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 and we are considering a derivative that may jump in value depending on some Cox process The drift , volatility , intensity of the Cox process and the size of the jump are all functions of . So this is a Markovian situation with the being the state.

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

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

 Notation. Index. Contents.