.³

The final two observations may partially explain volatility skew levels, in which equity put option prices often trade at higher implied volatility levels than equity call option prices of similar deltas.

19.3.2 Instruments Used by Volatility Arbitrage Funds

Managers of volatility arbitrage funds have substantial latitude in the choice of assets to trade in their funds. Broadly speaking, these funds may have positions in any instrument with volatility exposure. These assets include exchange-traded options, warrants, convertible bonds, other bonds with embedded options, over-the-counter (OTC) options, and OTC variance swaps. In recent years, a robust exchange-traded market has arisen in volatility futures and options, trading specifically on the Chicago Board Options Exchange (CBOE) Volatility Index (VIX), which measures the implied volatility of options on the S&P 500 Index. A given volatility arbitrage fund may focus on these assets within one market, such as equities, while others may mix instruments across currency, debt, equity, credit, and commodity markets. In addition to holding assets with option characteristics, volatility arbitrage funds also hold assets without option characteristics in order to hedge or reduce their net exposure to moves in the underlying markets. The simplest examples of positions taken by volatility arbitrage funds involve exchange-traded options and warrants, whose performance is tied to price moves in single-equity securities or futures contracts in the equity, commodity, currency, or debt markets. In order to focus trading on volatility, traders follow a strict delta-hedging process to hedge away moves in the underlying market.

Bonds with embedded options can also be attractive to managers of volatility arbitrage funds. Convertible bonds have an embedded long call option on the issuer's stock, whereas mortgage-backed securities (MBS) are short a put option on interest rates (which is a short call option on bond prices), meaning borrowers are allowed to prepay their mortgages without penalty. Complex or illiquid securities may offer higher expected returns and more frequent opportunities from mispricing. For example, valuation of MBS requires assumptions regarding future interest rate paths and volatility, as well as the potential prepayment rates of the borrowers under various interest rate scenarios.

Variance swaps are forward contracts in which one party agrees to make a cash payment to the other party based on the realized variance of a price or rate in exchange for receiving a predetermined cash flow. Variance swaps are OTC products and are commonly traded by volatility arbitrage funds. These contracts offer cash flows based on the annualized variance in the returns on a referenced asset. In a variance swap, one party (the variance buyer) pays a predetermined variance (referred to as a swap strike price or strike variance) and receives realized variance. The counterparty (the variance seller) has the opposite cash flow exposure, receiving a fixed variance and paying realized variance. The amount of the net cash flow is the difference between the realized variance and the strike variance multiplied by the variance notional value of the contract. The variance notional value of the contract simply scales the size of the cash flows in a variance swap. The annualized variance is simply the squared value of the annualized standard deviation. At maturity, a variance swap pays off according to the following formula:

(19.2a)

For example, consider a 30-day variance swap on the returns of the S&P 500 Index with a variance notional value of $100,000. The strike variance of the swap is 4.00 (corresponding to a 4% annualized variance). After the 30-day reference period is observed, the realized annualized variance in the index is, for example, 4.50. The payoff of the variance swap would be as follows:

A volatility swap mirrors a variance swap except that the payoff of the contract is linearly based on the standard deviation of a return series rather than the variance. In a volatility swap, the payoff is determined by multiplying the spread between the realized volatility and the strike volatility by the vega notional value. Similar to the variance notional value, the vega notional value of a contract serves to scale the contract and determine the size of the payoff in a volatility swap. The vega notional value provides a simple payoff formula for volatility swaps:

(19.2b)

For example, a volatility swap with a vega notional value of $50,000 would pay off $100,000 if the realized volatility was 22.00 when the strike volatility was 20.00.

The payoff to a variance swap in Equation 19.2a is often expressed using an expression that includes the vega notional value in place of the variance notional value. The variance notional value is equal to the vega notional value divided by 2 times the square root of the strike variance. Inserting the formula for variance notional based on vega notional value into Equation 19.2a offers the following more common but less simple and less intuitive payoff formula:

(19.3)

It should be noted that the exact computation methods are specified in the documentation but are not perfectly standardized.

The attraction to variance swaps is that they offer a pure play on asset return variance without exposure to the direction of moves in the underlying instrument. To speculate on the spread between implied and realized volatility in the exchange-traded options market without variance swaps, traders need to buy one set of options, sell another set of options, and frequently rebalance the hedges to keep exposures to the underlying markets close to delta-neutral. Options can be complex, exposing traders not only to volatility exposure (vega risk) but also to moves in the underlying assets (delta and gamma risk). Variance swaps give pure volatility exposure without the directional risk of moves in the underlying assets, which eliminates the obligation to continually rehedge the delta risk of the portfolio. As OTC products, variance swaps create counterparty risk, which must be monitored at all times.

19.3.3 Risks of Exchange-Traded versus OTC Derivatives

Standardized, exchange-traded derivatives and other instruments can be less risky than some OTC instruments. Exchange trading is physically or electronically centrally located. Each instrument traded on the exchange is listed by the exchange, a process that specifies and unifies the characteristics of each instrument. OTC instruments are typically traded by investment banks and fixed-income brokerage houses and vary from being uniform (e.g., shares of common stock) to being unique (e.g., currency swaps with specific delivery dates). Generally, there are three major risks that positions in OTC-traded instruments have relative to positions in exchange-traded instruments:

1. Exchange-traded instruments tend to offer less counterparty risk. Options involve an ongoing obligation by the party with the short position (the option writer) to pay cash or deliver assets to the other party (the option owner). Swaps offer an ongoing obligation by each party to pay cash to the other party. Counterparty risk is the potential dispersion in economic outcomes caused by the potential or actual failure of the other side of a contract to fulfill its obligations. In this case, the investor and the swap dealer have counterparty risk that the other might fail to fulfill the contract. By contrast, exchange-traded derivatives have clearinghouses that back the obligations of the members associated with each listed security.

