Content of present website is being moved to . Registration of will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
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) MATH . 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 $\tau_{i}$ be default time of underlying $X_{j}$ . 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 $n_{j}$ , $\sum_{j}n_{j}=1$ . We introduce the fractional defaulted notional MATH We assume that the recovery at default is the same across the entire portfolio. We denote such recovery rate by $R$ , $0<R<1$ . Hence, the portfolio loss from the defaults is given by MATH Each tranche is parametrized by "attachment" and "detachment" points that limit the fractional loss of the tranche $l_{t}^{a,b}$ as follows

MATH (Tranche loss)
MATH There are several tranches MATH , $b_{m}=a_{m+1}$ , $m=1,...,M-1$ , $a_{1}=0$ , $b_{M}=1$ . The tranche MATH is called "equity" tranche, the tranche MATH 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: MATH The senior tranche's notional $I_{t}^{a_{M}}$ is defined by the relationship
MATH (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 $n_{i}$ defaults and no longer generates coupon. According to the formula ( Tranche loss ) the equity tranche covers the loss MATH and the notional of the equity tranche decreases by MATH . However, the entire portfolio's notional decreases by $n_{i}$ . Hence, the senior tranche is equipped with the rule that tranche's notional decreases by $Rn_{i}$ 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: MATH The $l_{t}>a_{M}$ represents the situation when the senior tranche is the only remaining tranche and it covers the entire loss. The $l_{t}<a_{M}$ case allows further transformation MATH as claimed.


of CDO contract. Let $n_{j}$ and $\tau_{j}$ be the notional ratio and the default time of the obligor $j$ , $j=1,...,J$ . By definition of $n_{j}$ MATH Let MATH and MATH be the attachment and detachment points of the tranches. The defaulted notional is MATH and the total loss is MATH where the $R$ is the recovery rate. The loss covered by the tranche MATH is MATH and the coupon of the tranche is calculated from the notional of the tranche MATH for $m=1,...,M-1$ and MATH for the senior tranche MATH .

Notation. Index. Contents.

Copyright 2007