# Boris Albul Shankar Narayanan Margarita Tsoutsoura Carles Vergara

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1 Credit Default Swaps: Pricing and Finding the Sensitivity (of the Deviation of Theoretical Price from the Market Price) with Respect to Different Factors Boris Albul Shankar Narayanan Margarita Tsoutsoura Carles Vergara

3 that the bondholder recovers a fraction of 1 w of the par value of the bond in the event of default. We shall assume w to be 0.5 (50% recovery rate). We are also assuming a mean-reversion model for the intensity process l t, as in the paper of Longstaff (2002), Mithal (2002), and Neis (2002), i.e. dl = (a- bl) dt + s l 1/2 dz (1) where a, b and s are positive constants and Z T is a standard Brownian motion. These dynamics also allow for mean reversion & hetroskedasticity, in corporate spreads and guarantee that the intensity process is always nonnegative. Without going into the details of the gory mathematics, which can be found in the paper [1], the credit spread is given by: (2) We are planning to use Monte Carlo simulation technique to evaluate both the numerator and the denominator of equation (2) and then find the value of the spread. At this stage, we are considering the initial value of the default intensity to be based on the credit rating of the corporate bonds and hence it would be different for each kind of firm. The factors a, b and s for equation (1), shall be evaluated empirically as given in the next section. We can alternatively use the closed form solution to find the CS as mentioned in the paper of Longstaff, Mithal, and Neis [1]. 3. Estimation of the parameters & Data requirement The parameters a, b and s are evaluated from the yields of bonds having the maturity in the range of 3 8 years but the yields/prices of the bonds are observed at dates when the CDS is issued. We will try to get as many data points as possible for these bonds. After that, we first make an assumption about the three parameters a, b and s, and then find a value of the bond yield using the closed form solution mentioned in the paper [1]. This is compared to the observed yield/price of the bond and then we shall try to minimize the error between the calculated yield and the observed yield by making another assumption of the three parameters. This is done until the error is minimized to an acceptance level. It s worth mentioning that the notional amount of these bonds should not be less than that of the notional amount of CDS and the bonds chosen should be registered with SEC, having no embedded options. For a firm to be considered for pricing in our project, we need at least two such bonds [data points]. Some of this data is available in Bloomberg. We also contacted Credittrade for seeking this data. Once we obtain the values of a, b and s, we can easily find the default intensity λ T as say: dl = ( l) dt l 1/2 dz Then, those values of a, b and s shall are used to price the credit spread from the equation no. (2.)

5 5. References 1. Longstaff Francis, Mithal Sanjay, and Eric Neis, The Credit-Default Swap Market: Is Credit Protection Priced Correctly? The Journal of Finance (December 2002). 2. Hull, John, and Alan White, 2001, Valuing Credit Default Swaps II: Modeling Default Correlations, The Journal of Derivatives 8, No.3, Spring, Hull, John, and Alan White, 2000, Valuing Credit Default Swaps I: No Counterparty Default Risk, The Journal of Derivatives 8, No.1, Fall, 29-40

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