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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.

## Distribution of defaulted notional of CDO tranches.

e saw in the previous section that the present value of a CDO tranche is expressed in terms of quantities of the form for a variety of functions . Hence, we would like to compute distribution of the quantity for all payment dates .

We introduce discretization of the notional. Let be a small notional amount. We introduce the hut operation: and the quantity We perform the separation trick for as described in the section ( Credit correlation ). Hence, we will be assuming that the credit events of obligors are (conditionally) independent. We denote the defaulted notional of portfolio of obligors.

We start induction in with consideration of a portfolio consisting of only one obligor. Since is the total event we continue We apply the formula ( Bayes formula ) We introduce the notation Prob and use the delta symbol (in C++ notation) then we conclude where the is the fractional notional of the first obligor. The comes from the observation that if then the first (and only) obligor defaulted. Hence, .

We proceed with the general induction steps using the same tools.

Summary

We denote the number of obligor in the portfolio as , the number of coupon payments before maturity as and the size of notional mesh as . We calculate the quantities according to the formulas where the is the notation for Consequently, any expectation of the form is evaluated as

Note that the probabilities Prob were considered in the section ( CDS ). These probabilities come from calibration to observed market quotes for CDSs.

 Notation. Index. Contents.