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- 1. AY 2010-12FIRST AMERICAN BANK:CREDIT DEFAULT SWAPSAdvanced Fixed Income securities<br />CASE INTRODUCTION:<br />CapEx Unlimited is banking customer of Charles Bank International and is going through tough times with a loss of 82 million for the year 2000-01. It is in middle of an industrial shake-out and requires $ 50 to finance the expansion of its network. The company has already accumulated $100 million in previous loans from CBI and was depending on relations with bank for additional funding. CBI’s exposure to CEU would exceed permissible limits if it agreed to lend the amount. Kittal managing director of FAB envisioned helping CBI by mitigating credit risk through single-name credit default swap.<br />First American Bank (FAB) is one of the largest bank in America with asset value of $ 50 billion and businesses in 50 countries. First American Credit Derivatives was an independent business unit housed within Firrst American’s structured products branch. <br />CapEx Unlimited (CEU) is a company primarily into telecommunication focused on Northeast and Midwest United States, providing services like high-speed internet, Web hosting, Web hosting, data networking, voice communication, and video/tele-conferencing. <br />Kittal was contemplating using credit default swap for the current situation, because it made credit risk accessible to a broad range of investors in a way that was simple and confidential. In credit default CBI would make periodic fee payment to FAB in exchange for receiving credit protection. Counterparty risk is lower here given high rating of FAB, another way to reduce counterparty risk was to issue credit linked notes. In a way CBI would be playing the role of intermediate between FAB and CEU.<br />Through a single name credit default swap, party could buy protection from another party with respect to various predefined credit events occurring in certain reference identity. Party buying protection was called “protection buyer” and party providing protection was called “protection seller”. In case of a credit event protection seller makes payment to protection buyer. <br />CEU has total outstanding long term debt of $ 5 billion. Additional lending of 50 million won’t impact the credit rating of CEU. CEU’s publicly traded debt was already below investment grade (B2 rating from Moody’s). The term of new loan included a coupon rate of approximately 9.8% and a maturity of two years. CEU’s existing debt had an average maturity of five years, with average semiannual coupon of $ 130 million. In exchange for protection against a CEU credit event, CBI would make semiannual swap fee payments to First American Bank that coincided with the interest payments it received on the CEU loan. <br />Kittals was confused whether to keep the credit risk in-house or find some investors ready to take on the credit risk. In case of transferring the risk to some other party with low rating there was high counterparty risk, as the hedging is unfunded (high collaterals from low credit entities was not possible). Another way he was contemplating to transfer the risk was issuing credit-linked note which was attractive owing to its funded nature. <br />Pricing a CDS:<br />Calculations:<br />Valuation of CDS involves finding out the spread which is to be added to the reference rate of the derivative market. To calculate this spread we need to calculate the default probabilities each year and this can de done in two ways:<br /><ul><li>Using the historical data and
- 2. Using Bond Prices</li></ul>Both these methods have been demonstrated in the calculations below.<br />Default Probabilities based on Historical dataAs per Exhibit 10(b) TimeCummulative Default ProbabilityUnconditional Default ProbabilitySurvival Rate16.23%6.23%93.77%213.70%7.47%86.77%<br />Default Probabilities based on Bond Prices Risk Free Rate0.045 Yield on B2 bond0.098 Spread5.30% Recovery Rate82.00%From Exhibit 14 Hazard Rate29% <br />TimeCummulative Default ProbabilityUnconditional Default ProbabilitySurvival Rate126%26%74%245%19%60%<br />Qt=1-e-γt<br />Where<br />Q(t)= Cumulative default probability<br /> γ=s1-R <br />S= spread over the risk free bond<br />R=Recovery rate<br />Calculation of Spreads:<br />Using Bond prices to calculate the Default probability <br />Calculation of the present value of expected paymentsTimeProbability of SurvivalExpected PaymentDiscount FactorPV of expected payment10.740.740.9559974820.71216586120.600.600.9139311850.551470088 Total Payment1.263635949<br />TimeProbability of DefaultRecovery RateExpected PayoffDiscount FactorPV of expected payoff10.2550546690.820.045909840.9559974820.0438920.1900017850.820.0342003210.9139311850.031257 Total Payoff0.075146<br />So value of spread = (Total PV of expected payoff)/ (Total PV of expected payment)<br />=5.5% <br />Using Historical data to calculate the Default probability <br />TimeProbability of SurvivalExpected PaymentDiscount FactorPV of expected payment10.940.940.9559974820.89643883920.870.870.9139311850.792975875 Total Payment1.689414714<br />Calculation of the present value of expected payoffs TimeProbability of DefaultRecovery RateExpected PayoffDiscount FactorPV of expected payoff10.06230.820.0112140.9559974820.01072120.07470.820.0134460.9139311850.012289 Total Payoff0.023009<br />So value of spread = (Total PV of expected payoff)/ (Total PV of expected payment)<br />=1.3% <br />So the spread should be in the range of 1.3% to 5.5%<br />Assumptions:<br /><ul><li>Risk free rate is 4.5% which is for 5 years and not 2 years