   Clearinghouses have capital and the incentives and powers to demand collateral and creditworthiness of market participants, greatly mitigating the concerns with regard to the integrity of each contract. Moreover, clearinghouses diversify risk away from a single dealer and spread the risk across multiple members (or broker-dealers), making it less exposed to a single counterparty.

2. Exchange-traded instruments tend to offer higher price transparency and less pricing risk. Price transparency is information on the prices and quantities at which participants are offering to buy (bid) and sell (offer) an instrument. Pricing risk is the economic uncertainty caused by actual or potential mispricing of positions. For example, complex and unique derivative OTC instruments might have no information on prices other than estimations derived through complex models or price indications offered by dealers. Conversely, exchange-traded instruments have easily observable prices at which trades have taken place, and bids and offers of prices at which participants are currently willing to transact.
3. Exchange-traded instruments tend to offer higher liquidity. Owing to price transparency, standardization of the terms of a security, reduced counterparty risk, and centralized trading, exchange-traded instruments tend to offer substantially higher liquidity than do OTC instruments. Liquidity provides market participants with the ability to manage their risks more effectively by being able to transact without substantially affecting market prices.

These three major risks were vividly illustrated during the global financial crisis of 2007 to 2009. For example, counterparty risk was experienced in 2008. Traders with counterparty risk exposure to particular subsidiaries of Lehman Brothers were not paid the gains on their derivative positions when Lehman Brothers defaulted. Further losses to counterparties were avoided when the U.S. government was called on to guarantee the payment of OTC derivative contracts that had been sold by American International Group, Inc.

The price transparency of exchange-traded products facilitates the use of mark-to-market pricing. Marking-to-market refers to the use of current market prices to value instruments, positions, portfolios, and even the balance sheets of firms. The use of OTC derivatives often partially or fully relies on pricing based on a mark-to-model methodology. Marking-to-model refers to valuation based on prices generated by pricing models. The pricing models generally involve two components. An instrument that is not frequently traded, and therefore does not offer price transparency, is modeled as being related to one or more market prices, rates, or factors. The current values of the determinants of the model price are then input into the model to approximate the value of the instrument. Thus, marking-to-model requires the specification of a model and its inputs. A problem with marking-to-model is that different investors holding similar securities may report widely different valuations to their investors based on the assumptions underlying their proprietary pricing model or the inputs used. In comparison, exchange-traded products are marked-to-market, wherein all investors value their holdings at a single, exchange-disseminated price.

19.3.4 Volatility Arbitrage Strategies

An essential concept to understanding volatility arbitrage strategies is vega. As previously defined, vega is the sensitivity of an option or a security with an embedded option to changes in the volatility of the price or the returns of the asset underlying the option. A long position in an option has a positive vega, a short position in an option has a negative vega, and a position without option characteristics has a vega of zero. Note that vega indicates the sensitivity of an asset to changes in volatility assuming all other values are held constant. In practice, when volatility changes, there are usually changes in price levels.

Volatility arbitrage funds trade a variety of assets, typically taking long positions in instruments in which volatility is underpriced (or underestimated), and short positions in instruments in which volatility is overpriced (or overestimated). Other market risks are often hedged out, leaving the fund with less directional risk to the underlying markets. Instead, the positions are exposed to volatility risk and correlation risk. Volatility risk is dispersion in economic outcomes attributable to changes in realized or anticipated levels of volatility in a market price or rate. Correlation risk is dispersion in economic outcomes attributable to changes in realized or anticipated levels of correlation between market prices or rates.

As markets move, the fund manager needs to continue to implement rebalancing trades to remain delta-neutral. These rebalancing trades are profitable for long volatility positions that have positive gamma and unprofitable for short volatility positions that have negative gamma. However, positions long in vega are usually exposed to theta risk (negative theta), such that as time passes in a period with low asset volatility, the positions decline in value.

There are two main types of volatility arbitrage funds: those that are market (volatility) neutral and those that are intentionally exposed—typically long—to volatility. An example of a long volatility strategy is a variance buyer in a variance swap. The position either generates a payoff or requires a payoff based entirely on realized volatility. Long volatility funds can provide valuable tail risk protection during times of rising volatility, when markets are likely to decline. Market-neutral volatility funds seek to earn a profit without exposure to changes in volatility levels. An example of a market-neutral volatility strategy would be offsetting positions in two options with different implied volatilities in the same or similar underlying assets. The profit or loss is primarily driven by changes in the relationship between the two implied volatilities rather than the level of volatilities.

19.3.5 Market-Neutral Volatility Funds

The most common strategy pursued by market-neutral volatility funds has been to make the assumption that there is an arbitrage opportunity between the higher implied volatility and the lower realized volatility for some options. In other words, the assumption is that some options are overpriced, and the trading strategy involves writing those options. The fund hedges the overall exposure of short positions in the options perceived as being overpriced by taking one or more offsetting positions in securities deemed to be more appropriately priced. As an example, a fund may sell equity index options and hedge the risk with a dynamically adjusted replicating portfolio of equity index futures that approximates the returns of the realized variance of the underlying equity market.

One example of why implied volatility of some options might consistently overestimate realized volatilities involves out-of-the-money index puts. Due to the demand for index put options to serve as protection from downside risk, implied volatility of out-of-the-money index put options is frequently believed to trade higher relative to realized volatility. The spread between implied and realized volatility compensates volatility sellers for providing insurance against rising volatility and falling markets. This spread is likely to continue as long as sellers of index volatility continue to demand a risk premium for providing insurance coverage to other market participants and as long as insurance buyers continue to be willing to pay for the protection.

