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 I. Basic math.
 II. Pricing and Hedging.
 1 Basics of derivative pricing I.
 2 Change of numeraire.
 3 Basics of derivative pricing II.
 4 Market model.
 5 Currency Exchange.
 6 Credit risk.
 A. Delta hedging in situation of predictable jump I.
 B. Delta hedging in situation of predictable jump II.
 C. Backward Kolmogorov's equation for jump diffusion.
 D. Risk neutral valuation in predictable jump size situation.
 E. Examples of credit derivative pricing.
 F. Credit correlation.
 G. Valuation of CDO tranches.
 a. Definitions of CDO contract.
 b. Present values of CDO tranches.
 c. Distribution of defaulted notional of CDO tranches.
 7 Incomplete markets.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Definitions of CDO contract.

e consider a portfolio of several correlated underlyings (obligors) . Each underlying (obligor) pays coupon and may default. A specially created company (Special Purpose Vehicle) holds such portfolio and issues contracts called "CDO tranches" that passes both coupons and credit risk on the buyers of the tranches. We explain below the terms of the contracts.

Let be default time of underlying . Each underlying (=component of the portfolio) is represented by a certain notional in the portfolio. The fraction of the notional with respect to the total initial notional is denoted , . We introduce the fractional defaulted notional We assume that the recovery at default is the same across the entire portfolio. We denote such recovery rate by , . Hence, the portfolio loss from the defaults is given by Each tranche is parametrized by "attachment" and "detachment" points that limit the fractional loss of the tranche as follows

 (Tranche loss)
There are several tranches , , , , . The tranche is called "equity" tranche, the tranche is called "senior tranche" and the rest of the tranches are called "mezzanine" tranches. For purposes of calculation of coupon payments we introduce the notional of the tranche: The senior tranche's notional is defined by the relationship
 (Senior tranche loss)
The above relationship is necessary to maintain the balance between the coupon cashflow from the underlying portfolio and the payments to the tranches. For example, consider the first default in the underlying portfolio. Some notional defaults and no longer generates coupon. According to the formula ( Tranche loss ) the equity tranche covers the loss and the notional of the equity tranche decreases by . However, the entire portfolio's notional decreases by . Hence, the senior tranche is equipped with the rule that tranche's notional decreases by when the equity and mezzanine tranches are taking losses. We confirm that the formula ( Senior tranche loss ) indeed has such effect with the following calculation: The represents the situation when the senior tranche is the only remaining tranche and it covers the entire loss. The case allows further transformation as claimed.

Summary

of CDO contract. Let and be the notional ratio and the default time of the obligor , . By definition of Let and be the attachment and detachment points of the tranches. The defaulted notional is and the total loss is where the is the recovery rate. The loss covered by the tranche is and the coupon of the tranche is calculated from the notional of the tranche for and for the senior tranche .

 Notation. Index. Contents.