Spread Pricing Models

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). 

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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. 

The option-adjusted spread (OAS) is the most important measure of risk for bonds with embedded options. It is the average spread required over the yield curve in order to take into account the embedded option element. This is, therefore, the difference between the yield of a bond with embedded option and a government benchmark bond. The spread incorporates the future views of interest rates and it can be determined with an iterative procedure in which the market price obtained by the pricing model is equal to expected cash flow payments (coupons and principal). Also a Monte Carlo simulation may be implemented in order to generate an interest rate path. Note that the option-adjusted spread is influenced by the parameters implemented into the valuation model as the yield curve, but above all by the volatility level assumed. This is referred to volatility dependent. The higher the volatility, the lower the option-adjusted spread for a callable bond and the higher for a putable bond. 

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. 

The pricing of a spread option is dependent on the underlying process. As an example we compare the pricing results for a spread option model, including mean reversion to the pricing results from a standard Black-Scholes model in Exhibit 21.14 and Exhibit 21.15. 

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... 

The fair price of a convertible bond is the one that provides no opportunity for arbitrage profit that is, it precludes a trading strategy of running simultaneous but opposite positions in the convertible and the underlying equity in order to realize a profit. Under this approach we consider now an application of the binomial model to value a convertible security. Following the usual conditions of an option pricing model such as Black-Scholes (1973) or Cox-Ross-Rubinstein (1979), we assume no dividend payments, no transaction costs, a risk-free interest rate, and no bid-offer spreads. 

An example of a two-factor RF model could e.g. be enforced by a separate modeling of bond prices for corporate bonds and default spreads. 

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 . 

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) ... 

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 binomial tree model evaluates the return of a bond with embedded option by adding a spread to the risk-free yield curve. Generally, the price obtained by the model is compared to the one exchanged in the market. If the theoretical price is different, the model can be calibrated with three key elements. The first ones are the volatility and drift factor. They allow to calibrate the model interest rate path in order to obtain the equality with the market yield curve. The third one is the spread applied over the yield curve. Generally, when volatility and drift are correctly calibrated, the last element to select in order to obtain the market parity is the spread. Conventionally, banks define it in the following way ... 

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. 

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. 

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... 

In the single-factor HJM model, forward rates of all maturities move in perfect correlation. For actual market applications— pricing an interest rate instrument that is dependent on the spread between two points on... 

The binomial model evaluates a bond s return by measuring the extent to which it exceeds those determined by the risk-free short rates in the tree. The spread between these returns is the bond s incremental return at a specified price. Determining the spread involves the following steps ... 

To establish a framework for supply chain intermediary analysis, we focus on the economic incentives of three types of players suppliers, buyers, and intermediaries. All players are self interested, profit seeking, and risk neutral. In the simplest form, each supplier has an opportunity cost s, each buyer has a willingness to pay level v that could be public or private information depending on the model assumptions. The intermediary offers an asked price w to the supplier and a bid price p to the buyer while creating a non-negative bid-ask spread p — w) to support her operation. The intermediary has the authority to determine whether a particular trade is to take place using control p. Adopting some mechanism T P,p, w), the intermediary optimizes her own profit. 

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