- 3. Default probability annually is used and not semi-annually which is the actual coupon payment schedule.</li></ul>Hedging Credit Default Swaps<br />The CDS can be hedged through two main mechanism s<br />Create synthetic assets<br />Hedging using Cash Assets<br />Create synthetic assets:<br />For unleveraged investors, the generic synthetic asset strategy is to write default protection, post the required margin and invest the remaining principal in a near-money-market equivalent asset. Triple-A-rated floating-rate credit-card asset-backed securities are usually the cheapest type of asset for creating synthetic assets. These assets have negligible default risk because of early amortization features and credit enhancement achieved through subordination (12 percent to 15 percent) and excess servicing (3 percent to 6 percent). In addition, the potential loss of premium associated with early amortization events is mitigated by the floating-rate structure of the instruments. The combination of a floater and a default swap equates to a synthetic floating-rate note.<br />Investors are motivated to use default swaps to create synthetic assets for two reasons. First, relative value. There are times when a synthetic asset is cheaper than the cash-market equivalent. This is especially true when the implied repo rate in the default swap is trading at Libor. As a result, an investor can monetize the repo premium implied in the default swap, without having to finance the trade. Meanwhile, since out-of-favor or volatile credits tend to trade at higher repo premiums, investors can use default swaps to take views relative to the forward credit spreads implied by the default swap market. <br />A second motivation to use credit default swaps is that the instruments enable investors to tap into a market that's bigger than that of tradable securities. A desired credit exposure that is not available in the cash market can be synthetically created via a default swap. Given the historically low levels of interest rates and the flatness of the yield curve, a disproportionate share of new-issue volume has been both fixed and dated. As a result, the supply of corporate floaters and short-dated fixed-rate bonds has been concentrated in a handful of credits—generally in the financial services.<br />Hedge cash assets:<br />One of the most important applications of default swaps is hedging. All hedges incur basis risk; the basis risk in a default swap stems from the volatility in the implied repo premium. Since this premium will be more volatile for low-rated and distressed credits, these types of credits will be subject to more basis risk than their investment-grade counterparts. As a rule, the cheapest time to implement a hedge is when the market is not concerned about the risk.<br />One way to illustrate the effectiveness of default swaps in hedging is to assess how a hedge performed in the past. Consider a hypothetical hedge employed by a money manager benchmarked to the Merrill Lynch Corporate Aggregate, who held 10 percent of the portfolio in Hilton Hotels (Baa1/BBB). In September 1997, this $500 million portfolio had $50 million in Hilton five-year bonds, which were originally purchased at a discount and now have a four-point gain. The remaining 90 percent of the portfolio matches the index in terms of duration and credit quality. Note that this hedge can be viewed generically. One way to reduce a portfolio's exposure to the REIT market, for instance, would be to buy default protection on the most representative credit.<br />The exposure of the portfolio can be brought back to index levels with either an outright sale of the bonds or a hedge using a default swap. There are three reasons why the portfolio manager might opt to hedge rather than sell: because of adverse tax events (four points of capital gain position), because the cost of hedging is relatively inexpensive (basis could work in favor of hedge), or because of the high transaction cost resulting from low liquidity in the cash market (credit is out-of-favor).