3. Default probability annually is used and not semi-annually which is the actual coupon payment schedule.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 quot;
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 quot;
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 />