Appendix - Capital Structure Arbitrage
Derivatives-based Portfolio Solutions


When Credit Default Swaps were created in the late 1990’s, they traded independently of the equity derivative market. The high levels of volatility and credit spreads during the bursting of the TMT bubble demonstrated the link between credit spreads, equity, and implied volatility. We examine four models that demonstrate this link (Merton model, jump diffusion, put vs CDS, and implied no-default volatility).

Capital structure arbitrage models can link the price of equity, credit and implied volatility.  However, the relatively wide bid-offer spreads of equity derivatives mean trades are normally carried out between credit and equity  (or between different subordinations of credit and preferred shares vs ordinary shares). The typical trade is for an investor to go  long the security that is highest in the capital structure, for example, a corporate bond (or potentially a convertible bond), and short  a security that is lower in the capital structure, for example, equity. Reverse trades are possible, for example, owning a subordinated  higher yielding bond and shorting a senior lower yielding bond (and earning the positive carry as long as bankruptcy does not occur).  Only for very wide credit spreads and high implied volatility is there a sufficiently attractive opportunity to carry about an arbitrage  between credit and implied volatility. We shall concentrate on trading credit vs equity, as this is the most common type of trade.

Credit spread is only partly due to default risk
The OAS (Option Adjusted Spread) of a bond over the risk-free rate can be divided into three categories. There is the expected loss from default; however, there is also a portion due to general market risk premium and additionally a liquidity cost. Tax effects can also have an effect on the corporate bond market. Unless a capital structure arbitrage model takes into account the fact that not all of a bond’s credit spread is due to the risk of default, the model is likely to fail. The fact that credit spreads are higher than they should be if bankruptcy risk was the sole risk of a bond was often a reason why long credit short equity trades have historically been more popular than the reverse (in addition to the preference to being long the security that is highest in the capital structure in order to reduce losses in bankruptcy).

CDS usually better than bonds for credit leg, as they are unfunded and easier to short
Using CDS rather than corporate bonds can reduce many of the discrepancies in spread that a corporate bond suffers and narrow the difference between the estimated credit spread and the actual credit spread. We note that CDS are an unfunded trade (ie, leveraged), whereas corporate bonds are a funded trade (have to fund the purchase of the bond) that has many advantages when there is a funding squeeze (as occurred during the credit crunch). CDS also allow a short position to be easily taken, as borrow for corporate bonds is not always available, is usually expensive and can be recalled at any time. While borrow for bonds was approx. 50bp before the credit crunch it soared to approx. 5% following the crisis.

Credit derivatives do not have established rules for equity events
While credit derivatives have significant language against credit events, they have no language for equity events, such as special dividends or rights issues. Even for events such as takeovers and mergers, where there might be relevant documentation, credit derivatives are likely to behave differently than equity (and equity derivatives).

We note that there are occasions when corporate bond prices lag a movement in equity prices, simply as traders have not always updated levels (but this price would be updated should an investor request a firm price). CDS prices suffer less from this effect, and we note for many large companies the corporate bond market is driven by the CDS market and not vice versa (the tail wags the dog). Although intuitively the equity market should be more likely to lead the CDS market than the reverse (due to high frequency traders and the greater number of market participants), when the CDS market is compared to the equity market on average neither consistently leads the other. Even if the CDS and equity on average react equally as quickly to new news, there are still occasions when credit leads equity and vice versa. Capital structure arbitrage could therefore be used on those occasions when one market has a delayed reaction to new news compared to the other.

In order for capital structure arbitrage to work, there needs to be a strong correlation between credit and equity. This is normally found in companies that are rated BBB or BB. The credit spread for companies with ratings of A or above is normally more correlated to the general credit supply and interest rates than the equity price. For very speculative companies (rated B or below), the performance of their debt and equity is normally very name-specific, and often determined by the probability of takeover or default.

Capital structure arbitrage works best when companies don’t default
Capital structure arbitrage is a bet on the convergence of equity and credit markets. It has the best result when a company in financial distress recovers, and the different securities it has issued converge. Should the company enter bankruptcy, the returns are less impressive. The risk to the trade is that the company becomes more distressed, and as the likelihood of bankruptcy increases the equity and credit markets cease to function properly. This could result in a further divergence or perhaps closure of one of the markets, potentially forcing aliquidation of the convergence strategy.

Capital structure arbitrage assumes equity and credit markets move in parallel. However, there are many events that are bullish for one class of investors and bearish for another. This normally happens when the leverage of a company changes suddenly. Takeovers and rights issues are the two main events that can quickly change leverage. Special dividends, share buybacks and a general reduction of leverage normally have a smaller, more gradual effect.

