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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
1. Black-Scholes formula.
2. Change of variables for Kolmogorov equation.
3. Mean reverting equation.
4. Affine SDE.
A. Ricatti equation.
B. Evaluation of option price.
C. Laplace transform.
D. Example: CDFX model.
a. Definition of CDFX model.
b. The martingale normalization (CDFX).
c. Fourier transform (CDFX).
d. Calculation of Fourier transform (CDFX).
e. Calculation of Premium Leg of CDS.
f. Calculation of the protection leg of the CDS.
5. Heston equations.
6. Displaced Heston equations.
7. Stochastic volatility.
8. Markovian projection.
9. Hamilton-Jacobi Equations.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Calculation of Premium Leg of CDS.

e would like to calculate the expression MATH Observe that $P_{t}$ is not a martingale. Indeed, MATH Hence, MATH We conclude that MATH Therefore, MATH for some martingale $M_{t}$ . We want to replicate the $P_{t}$ with some function of state variables $U_{t}$ . If successful, such function should satisfy MATH for some martingale $\hat{M}_{t}$ . If both martingales also coincide at final time $T$ then we will have the equality: MATH

We will seek for $U_{t}$ in the form MATH where the $\alpha$ and $\beta$ are some deterministic functions. Hence, MATH We introduce notation MATH with

MATH then MATH hence we calculate the drift part as MATH We want the above expression to be MATH Hence, it suffices to have MATH where MATH Consequently, MATH We separate the coordinates MATH We substitute the expressions for the $g,h,q$ . The equations resolve to MATH These coincide with the equations (17) in the article because (in article [CarrWu2006a] 's notation) MATH

Notation. Index. Contents.

Copyright 2007