But is there evidence that implied volatility consistently overestimates realized volatility? The VIX tracks the implied volatility on various S&P 500 options. Rampart Investment Management estimates that the implied volatility of the S&P 500 Index, as measured by the VIX, exceeded the realized volatility over the subsequent month in 87% of all months from December 2002 to June 2014.⁴

19.3.6 Challenges of Estimating Dispersion

Care is necessary in interpreting measures of dispersion in the context of volatility derivatives. First, there are numerous conventions for calculating dispersion, so implied versus realized computations should be compared only when both series are calculated with consistent methodologies. Second, the payoff of variance swaps is linearly related to the square of volatility (i.e., variance is the square of standard deviation) and is therefore highly nonlinear relative to volatility, or standard deviation. Estimates of implied standard deviation based on observation of derivative prices with payouts linearly related to variance, such as those shown in Equation 19.3, are biased as predictors of volatility.

19.3.7 Tail Risk Strategies

Tail risk is the potential for very large loss exposures due to very unusual events, especially those associated with widespread market price declines. Entities with undesirably high exposures to tail risk may seek protection from tail risk that is often termed portfolio insurance. Portfolio insurance is any financial method, arrangement, or program for limiting losses from large adverse price movements. Portfolio insurance can be provided through dynamic trading strategies that hedge losses, such as taking short positions in corresponding futures contracts that are adjusted in size based on market levels. Portfolio insurance can also be provided by establishing positions in investments that thrive during periods associated with tail risk. A straightforward solution is to purchase long positions in put options that are very far out-of-the-money. The problem with buying puts that are far out-of-the-money is that they are often viewed as being priced very high. In other words, market participants often view those options as having implied volatility that substantially exceeds expectations of realized volatility. Purchasing out-of-the-money put options at high implied volatilities can be a substantial drag on portfolio performance when, as usually happens, there are no crises and the options expire worthlessly. The explanation for the high implied volatilities is that there is tremendous demand from institutional investors to hold the options for protection against tail risk and a limited number of market participants with the financial resources and desire to provide such protection by writing the puts.

Tail risk strategies may be viewed as attempts to fill the market need for portfolio protection without the potentially large costs of purchasing put options that regularly expire worthlessly and that therefore generate losses during normal market conditions. Some volatility arbitrage funds attempt to design tail risk strategies that earn substantial profits during times of stress, panics, crises, and widespread losses, and generate only small losses or perhaps even small gains during most other market conditions.

For example, a fund may develop a strategy of taking long positions in options with implied volatilities that are deemed low and writing options with implied volatilities that are deemed high. The fund may believe that these particular option positions will permit large profits in the event of a major market decline while having very limited losses in normal markets. It should be noted that option-like payoffs can be attempted using non-option securities through dynamic rebalancing strategies. For example, a strategy that buys additional assets when asset prices rise and liquidates positions when asset prices decline exhibits high upside potential and low downside risk, similar to that of a long call option position. However, such strategies are likely to fail during periods of extreme market stress or over times when markets are closed, since prices and volatility can jump before the necessary dynamic adjustments can be implemented.

The payoff profile of tail risk funds is designed to be negatively correlated to price levels in major markets, especially equity and credit markets. As equity markets decline and credit spreads widen, volatility and correlation tend to increase. Tail risk funds that can profit during these times of crisis can serve as a hedge to the risk exposures of traditional equity and credit market investors. These tail risk funds are used by investors who mix the tail risk strategies into portfolios with substantial long exposure to equity and credit investments to provide a combined portfolio with the goal of profitability during normal market conditions and little or no downside risk during periods of market stress.

Funds that offer the attractive payoff profile of providing tail risk protection may be relatively delta-neutral for small changes in market conditions but lose their delta neutrality for large changes so that they can generate gains during large market drops. In other words, the strategies are long gamma. The strategies are also long vega, since they benefit from rising levels of volatility.

Correlation among assets is a crucial issue in tail risk strategies. During normal market conditions it is observed that, for example, stocks have modest correlation with each other, simultaneously rising or falling by different amounts. However, during periods of market stress, it is often said that correlations go to one. The term correlations go to one means that during periods of enormous stress, stocks and bonds with credit risk decline simultaneously and with somewhat similar magnitudes. However, some analysts prefer to describe this phenomenon by breaking movements in risky assets into market risks and idiosyncratic risks. During normal market conditions, price changes due to idiosyncratic factors are not dominated by changes due to market factors, so correlations between risky assets are modest. The reason is that idiosyncratic movements are uncorrelated by definition. However, during periods of stress, the market factors dominate the idiosyncratic factors, causing risky assets to have highly correlated returns. The reason is that the market-related movements of the individual assets are perfectly correlated by definition. The point of this analysis is that the underlying correlations, parameters, and processes do not change during periods of market stress. Rather, during these periods of stress, market factors experience larger volatility and therefore exert larger effects than do idiosyncratic factors. However, not all assets have returns that increase in correlation with each other during a market crisis. There are defensive assets that have historically been able to maintain their value or even post profits in a market crisis, actually moving toward negative one in terms of correlation with risky assets. These defensive assets may include long put options, long call options on volatility, sovereign debt, and even some hedge fund strategies, such as global macro or managed futures.

Tail risk funds tend to be less focused on pure arbitrage and therefore take positions across markets, attempting to sell overpriced volatility and buy underpriced volatility in whatever markets can be found. If tail risk strategies are able to post large gains during times of market crisis, owners of these funds gain access to valuable cash when other investors may have constrained liquidity in their portfolios. This improved cash position can provide substantial benefits to investors. Investors may be able to avoid losses due to liquidity concerns, such as by being able to fund capital calls to real estate and private equity funds without selling equity and credit investments after sharp market declines. The cash generated from the tail risk portfolio can also be used to opportunistically purchase assets at fire-sale prices from distressed investors who need to raise cash. Although the benefits of a tail risk fund are clear, the challenge to the fund manager is to provide protection during crisis markets without paying too much in option premiums during normal market conditions, which can persist for a very long time. In essence, the strategy attempts to mimic the payouts to out-of-the-money put protection at a lower cost through the implementation of sophisticated trading strategies.