Rights issue. A rights issue will always reduce leverage, and is effectively a transfer of value from equity holders to debt holders (as the company is less risky, and earnings are now divided amongst a larger number of shares).

Takeover bid (which increases leverage). When a company is taken over, unless the acquisition is solely for equity, a portion of the acquisition will have to be financed with cash or debt (particularly during a leveraged buyout). In this case, the leverage of the acquiring company will increase, causing an increase in credit spreads and a reduction in the value of debt. Conversely, the equity price of the acquiring company is more stable. For the acquired company, the equity price should jump close to the level of the bid and, depending on the structure of the offer, the debt could fall (we note that if the acquired company is already in distress the value of debt can rise; for example, when Household was acquired by HSBC).

On May 4, 2005, Kirk Kerkorian announced the intention to increase his (previously unknown) stake in GM, causing the troubled company’s share price to soar 18% intraday (7.3% close to close). The following day, S&P downgraded GM and Ford to ‘junk’, causing a collapse in the credit market and a 122bp CDS rise in two days. As many capital structure arbitrage investors had a long credit short equity position, both legs were loss making and large losses were suffered.

For many companies the correlation between equity and credit is not particularly strong, with a typical correlation between 5% and 15%. Hence it is necessary for a capital structure arbitrage investor to have many different trades on simultaneously. The correlation of a portfolio of bonds and equities is far higher (approx. 90%).

While there are many models that show the link between the equity, equity volatility and debt of a company, we shall restrict ourselves to four of the most popular.
  • Merton model. The Merton model uses the same model as Black-Scholes, but applies it to a firm. If a firm is assumed to have only one maturity of debt, then the equity of the company can be considered to be a European call option on the value of the enterprise (value of enterprise = value of debt + value of equity) whose strike is the face value of debt. This model shows how the volatility of equity rises as leverage rises. The Merton model also shows that an increase in volatility of the enterprise increases the value of equity (as equity is effectively long a call on the value of the enterprise), and decreases the value of the debt (as debt is effectively short a put on the enterprise, as they suffer the downside should the firm enter bankruptcy but the upside is capped).
  • Jump diffusion. A jump diffusion model assumes there are two parts to the volatility of a stock. There is the diffusive (no-default) volatility, which is the volatility of the equity without any bankruptcy risk, and a separate volatility due to the risk of a jump to bankruptcy. The total volatility is the sum of these two parts. While the diffusive volatility is constant, the effect on volatility due to the jump to bankruptcy is greater for options of low strike than high strike causing ‘credit induced skew’. This means that as the credit spread of a company rises, this increases the likelihood of a jump to bankruptcy and increases the skew. A jump diffusion model therefore shows a link between credit spread and implied volatility.
  • Put vs CDS. As the share price of a company in default tends to trade close to zero, a put can be assumed to pay out its strike in the even of default. This payout can be compared to the values of a company’s CDS, or its debt market (as the probability of a default can be estimated from both). As a put can also have a positive value even if a company does not default, the value of a CDS gives a floor to the value of puts. As 1xN put spreads can be constructed to never have a negative payout, various caps to the value of puts can be calculated by ensuring they have a cost. The combination of the CDS price floor, and put price cap, gives a channel for implieds to trade without any arbitrage between CDS and put options.
  • No-default implied volatility. Using the above put vs CDS methodology, the value of a put price due to the payout in default can be estimated. If this value is taken away from the put price, the remaining price can be used to calculate a no-default implied volatility (or implied diffusive volatility). The skew and term structure of implied no-default implied volatilities are flatter than Black-Scholes implied volatility, which allows an easier comparison and potential for identifying opportunities.


The Merton model assumes that a company has an enterprise value (V) whose debt (D) consists of only one zero coupon bond whose value at maturity is K. These assumptions are made in order to avoid the possibility of a default before maturity (which would be possible if there was more than one maturity of debt, or a coupon had to be paid. The company has one class of equity (E) that does not pay a dividend. The value of equity (E) and debt (D) at maturity is given below.
Enterprise value = V = E + D
Equity = Max (V – K, 0) = call on V with strike K
Debt = Min (V, K) = K – Max(K – V, 0) = Face value of debt K – put on V with strike K

Enterprise value of a firm at maturity has to be at least K or it will enter bankruptcy
Before the maturity of the debt, the enterprise has obligations to both the equity and debt holders. At the maturity of the debt, if the value of the enterprise is equal to or above K, the enterprise will pay off the debt K and the remaining value of the firm is solely owned by the equity holders. If the value of the enterprise is below K then the firm enters bankruptcy. In the event of bankruptcy, the equity holders get nothing and the debt holders get the whole value of the enterprise V (which is less than K).

