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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.
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.

Delta hedging in situation of predictable jump I.

e are assuming that the price of some underlying asset is given by MATH and we are considering a derivative $V_{t}$ that may jump in value depending on some Cox process $N_{t}.$ The drift $\mu$ , volatility $\sigma$ , intensity of the Cox process $dN$ and the size of the jump $\left[ V\right] $ are all functions of $S_{t}$ . So this is a Markovian situation with the MATH being the state.

We form a portfolio MATH where $U$ is another derivative. In reality we would, of course, sell the derivative $V$ and hedge it with the underlying $S$ and more liquid derivative $U$ : MATH However, the result would be the same and our unnatural way keeps expressions symmetrical.

We calculate the differential MATH and choose the $\alpha$ and $\beta$ to remove both sources of randomness: MATH We solve for the $\alpha,\beta:$ MATH MATH MATH For such $\alpha,~\beta$ we have MATH MATH MATH Note that the expression on the LHS only depends on the function $V$ while the expression on the RHS only depends on $U$ . The functions $U$ and $V$ are not related. Therefore, both parts of the equation do not, in fact, depend on $U$ or $V$ . We conclude that there is a function $\phi,$ depending on the state, such that MATH The probabilistic meaning of the above equation is provided in the section ( Backward_equation_for_jump_diffusion ).

Notation. Index. Contents.

Copyright 2007