19.3.8 The Dispersion Trade

The classic dispersion trade is a market-neutral short correlation trade, popular among volatility arbitrage practitioners, that typically takes long positions in options listed on the equities of single companies and short positions in a related index option. For example, a fund may buy options on 50 different large-capitalization, U.S.-listed firms and take a short position in options listed on the S&P 500 Index. Typically, the goal is to create a basket of options on individual assets that mimics the composition of the index closely, perhaps by matching the industry weights of the portfolio.

The key to the dispersion trade is the relationship between a portfolio of options and a single option on a portfolio. That relationship is driven by volatility, which in turn is driven by correlations across assets. Portfolio variance is lower when the constituent stocks have lower volatility and lower correlation with each other. Conversely, as the correlations between stocks rise, portfolio variance increases, as there are fewer stocks experiencing offsetting price moves. Thus, the relative returns of options on indices and options on individual assets are driven by changes in the anticipated correlation among the assets. In practice, individual assets are not highly correlated with each other, so the realized volatility of individual assets tends to be substantially higher than the realized volatility of a related index. Therefore, the implied volatilities of options on individual assets tend to be higher than the implied volatility of an option on a related index. Equation 19.4 expresses the variance of the return of a portfolio as depending on the variance of the constituent assets and their correlations:

(19.4)

where V(R_p) is the variance of the portfolio, R_p is the return on the portfolio, n is the number of assets in the portfolio, w_i is the weight of asset i in the portfolio, σ_i is the standard deviation of returns for asset i, and ρ_ij is the correlation coefficient between returns on assets i and j. When i = j, ρ_ij = 1.

In summary, correlation drives the magnitude of the differences between the volatilities of individual assets and portfolios. Lower values of correlation generate lower portfolio risk through diversification, whereas higher correlation inhibits diversification. Dispersion trades are speculations on correlation. The classic dispersion trade is that realized correlations between assets will be lower than the correlation implied by the pricing of index options relative to options on individual assets. Therefore, the classic dispersion trade is referred to as a short correlation trade because the trade generates profits from low levels of realized correlation and losses from high levels of realized correlation.

Profits from the classic dispersion trade (long individual asset options and short index options) are the greatest during times of declining correlation, and losses occur when correlations rise significantly. The logic and terminology of dispersion trades parallel those of most option trading. Fund managers focus on the difference between implied correlations and realized correlations rather than implied volatility and realized volatility. The ideal condition for a classic dispersion trade is when implied correlation between stocks is high and the fund manager can consistently predict when realized correlation is going to be lower. Conversely, traders may implement a reverse dispersion trade—buying the index options and writing the single stock options—when implied correlation is lower than the trader's expectation for realized correlation.

As an example, consider a basket of four stocks (stocks A, B, C, and D), each of which is one-quarter of the weight of an index. Begin by assuming that the classic dispersion trade is implemented by purchasing equal quantities of four call options that are near-the-money on the four individual stocks and writing call options on the index. The short position in the index calls is assumed to have an aggregated magnitude in terms of underlying asset value equal to the sum of the underlying asset values of the four individual options. Assume that the options have three months to expiration, that the implied correlation among the four individual stocks in the index is equal to 0.30, and that the implied annualized volatilities of the individual options are all 0.40.

The profits from the classic dispersion trade are high when the realized correlation is lower than the implied correlation (i.e., when realized volatilities on individual stocks are relatively high, and realized volatilities on the index are relatively low). For example, if stocks A and B rise in value by 50% during the lifetime of the options, and stocks C and D fall by 50% over the same time period, the profitability of the single stock call options on A and B is extremely high due to the large upward movement in the underlying stocks. The call options on stocks C and D are worthless. The positive gamma ensures that the profits on the options on A and B will exceed the losses on C and D, so that the aggregated long positions in the individual options perform very well. Note that although the stocks experienced large moves, the stock market index was unchanged, as the positive returns on A and B were offset by the negative returns on C and D. The correlations among the assets in the index were a mix of positive and negative values. Since the stock market index was unchanged, the index options that had been written expire worthless, making for an extremely profitable dispersion trade. In a nutshell, the realized correlation was lower than the implied correlation.

Note that the dispersion trade in this example would also be very profitable using put options instead of call options. In that case, the put options on C and D would pay off well, while the losses on the put options on A and B would be limited. Further, the short position in the index put would generate a profit by expiring worthlessly. The reason that either calls or puts would generate profits is that the market remained unchanged.

Having analyzed the profitability of the classic dispersion trade, it is easy to compute the profitability of the reverse trade (buying the index option and writing the individual options). The reverse trade would have lost money using calls or puts, since the classic trade and reverse trade are mirror images of each other. Delta neutrality can be pursued either by mixing calls and puts (i.e., using straddles and strangles) or by hedging with the underlying assets of the options.

Now consider what would happen in a classic dispersion trade using call options if all four stocks moved up 50% together or down 50% together. In other words, what would happen if the realized correlation was 1.00? If all four stocks rose 50%, all four call options on the individual stocks, as well as the index option, would pay approximately 50% of the value of the underlying assets; and given the weighting assumptions, the aggregated payoff would be zero, due to the loss on the short position in the index option. If all four stocks fell by 50%, all four call options as well as the index option would expire worthlessly, and the aggregated payoff would again be zero. However, the classic trade would generate losses in either scenario, since the positions required an initial outlay of capital. The reason that establishing the positions required an initial outlay was that the options on the individual stocks cost more than the income the writing of the index option generated; this is because the implied volatilities of the individual options exceeded the implied volatility of the index, which is always the case when the implied correlation is less than one.

Exhibit 19.8 summarizes the risks of the volatility arbitrage strategy.

