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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 II.

e present a negative result, to show how easy it is to have a situation when the delta hedging argument does not deliver a pricing equation.

We are assuming that a price of underlying asset is given by MATH where the drift $\mu$ , volatility $\sigma$ , intensity of the Poisson process $dN$ and the size of the jump $J$ are all functions of $S_{t}$ . So this is a Markovian situation with the MATH being the state.

Suppose we have two traded derivatives $V$ and $U$ depending on $S$ . We form a portfolio MATH where MATH is the trading strategy. 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 MATH MATH We see that, unfortunately, the functions do not separate. One may attempt to apply the risk neutral approach formally to try and see what kind of PDE should be expected and then attempt to separate the functions. It seems that this does not work. Apparently, this means that there is a significant room for difference of opinions about the price. Such opinions are dictated by the way of replication of the derivative.

Notation. Index. Contents.

Copyright 2007