Equity is long a call on the value of a firm
If the value of the enterprise V is below the face value of debt K at maturity the equity holders receive nothing. However, if V is greater than K, the equity holders receive V - K. The equity holders therefore receive a payout equal to a call option on V of strike K.

Debt is short a put on value of firm
The maximum payout for owners of debt is the face value of debt. This maximum payout is reduced by the amount the value of the enterprise is below the face value of debt at maturity. Debt is therefore equal to the face value of debt less the value of a put on V of strike K.


As the value of the short put has a delta, debt has a delta. It is therefore possible to go long debt and short equity (at the calculated delta using the Merton model) as part of a capital structure arbitrage trade.

If enterprise value is unchanged, then if value of equity rises, value of credit falls
As enterprise value is equal to the sum of equity and debt, if enterprise value is kept constant then for equity to rise the value of debt must fall. An example would be if a company attempts to move into higher-risk activity, lifting its volatility. As equity holders are long a call on the value of the company they benefit from the additional time value. However, as debt holders are short a put they suffer should a firm move into higher-risk activities.

Merton model assumes too high a recovery rate

Using the vanilla Merton model gives unrealistic results with credit spreads that are too tight. This is because the recovery rate (of V/K) is too high. However, using more advanced models (eg, stochastic barrier to take into account the default point is unknown), the model can be calibrated to market data.

The volatility of an enterprise should be based on the markets in which it operates, interest rates and other macro risks. It should, however, be independent of how it is funded. The proportion of debt to equity therefore should not change the volatility of the enterprise V; however, it does change the volatility of the equity E. It can be shown that the volatility of equity is approximately equal to the volatility of the enterprise multiplied by the leverage (V/E). Should the value of equity fall, the leverage will rise, lifting the implied volatility. This explains skew: the fact that options of lower strike have an implied volatility greater than options of high strike.
σE ≈ σV × V / E (= σV × leverage)

Firms with a small amount of debt have equity volatility roughly equal to firm volatility
If a firm has a very small (or zero) amount of debt, then the value of equity and the enterprise are very similar. In this case, the volatility of the equity and enterprise should be very similar.

Firms with high value of debt to equity have very high equity volatility
For enterprises with very high levels of debt, a relatively small percentage change in the value of the enterprise V represents a relatively large percentage change in the value of equity. In these cases equity volatility will be substantially higher than the enterprise volatility.

Proof equity volatility is proportional to leverage
The mathematical relationship between the volatility of the enterprise and volatility of equity is given below. The N(d1) term adjusts for the delta of the equity.
σE = N(d1) × σV × V / E

If we assume the enterprise is not distressed and the equity is ITM, then N(d1) or delta of the equity should be very close to 1 (it is usually approx. 90%). Therefore, the equation can be simplified so the volatility of equity is proportional to leverage (V / E).
σE ≈ σV × leverage


A jump diffusion model separates the movement of equities into two components. There is the diffusive volatility, which is due to random log-normally distributed returns occurring continuously over time. In addition, there are discrete jumps the likelihood of which is given by a credit spread. The total of the two processes is the total volatility of the underlying. It is this total volatility that should be compared to historic volatility or Black-Scholes volatility.

Default risk explained by credit spread
For simplicity, we shall assume that in a jump diffusion model the jumps are to a zero stock price as the firm enters bankruptcy, but results are similar for other assumptions. The credit spread determines the risk of entering bankruptcy. If a zero credit spread is used, the company will never default. The probability of default increases as the credit spread increases (approximately linearly).

To show how credit spread (or bankruptcy) causes credit-induced skew, we shall price options of different strike with jump diffusion, keeping the diffusive volatility and credit spread constant. Using the price of the option, we shall then calculate the Black-Scholes implied volatility. The Black-Scholes implied volatility is higher for lower strikes than higher strikes, causing skew.

Credit-induced skew is caused by ‘option on bankruptcy’
The time value of an option will be divided between the time value due to diffusive volatility and the time value due to the jump to zero in bankruptcy. High strike options will be relatively unaffected by the jump to bankruptcy, and the Black-Scholes implied volatility will roughly be equal to the diffusive volatility. However, the value of a jump to a zero stock price will be relatively large for low strike put options (which, due to put call parity, is the implied for all options). The difference between the Black-Scholes implied and diffusive volatility could be considered to be the value due to the ‘option on bankruptcy’.


