Credit spreads modeling

As shown in previous sections, the credit spread on a corporate bond takes into account its expected default loss. Structural approaches are based on the option pricing theory of Black Scholes and the value of debt depends on the value of the underlying asset. The determination of yield spread is based on the firm value in which the default risk is found as an option to the shareholders. Other models proposed by Black and Cox (1976), Longstaff and Schwartz (1995) and others try to overcome the limitation of the Merton s model, like the default event at maturity only and the inclusion of a default threshold. This class of models is also known as first passage models . 

The empirical evidence is that by including these assumptions, the Black Cox s model generates credit spread very similar to the ones observed in the market. 

The research compares the model spread to the one observed in the market. In order to determine the term structure of credit spread. Eons uses historical probabilities by Moody s database, adopting a recovery rate of 48.38%. The empirical evidence is that bonds with high investment grade have an upward credit spread curve. Therefore, the spread between defaultable and default-free bonds increases as maturity increases. Conversely, speculative-grade bonds have a negative or flat credit yield curve (Figure 8.7). 

In practice, assuming the discrete time case, the transition matrix includes the transition probabilities between the possible states. Therefore, in this model, market prices are used to find the credit spread and convert the matrix of transition probabilities to the time-dependent risk-neutral matrices Qt t+i- The credit spread is given by Equation (8.32) ... 

The risk-free rate affects both elements, option-free bond and embedded option. Conversely, the credit spread is applied to the risk-free rate in order to find the price of the option-free bond. If the credit spread is also included into the option pricing model, the option value rises. For instance, consider the scenario in which the risk-free rate is 1.04% and the option value is 0.46. If the risk-free rate is 7.04%, then the option value increases to 0.66. Figure 9.16 shows the effect of a different interest rate level. 

In Chapter 8, we described several models to measure the term structure of credit spread and we introduced the model proposed by Longstaff and Schwartz (1995) for pricing fixed-rate debt. The authors propose also a model to valuing floating-rate notes. The equation derived for pricing floating-rate bonds is given by (10.2) ... 

The appropriateness of either model (cash-flow-based versus market value-based) will depend on the asset manager s trading style as well as the particulars of the asset class the asset s market liquidity, duration profile, and credit spread volatility. In terms of mechanics, cash flow arbitrage CDOs are no different than balance sheet CDOs, (again, the only difference being their intended pnrpose and asset sourcing strategy). Consequently, one should see the section on Balance Sheet CDOs for further details. Now, we shift the discussion to market value CDOs. 

Robert Jarrow and David Lando, A Markov Model for the Term Structure of Credit Spreads, Review of Financial Studies 10 (1997), pp. 481-523 Darrell Duffie and Kenneth Singleton, Modelling Term Structures of Defaultable Bonds, Review of Financial Studies (1997). 

The Das-Tufano (DT) model is an extension of the JLT model. The model aims to produce the risk-neutral transition matrix in a similar way to the JLT model however, this model uses stochastic recovery rates. The final risk neutral transition matrix should be computed from the observable term structures. The stochastic recovery rates introduce more variability in the spread volatility. Spreads are a function of factors that may not only be dependent on the rating level of the credit as in practice, credit spreads may change even though credit ratings have not changed. Therefore, to some extent, the DT model introduces this additional variability into the risk-neutral transition matrix. 

Various credit derivatives may be priced using this model for example, credit default swaps, total return swaps, and credit spread options. The pricing of these products requires the generation of the appropriate credit dependent cash flows at each node on a lattice of possible outcomes. The fair value may be determined by discounting the probability-weighted cash flows. The probability of the outcomes would be determined by reference to the risk neutral transition matrix. 

The pricing of credit derivatives that pay out according to the level of the credit spread would require that the credit spread process is adequately modeled. In order to achieve this, a stochastic process for the distribution of outcomes for the credit spread is an important consideration. 

An example of the stochastic process for modeling credit spreads, which may be assumed, includes a mean reverting process such as... 