19.3.9 Historical Return Observations

Exhibit 19.9 summarizes the monthly returns of the HFRI Relative Value Volatility Index from January 2005 to December 2014, along with the overall HFRI Relative Value Index and several major market indices, following the standard format used throughout this book and detailed in the appendix. Exhibit 19.9a indicates that the cross-sectionally averaged returns of volatility funds were low but with modest volatility and therefore a modest Sharpe ratio. Volatility funds experienced a negative skew as well as modest leptokurtosis. Both the volatility fund index and the overall relative value index exhibited relatively small minimum monthly returns and small maximum drawdowns.

Exhibit 19.9b indicates the low volatility and modest overall returns of volatility funds through a cumulative wealth index. Exhibit 19.9c fully meets expectations by showing low correlations and betas for volatility funds with other indices except for the changes in the VIX. Volatility funds were strongly inversely correlated with market volatility, indicating a positive correlation to major market events.

Finally, Exhibit 19.9d illustrates the low correlation of volatility fund returns and world equity returns through a scatter diagram. Volatility funds provided nice diversification over the period but exhibited poor average returns. However, Exhibit 19.9 does not provide convincing evidence that future average returns will be similarly disappointing.

19.4.1 The Core of Fixed-Income Arbitrage Strategies

At the core of any arbitrage strategy is a model of how prices should behave. This model may be based on theory, empirical observations, or both. The arbitrage is often performed on a pair of securities with a long position in one security offset by a short position in the other security. However, the arbitrage can involve any number of longs and shorts.

An example of a three-security trade is as follows: Assume that based on theoretical reasons or past observations, a fund manager predicts that the yield on 9-month debt will trade at a particular relationship to the yields on 6-month and 12-month debt. Assume that the fund manager predicts that the 9-month yield will trade within five basis points of the mean between the other two yields in a particular market. The fund manager might take a long position in the 9-month debt whenever its yield trades above this relationship, while taking offsetting short positions in the 6-month and 12-month bonds. The fund manager is speculating that the yield on the 9-month debt will decline relative to the average yields of the other two bonds as its yield returns toward the long-term relationship that the fund manager predicts. Note that the manager is not speculating necessarily that the 9-month yield is absolutely high or that the 6- and 12-month yields are absolutely low. Rather, the manager is speculating on the relative values and, in particular, that the relative values will converge as predicted by the manager's model.

Fixed-income arbitrage managers search continuously for pricing inefficiencies across all fixed-income markets. These arbitrage strategies are similar to the traditional goal of buying low and selling high. However, in arbitrage, the trade is based on relative value rather than absolute value, and the goal is to hedge the aggregated position against all risks other than the specific behavior on which the manager is speculating. The arbitrageur hedges the positions against market factors such as credit risks and general interest rate risks, then waits for the relatively undervalued security (or securities) to increase in value, the relatively overvalued security (or securities) to decline in value, or both to occur.

In most cases, trades are designed to be duration-neutral. Duration is a measure of the sensitivity of a fixed-income security to a change in the general level of interest rates. A duration-neutral position means that the returns to the position are relatively insensitive to changes in the general level of market interest rates. However, fixed-income positions can also be exposed to other risks, such as changes in credit spreads, changes in yield curve shapes, changes in volatility, and changes in liquidity.

Generally, the perceived relative mispricing between fixed-income securities is small. Thus, the potential profit of the fixed-income arbitrageur is typically small relative to the sizes of the long and short positions. By controlling for other risks, the hedge fund manager attempts to generate returns driven solely by the behavior of the pricing discrepancy. If the pricing discrepancy converges over time, the strategy should generate a profit. If the pricing discrepancy diverges further, the positions generate losses.

Given the relatively small potential profits as a proportion of position sizes, hedge fund managers typically add more profit potential through leveraging their portfolios with direct borrowings from their prime brokers or with swaps and other derivative securities. This leverage can lead to substantial positive returns when prices return to their predicted levels, which typically happens in normal markets but can create disastrous losses in turbulent environments. Key issues in such arbitrage strategies are managing liquidity and adjusting the size of the positions as perceived price discrepancies diverge further and further in turbulent markets. If positions are reduced, the fund may have reduced its profit potential when the prospects for future profits are at their highest. However, if positions are maintained or increased as losses mount, the firm runs the risk of being forced to liquidate when price discrepancies and losses are at their highest levels.

19.4.2 Types of Fixed-Income Arbitrage Strategies

There are numerous ways to categorize fixed-income arbitrage strategies. Within a particular bond market, positions may be established by anticipating various changes in relationships. These strategies include speculations that the yield curve will become less steeply sloped (yield flattener), that the yield curve will become more steeply sloped (yield steepener), or that portions of the curve will become more curved or less curved (yield butterflies). These are examples of intracurve arbitrage positions because they are based on hedged positions within the same yield curve.

A yield curve is the relationship between the yields of various securities, usually depicted on the vertical axis, and the term to maturity, usually depicted on the horizontal axis. The terms yield curve and term structure of interest rates are often used interchangeably. Sometimes the term structure of interest rates is distinguished from the yield curve because the yield curve plots yields to maturity of coupon bonds, whereas the term structure of interest rates plots actual or hypothetical yields of zero-coupon bonds.

There are also intercurve arbitrage positions, which means arbitrage (hedged positions) using securities related to different yield curves. Examples include swap-spread trading (arbitraging differences in swap rates) and carry trades. Carry trades attempt to earn profits from carrying or maintaining long positions in higher-yielding assets and short positions in lower-yielding assets without suffering from adverse price movements. For further examples, see Duarte, Longstaff, and Yu's “Risk and Return in Fixed-Income Arbitrage: Nickels in Front of a Steamroller?”⁵ They discuss swap-spread arbitrage, yield-curve arbitrage, mortgage arbitrage, volatility arbitrage, and capital-structure arbitrage.