The probability distribution of a stock price can be decomposed into the probability of a jump close to zero due to credit events or bankruptcy, and the log-normal probability distribution that occurs when a company is not in default. While the value of a put option will be based on the whole probability distribution, the value of a CDS will be driven solely by the probability distribution due to default. The (bi-modal) probability distribution of a stock price due to default, and when not in default, is shown below.

Puts can be used instead of CDS (as puts pay out strike price in event of bankruptcy)
When a stock defaults, the share price tends to fall to near zero. The recovery rate of equities can only be above zero if debt recovers 100% of face value, and most investors price in a c40% recovery rate for debt. A put can therefore be assumed to pay out the maximum level (ie, the strike) in the event of default. Puts can therefore be used as a substitute for a CDS. The number of puts needed is shown below.
Value of puts in default = Strike × Number of Puts
Value of CDS in default = (100% – Recovery Rate) × Notional
In order to substitute value of puts in default has to equal value of CDS in default.
  • Strike × Number of Puts = (100% – Recovery Rate) × Notional
  • Number of Puts = (100% – Recovery Rate) × Notional / Strike

As a put can have a positive value even if a stock is not in default, a CDS must be cheaper than the equivalent number of puts (equivalent number of puts chosen to have same payout in event of default, ie, using the formula above). If a put is cheaper than a CDS, an investor can initiate a long put-short CDS position and profit from the difference. This was a popular capital structure arbitrage trade in the 2000-03 bear market, as not all volatility traders were as focused on the CDS market as they are now, and arbitrage was possible.

CDS in default must have greater return than put in default (without arbitrage)
As a CDS has a lower price for an identical payout in default, a CDS must have a higher return in default than a put. Given this relationship, it is possible to find the floor for the value of a put. This assumes the price of a CDS is ‘up front’ ie, full cost paid at inception of the contract rather than quarterly.
Puts return in default = Strike / Put Price
CDS return in default = (100% – Recovery Rate) / CDS Price
As CDS return in default must be greater than or equal to put return in default.
  • (100% – Recovery Rate) / CDS Price ≥ Strike / Put Price
  • Put Price ≥ Strike × CDS Price / (100% – Recovery Rate)

As the prices of the put and CDS are known, the implied recovery rate can be backed out using the below formula. If an investor’s estimate of recovery value differs significantly from this level, a put vs CDS trade can be initiated. For a low (or zero) recovery rate, the CDS price is too high and a short CDS long put position should be initiated. Conversely, if the recovery rate is too high, a CDS price is too cheap and the reverse (long CDS, short put) trade should be initiated.
Put Price = Strike × CDS Price / (100% – Implied Recovery Rate)

CDSs provide a floor to the price of a put. It is also possible to cap the price of a put by considering ratio put spreads. For example, if we have the price for the ATM put, this means we know that the value of a 50% strike put cannot be greater than half the ATM put price. If not, we could purchase an ATM-50% 1×2 put spread (whose payout is always positive) and earn a premium for free. This argument can be used for all strikes K and all 1xN put spreads, and is shown below:
N × put of strike N / K ≤ put of strike K

The combination of CDS prices providing a floor, and put prices of higher strikes providing a cap, gives a corridor for the values of puts. The width of this corridor is narrowest for low strike long maturity options, as these options have the greatest percentage of their value associated with default risk. As for all capital structure arbitrage strategies, companies with high credit spreads are more likely to have attractive opportunities and arbitrage is potentially possible for near-dated options.


The volatility of a stock price can be decomposed into the volatility due to credit events or bankruptcy and the volatility that occurs when a company is not in default. This is similar to the volatility due to jumps and the diffusive volatility of a jump diffusion model. As the value of a put option due to the probability of default can be calculated from the CDS or credit market, if this value was taken away from put prices this would be the ‘no-default put price’ (ie, the value the put would have if a company had no credit risk). The implied volatility calculated using this ‘no-default put price’ would be the ‘no-default implied volatility’. Nodefault implied volatilities are less than the vanilla implied volatility, as vanilla implied volatilities include credit risk).

No-default implied volatilities have lower skew and term structure
While we derive the no-default implied volatility from put options, due to put call parity the implied volatility of calls and puts is identical for European options. As the value of a put associated with a jump to default is highest for low-strike and/or long-dated options, no-default implied volatilities should have a lower skew and term structure than vanilla Black-Scholes implied volatilities. A no-default implied volatility surface should therefore be flatter than the standard implied volatility surface and, hence, could be used to identify potential trading opportunities.

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