Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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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 MATH for a variety of functions MATH . Hence, we would like to compute distribution of the quantity $d_{T_{p}}$ for all payment dates MATH .

We introduce discretization of the notional. Let $\delta n$ be a small notional amount. We introduce the hut operation: MATH and the quantity MATH MATH We perform the separation trick for $\QTR{cal}{G}_{t}$ as described in the section ( Credit correlation ). Hence, we will be assuming that the credit events of obligors are (conditionally) independent. We denote $d_{T_{p}}^{j}$ the defaulted notional of portfolio of $j$ obligors.

We start induction in $j$ with consideration of a portfolio consisting of only one obligor. MATH Since MATH is the total event we continue MATH We apply the formula ( Bayes formula ) MATH We introduce the notation $Q_{p}^{j}=$ Prob MATH and use the delta symbol MATH (in C++ notation) then we conclude MATH where the $n_{1}$ is the fractional notional of the first obligor. The MATH comes from the observation that if MATH then the first (and only) obligor defaulted. Hence, MATH .

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



We denote the number of obligor in the portfolio as $J$ , the number of coupon payments before maturity as $P$ and the size of notional mesh as MATH . We calculate the quantities MATH according to the formulas MATH where the $Q_{p}^{j}$ is the notation for MATH Consequently, any expectation of the form MATH is evaluated as MATH

Note that the probabilities $Q_{p}^{j}=$ Prob MATH were considered in the section ( CDS ). These probabilities come from calibration to observed market quotes for CDSs.

Notation. Index. Contents.

Copyright 2007