<br />Delta hedging of reverses knock-outs<br />The first hedging strategy used by the seller of the reverse knock-out call will be to invest in the underlying asset and continuously readjust that position according to the delta of the option. This will protect the hedge against directional moves of the asset price. This hedge still leaves him with gamma risk, however—a residual risk linked to the amplitude of spot moves, whose impact on his P&L depends on the convexity of the option. When the option profile is convex, he will be hurt by spot moves being higher than those given by his volatility assumption. This could happen at any level when a trader is short a European option and around the strike when he is short a barrier option. When the option profile is concave, which is the case around the barrier in our example, the trader will be hurt by asset price moves being lower than anticipated.<br />The following two scenarios explain the consequences of these concepts and help us understand the problem of pricing, hedging, and marking-to-market exotic options under a Black-Scholes regime. We assume that a skew exists whereby vanilla options struck at or near the barrier are trading at 15 percent rather than the 20 percent implied volatility of at-the-money options.<br />First scenario: The trader chooses to price at the strike volatility (20 percent). If the spot ends up around the barrier and its volatility is lower than 20 percent (as anticipated by the market in that implied volatility at the barrier is lower), he will lose money.<br />Second scenario: The trader chooses to price at the barrier volatility (15 percent). If the spot ends up around the strike and its volatility is 20 percent (as anticipated by the market), he will lose money.<br />Whatever his choice, he will be dependent on the underlying directional moves, which is to say that his pricing and delta hedge are wrong and incompatible with the market expectations. A better model is needed.<br />The "smile” model<br />While Black-Scholes assumes volatility has to be constant, the simplest extension of Black-Scholes that is compatible with market prices of European options and market expectation of volatility at various levels of the underlying is a "smile” model. This model assumes that local volatility is a function of current level and possibly time. In one form or another, it has been implemented by many banks since the mid-1990s and is a major improvement for mark-to-market pricing and risk management. Also, for our up-and-out reverse knock-out option, this model will be able to incorporate different volatilities around the strike and the barrier: a large volatility around the strike (where the option tends to be convex) will increase the price more than Black-Scholes. A lesser volatility around the barrier (where the option tends to be concave) will also increase the price. The compounding of those two effects often leads to a price higher than any Black-Scholes price.<br />Vega hedging<br />We have seen that properly taking into account the negative volatility skew can result in a higher price to the reverse knock-out call than using the constant at-the-money volatility. Another knock-out case could be built with a symmetrical profile with respect to the at-the-money option (100 percent). Let's assume we have a three-month option that is a reverse knock-out put struck at 100 percent with a barrier at 80 percent. Let's further assume that the market smile is as follows:<br />20 percent out-of-the-money vanilla puts are priced at 25 percent implied volatility;<br />at-the-money vanilla options are priced at 20 percent implied volatility;<br />20 percent out-of-the-money vanilla calls are priced at 17 percent implied volatility.<br />Implications<br />Three important properties of this strategy will impact risk management of barrier options in real life:<br /> This hedge is unfortunately not static and will vary as conditions such as the underlying asset price and volatility surface change. There will often be a systematic rebalancing cost associated with the strategy because of convexity in volatility, and hence the need for a stochastic volatility model for even more accurate pricing.<br />Trading several European options for each barrier option will certainly have a tendency for high transaction costs over time. Therefore, vega hedging has to be performed at the portfolio level in order to benefit from any cancellation of risk between the exotics contained in a market-maker's book.<br />We have only described hedging the proper volatility pricing and hedging of an exotic option. It turns out, of course, that with knock-out barriers, when spot approaches the barrier level, gamma becomes the main source of risk. At the extreme—on an expiration day, for example—this risk will be unhedgeable. In other instances in which an option still has a period of time to run, other hedging techniques, which are beyond the scope of this article, have to be used as well.<br />

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