The pricing of a European spread option requires the distribution of the credit spread at the maturity (T) of the option. The choice of model affects the probability assigned to each outcome. The mean reversion factor reflects the historic economic features overtime of credit spreads, to revert to the average spreads after larger than expected movements away from the average spread. 

More complex models for the credit spread process may take into account factors such as the term structure of credit and possible correlation between the spread process and the interest process. 

Expiry in Six Months Risk-free rate = 10% Strike = 70 bps Credit spread = 60 bps Volatility = 20% Mean Reversion Model Price Standard Black Scholes Price Difference Between Standard Black Scholes and Mean Reversion Model Price... 

First generation pricing models for credit spread options may use models as described in the section on spread models. The key market parameters in a spread option model include the forward credit spread and the volatility of the credit spread. 

A key issue with credit spread options is ensuring that the pricing models used will calibrate to the market prices of credit risky reference assets. The recovery of forward prices of the reference asset would be a constraint to the evolution of the credit spread. More complex spread models may allow for the correlation between the level of the credit... 

For the following example, we utilize the Barclays Capital Portfolio Analytics System XQA, which incorporates the aforementioned multifactor model. Again, this model incorporates factors that include points on the yield curve as well as factors related to credit spreads. We took the yield curve data in the sterling model from gilts and for the euro model from Bunds. The credit spread factors consist of buckets by sector and rating, among other factors. 

Our sterling multifactor model consists of 32 factors reflecting changes in yield curve and credit spreads. We obtained historical monthly... 

Practitioners increasingly model credit risk as they do interest rates and use spread models to price associated derivatives. One such model is the Heath-Jarrow-Morton (HJM) model described in chapter 4. This analyzes interest rate risk, default risk, and recovery risk—that is, the rate of recovery on a defaulted loan, which is always assumed to retain some residual value. 

The models analyze spreads as wholes, rather than splitting them into default risk and recovery risk. Das (1999), for example, notes that equation (10.1) can be used to model credit spreads. Credit options can thus be analyzed in the same way as other types of options, modeling the credit spread rather than, say, the interest rate. 

For more detail on modeling credit spreads to price credit derivatives, see Choudhry (2004). 

Jarrow, R.A., Lando, D., Turnbull, S.M., 1997. A Markov model for the term structure of credit ride spreads. Rev. Financ. Stud. 10 (2), 481-523. 

The disadvantages of the model include the fact that it depends on the selected historical transition matrix. The applicability of this matrix to future periods needs to be considered carefully, whether, for example, it adequately describes future credit migration patterns. In addition it assumes all securities with the same credit rating have the same spread, which is restrictive. For this reason the spread levels chosen in the model are a key assumption in the pricing model. Finally, the constant recovery rate is another practical constraint as in practice, the level of recovery will vary. 

This same factor can also be used to compute spread risk in markets where there is not enough data to build a detailed credit block. It can also be used in markets where more detailed credit factors are available, but when there is not enough information to expose a bond to the appropriate credit factor. As we will see in what follows, this will be the case when a euro- or sterling-denominated corporate bond is not rated. Based on the observation that bonds with larger spreads are on average more risky, Barra s model assumes the following exposure to the swap factor ... 

At the time of this writing, corporate bonds denominated in currencies others than euro and sterling are only exposed to the local interest factors and if it exists, the swap factor. This swap factor is roughly equivalent to a financial AA spread factor, as the bulk of organizations that engage in swaps are AA-rated financial institutions. The swap model is coarser than the two local credit models discussed in the next section, but it performs adequately because spread changes are highly correlated within markets. 

The euro and sterling markets are broad and liquid markets. Accurately modeling spread risk in these two markets requires market-dependent, credit blocks. ... 

The model uses average spread levels observed within each rating category. Since these levels are market-dependent, so is specific risk. Another consequence is that this approach can only be implemented in highly liquid markets, where there are enough bonds to robustly estimate average spread levels—in practice, markets for which we can construct sector-by-rating credit factors. 

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