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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.

## Valuation of CDO tranches.

uppose we collect a portfolio of 125 high coupon CDSs with equal notional. We mean to issue a contract that covers the last 70% of possible loss on such portfolio. The contract does not impose any negative cashflow to its holder as long as less than 30% of the portfolio reported a credit event. If all the CDSs are uncorrelated then the probability of loss on such contract is rather slim. Hence, we offer a low coupon on it. The holder of such contract hardly ever have to do anything but she earns a small coupon. Hence, the contract is attractive. Such contract is called "the senior tranche of CDO".

We also mean to issue a contract that covers first 5% of the loss on the portfolio. We call such a contract "the equity tranche". The probability of loss on the equity tranche is significant. However, since we have a pool of high coupon CDSs and we mean to pay only a small coupon to the senior tranche, we are capable to reward any holder of the equity tranche with such a high coupon that may not be found among securities of other types. Hence, the equity tranche is attractive as well. Similarly, we assign some modest coupons to contracts that cover 5%-30% of the possible losses. Such contracts are called "mezzanine tranches".

The initial portfolio's risk exposure may be covered in such manner and variety of securities for every risk appetite may be offered on the market. Small part of coupon cashflow is diverted as management fee.

 a. Definitions of CDO contract.
 b. Present values of CDO tranches.
 c. Distribution of defaulted notional of CDO tranches.
 Notation. Index. Contents.