Fixed-income arbitrage funds are often differentiated by the markets in which they speculate. These markets fall into a number of categories, including sovereign debt and asset-backed or mortgage-backed securities.

19.4.3 Fixed-Income Arbitrage Strategies: Sovereign Debt

Sovereign debt is debt issued by national governments. Sovereign debt possesses distinct credit risks from corporate debt because governments can choose to default on their obligations even when they are technically able to meet them. Further, most national governments can use monetary policy to alter the value of their currency and thereby change the real value of their outstanding obligations. In other words, most national governments can literally print money to pay their debts but can choose to default anyway. Sovereign debt ranges in creditworthiness from the low-credit-risk obligations of the largest and most secure nations to the obligations of the least creditworthy nations. Fixed-income arbitrage and hedging using the obligations of the U.S. government are illustrated here.

Fixed-income arbitrage does not need to use exotic securities. For example, it can be nothing more than buying and selling U.S. Treasury securities. In the U.S. bond market, the most liquid securities are on-the-run U.S. Treasury bonds. On-the-run Treasury bonds are the most currently issued bonds for each common maturity issued by the U.S. Treasury Department (e.g., 3-month, 6-month, and 12-month Treasury bills; 10-year notes; and so forth). There are other U.S. Treasury bonds outstanding (known as off-the-run) that have similar maturities and coupons to the on-the-run Treasury bonds. However, off-the-run bonds were issued much earlier than on-the-run bonds and are now less liquid, as dealers are less actively trading them and many of them have been bought and held by long-term investors. As a result, price discrepancies occur among off-the-run issues, as well as between on-the-run and off-the-run issues. The difference in prices may be very small, just a few 32nds of 1%, but can increase in times of high uncertainty, when there are high and erratic levels of trading as investors shift money into and out of the most liquid U.S. Treasury bonds in response to the market crisis.

Another form of fixed-income arbitrage involves trading among maturity ranges of fixed-income securities, especially those that are relatively close to maturity. This is a form of yield-curve arbitrage. These types of trades are driven by temporary imbalances in the supply of and demand for the securities that apparently cause temporary distortions in the yield curve. Kinks in the yield curve can happen at any maturity and usually reflect a change in liquidity demand around the focal point. These kinks provide an opportunity to speculate on changes in the shape of the yield curve by purchasing and selling Treasury securities that are similar in maturity.

Investors who hold bonds can view their returns as being driven not just by shifts in the yield curve but also by the change in a bond's yield if the yield curve remains constant and the maturity of the bond shortens. The process of holding a bond as its yield moves up or down the yield curve due to the passage of time is known as riding the yield curve. Consider a yield curve with an upward slope between the two-year and five-year maturities. The holder of the five-year Treasury bond can profit by rolling down or riding down the yield curve toward the two-year rate if the yield curve does not shift. Rolling down the yield curve is the process of experiencing decreasing yields to maturity as an asset's maturity declines through time in an upward-sloping yield curve environment. In other words, if the yield curve remains static, the five-year Treasury note ages into a lower-yielding part of the yield curve.

Continuing the example of a yield curve that slopes upward, the investor might buy a five-year note at a yield of 5.2% and hold it for three years. If the yield curve has not changed over this holding period, the resulting two-year note position will now fall to a yield of perhaps 5.1%. As the bond's yield falls from 5.2% to 5.1% with the passage of time, the owner of this bond has a profit from rolling down the curve. Moving down the yield curve generally means positive price appreciation as a bond's yield declines. Conversely, Treasury bonds with maturities in a downward sloping range of the yield curve roll up the yield curve to higher yields if the yield curve remains static. This means that the bond prices would underperform if the yield curve remains static and the bond ages into a higher-yielding maturity range.

The slope of the yield curve usually differs across various maturity ranges. Based on differences in the slopes along the yield curve, an arbitrage trade might be to purchase bonds in an upward-sloping maturity range and short bonds in a downward-sloping maturity range. As the short bond positions roll up the yield curve, their values should decline as yields rise, while the long bond positions should increase in value as they roll down the yield curve. This arbitrage trade will work as long as the yield curve is static. In an efficient market, the yield curve could be expected to shift in a manner to make expected risk-adjusted returns equal.

Attempts to arbitrage yield curves have risks. First, shifts in the yield curve up or down can affect the profitability of the trade if it is not duration-neutral. A duration-neutral position is a portfolio in which the aggregated durations of the short positions equal the aggregated durations of the long positions weighted by value. A duration-neutral position is protected from value changes due to shifts in the yield curve that are small, immediate, and parallel. A parallel shift in the yield curve happens when yields of all maturities shift up or down by equal (additive) amounts. However, a hedge that is duration-neutral does not necessarily provide perfect interest rate immunization. Interest rate immunization is the process of eliminating all interest rate risk exposures. Duration-neutral positions may still be exposed to the risks of large or nonparallel interest rate shifts. To provide immunization against more general interest rate behavior, the hedge fund manager needs to regularly adjust the positions to maintain duration neutrality and possibly needs to introduce other positions to provide protection from other sources of risk, such as large and nonparallel yield curve shifts.

For fixed-income securities without option characteristics, duration is calculated as the value-weighted average time to maturity of the security's principal and coupon cash flows. A zero-coupon bond pays only the principal value at maturity with no coupon payments, so its duration equals its maturity. Thus, the duration of a five-year zero-coupon bond is five. The derivative of that bond's log price with respect to its continuously compounded yield to maturity is minus five. So for each small change in its continuously compounded yield, the price moves in the opposite direction with a magnitude of five. If the bond's continuously compounded yield instantaneously falls by 0.1% (e.g., from 4.0% to 3.9%), the bond's price would rise by approximately 0.5%. Rather than expressing the relationship with continuous compounding, the sensitivity of a bond price with respect to discretely compounded yields can be expressed as the modified duration. Modified duration is equal to traditional duration divided by the quantity [1 + (y/m)], where y is the stated annual yield, m is the number of compounding periods per year, and y/m is the periodic yield. With continuous compounding, m is infinity, and traditional duration equals modified duration.

Although duration can be used as a linear approximation of a bond price's change to small yield changes, bond prices have nonlinear relationships to their yields, making the approximation inaccurate for large yield changes. The nonlinear relationship between a bond's price and its yield is measured by its convexity.

Consider a two-year note with a 2% yield to maturity and a five-year note with a 3% yield to maturity, both paying semiannual coupon interest. The two-year note has a duration of 1.97 years, and the five-year note has a duration of 4.68 years. Because the five-year note is expected to be 2.376 times (i.e., 4.68/1.97) more volatile than the two-year note for a given change in yield, a trade that equally weights the long five-year note positions and the short two-year note positions will be exposed to the risk of increases in the market level of interest rates. To make this trade market-neutral to a parallel shift in the yield curve (such as yields rising by 0.1% at both maturities), a duration-neutral weighting must be used. The trader would sell short $2.376 million of the two-year note for each $1 million held long in the five-year note. The total profit or loss of the position would depend on interest rate behavior. For example, the potential benefits of rolling up the yield curve with the short position and down the yield curve with the long position could add considerably to the final profits.

There is a strong parallel between duration hedging in fixed-income securities and delta hedging in options. Both are linear approximations to nonlinear relationships; therefore, they hold only as approximations, with increasing inaccuracy when there are large shifts. The nonlinearity is addressed in both cases by second-order risk measures: convexity for bonds and gamma for options.

19.4.4 Asset-Backed and Mortgage-Backed Securities Strategies

Still another subset of fixed-income arbitrage trades is asset-backed securities (ABS), which are securitized products created from pools of underlying loans or other assets. ABS can diversify the idiosyncratic risk of the underlying assets through the use of pooling, while the securitization or structuring of such a pool can create a security that meets the risk and return preferences of investors. Moreover, ABS transform assets that are not easily traded into securities that can be much more easily traded. These loans are originally issued for a variety of purposes, including credit cards, tuition, automobiles, and mortgages on residential and commercial properties. Banks and other financial institutions originate loans to individual borrowers and then sell the loans into the financial markets through the pooling and securitization process. After loan originators sell these loans into the securitized pools, capital is returned to the banks or other institutions that issued the loans, restoring their capacity to make new loans.

Cash flows from ABS are difficult to predict due to the borrowers' option to prepay the loans and the probabilities of various default rates. Therefore, the valuation of ABS is complex, requiring advanced modeling and sophisticated analysis. The complexity of these securities and their valuations makes them a fertile area for fixed-income arbitrage.

A 2013 report from Guggenheim Partners estimates the size of the U.S. structured finance market at $10 trillion.⁶ This number includes $1.2 trillion in asset-backed securities, including auto loans, student loans, credit cards, as well as commercial structured products, such as aircraft leases and collateralized loan obligations (CLOs). The residential and commercial MBS outstanding, then, total approximately $8.8 trillion.

19.4.5 Prepayment Risk and Option-Adjusted Spreads

Most consumer loans, including auto loans and mortgage loans, allow borrowers to make principal payments in excess of that required by the loan's amortization schedule. Although the loans have a stated maturity, unscheduled principal payments or prepayments cause the loans to be repaid ahead of schedule, leaving ABS and MBS with an uncertain duration. When securities have option characteristics that alter the interest rate risk, risk is usually measured as effective duration. Effective duration is a measure of the interest rate sensitivity of a position that includes the effects of embedded option characteristics. Thus, the effective duration of a 30-year mortgage, or any callable bond, is substantially lower than its traditional duration (i.e., the weighted average of the times to maturity of the mortgage's scheduled cash flows).

Chapter 14 provided details on the measurement of prepayment rates for mortgages. Modeling prepayment risk is a complex and important part of ABS and MBS investments. Investors who model ABS prices by assuming a prepayment speed that is too fast typically overvalue a security by underestimating its longevity. Those underestimating prepayment speeds project receiving payments too slowly, overestimate longevity, and typically undervalue the security.

Prepayment risk is typically to the detriment of ABS and MBS investors, since prepayment is a short option position to the investor. When interest rates rise, borrowers prepay more slowly, which leads to rising duration during times of falling bond prices. Conversely, when interest rates decline, consumers rush to refinance their mortgages and other debts, reducing the longevity of the payment streams received by ABS investors. The higher prepayment rates in falling interest rate environments increase the cash received by the investors in an interest rate environment with low reinvestment rates. In short, the option for borrowers to prepay their debt when interest rates fall is valuable to borrowers. Optimal exercise of those options benefits the borrowers and harms the investors in ABS. Investors in ABS are well aware of the embedded option and are therefore careful to price securities properly by taking into account the value of the embedded short positions in options.

Mortgage-backed securities arbitrage attempts to generate low-risk profits through the relative mispricing among MBS or between MBS and other fixed-income securities. For example, MBS arbitrage can be performed between fixed-income markets, such as buying MBS and selling U.S. Treasuries. This investment strategy is designed to capture inefficiencies between U.S. Treasuries and MBS while hedging underlying interest rate risk with short positions in U.S. Treasuries. To reflect the uncertainties associated with MBS, these securities trade at a spread over U.S. Treasuries. This spread reflects any credit risk of the MBS along with the value of the short call option (the prepayment option) embedded into the MBS.

MBS arbitrage can be quite sophisticated. Hedge fund managers use proprietary models to price the value of the prepayment options and to value the MBS. The short call option implicit in a prepayable fixed-income security causes the price of the security to be lower and the yield of the security to be higher than in an otherwise comparable security without the prepayment option. A key concept in pricing fixed-income securities with embedded prepayment options is the option-adjusted spread (OAS), which is a measure of the excess of the return of a fixed-income security containing an option over the yield of an otherwise comparable fixed-income security without an option after the return of the fixed-income security containing the option has been adjusted to remove the effects of the option. For example, a prepayable mortgage may have a yield of 7%. A Treasury security of comparable maturity and with no call features may have a yield of only 5.5%. Analysis indicates that 90 basis points, or 0.9% of the mortgage's yield, is attributable to the prepayment option. The OAS would be the remaining difference in yield, 0.6%, or 60 basis points. The difference in yield may be attributable to credit risk, to liquidity differences, to mispricing, or to taxability differences. More formally, the OAS is calculated as the spread over the Treasury spot curve that equates the present value of a bond's cash flows to its market price, incorporating the fact that the bond's cash flows may change under different interest rate environments. The calculations are based on a specific model, and thus OAS is model dependent. Hedge fund managers can use mortgage pricing models that rely on the concept of OAS to evaluate the market prices of ABS. In effect, the hedge fund manager estimates the option-adjusted price of various ABS using OAS and searches for relatively mispriced securities.

A hedge fund manager may attempt to arbitrage perceived pricing differentials within the ABS and MBS markets. The options embedded in ABS in general and MBS in particular are enormously complex. Some borrowers may make prepayments to exploit interest rate changes (i.e., refinancing when rates fall). However, other prepayments are made for idiosyncratic factors, such as when the homeowner moves or needs to refinance to withdraw equity from the house. Default represents a prepayment when the mortgage is covered by mortgage insurance. Prepayments due to default can be a benefit or a disadvantage to lenders, depending on the interest rate of the mortgage and current interest rate levels. The substantial cash flow timing uncertainty and highly complex option characteristics of ABS provide potential for security mispricing and arbitrage.

19.4.6 Risks of Asset-Backed and Mortgage-Backed Securities Arbitrage

Many risks are associated with MBS arbitrage. Mortgage-backed securities have complex risks that are driven not just by changes in interest rate levels but also by changes in the shape of the yield curve, the prepayment rates of the borrowers, and the default rates of the borrowers. Hedging these risks may require the purchase or sale of MBS derivative products or other derivative products—including exchange-traded products—and OTC products, such as interest rate forwards, swaps, and OTC options.

The use of OTC derivatives for hedging adds counterparty risk. If a hedging strategy is accomplished using exchange-traded futures and options, counterparty risk is negligible, as the exchange's clearinghouse stands behind every trade. However, if the hedge fund manager hedges with an OTC instrument such as a swap, it is a private transaction for which the hedge fund manager accepts the risk that the counterparty may not complete the transaction by paying cash flows according to the terms of the swap. Although this risk can be minimized through collateral and standardized contractual agreements, it is not foolproof, as the sudden collapse of Lehman Brothers in 2008 demonstrated.

As noted earlier, during a flight to quality, some investors tend to seek out the most liquid markets, such as the on-the-run U.S. Treasury market, and bid the prices of these securities up to induce their holders to sell them at a time of crisis. Conversely, some investors liquidate riskier positions and offer them at low prices in order to induce other investors to buy them at a time of crisis. The decline in Treasury yields and the increase in yields of risky assets cause credit spreads to temporarily increase beyond what is historically, or perhaps even economically, justified. In this case, sophisticated investors with sufficient liquidity may speculate that the MBS market is priced very cheaply compared to U.S. Treasuries. The arbitrage strategy would be to buy MBS and sell U.S. Treasury securities when the interest rate exposure of both instruments is sufficiently similar to eliminate most (if not all) of the risk with regard to Treasury yield levels. The expectation is that the credit spread between MBS and U.S. Treasuries will decline and that MBS will increase in value relative to U.S. Treasuries.

What should be noted about fixed-income arbitrage strategies is that they are generally designed to have profitability that is independent of the direction of the general financial markets. Arbitrageurs seek out pricing inefficiencies based on relative valuations between securities instead of making bets on the absolute pricing of the overall market. Exhibit 19.10 summarizes the major risks of fixed-income arbitrage funds.

19.4.7 Historical Return Observations for Fixed-Income Arbitrage Strategies

Exhibit 19.11 summarizes the monthly returns of the HFRI Relative Value Fixed-Income Corporate Index from January 2000 to December 2014, along with the overall HFRI Relative Value Index and several major market indices, following the standard format used throughout the book and detailed in the appendix. Exhibit 19.11a indicates that the cross-sectionally averaged returns of fixed-income arbitrage funds were moderate and volatility was modest, with a moderate Sharpe ratio. Fixed-income arbitrage funds experienced a substantial negative skew as well as substantial leptokurtosis. Exhibit 19.11b indicates the steady positive performance of fixed-income arbitrage prior to 2007 and subsequent to 2009. However, the large decline of fixed-income arbitrage funds during the first two years of the financial crisis was key in explaining the overall modest performance of this arbitrage strategy. Exhibit 19.11c indicates high correlations and betas for fixed-income arbitrage fund returns with most other indices. Fixed-income arbitrage funds benefited from gains in high-yield bonds but suffered from increases in credit spreads and equity volatility, consistent with expectations. Note that the high correlation between equity and fixed-income arbitrage is a proxy for the exposure of this strategy to increased financial distress. This can be seen by the low estimated multivariate betas in comparison to the high univariate betas. In other words, fixed-income arbitrage, as expected, does not have direct exposure to equities. Finally, Exhibit 19.11d illustrates the high correlation of volatility fund returns and world equity returns through a scatter diagram. Note that the very worst month for both world equities and fixed-income arbitrage is the same month. Exhibit 19.11d vividly depicts the strong correlation between the extreme returns of fixed-income arbitrage funds and world equities. Again, it is important to compare the high univariate betas to the relatively low multivariate betas to understand the true risk exposure of this strategy.