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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.
A. Forward LIBOR.
B. LIBOR market model.
C. Swap rate.
D. Swap measure.
5. Currency Exchange.
6. Credit risk.
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.

LIBOR market model.


e are considering a debt structure with payment dates MATH $.$ We introduce the following notation

MATH (Libor2)
We introduce the probability measure $Q_{k}$ associated with the price MATH taken as a numeraire, MATH . The MATH is an increment of the $s-$ th standard Brownian motion under the $Q_{k}$ . It is the assumption of the model that the process MATH is lognormal MATH under $Q_{k}$ for any $k$ , where the MATH are some deterministic functions. The increment MATH , $j\neq k$ , has a drift under $Q_{k}$ MATH The calculation of MATH is our next task. According to the formula ( Change of drift recipe 1 ) MATH




By the definition ( Libor definition ), MATH Hence, for $\dot{j}>k$ , MATH MATH We use the formula ( Change_of_drift_recipe_1 ), MATH Therefore MATH MATH where the $\rho_{sj}$ are correlations and $\sigma_{j}$ are volatilities of the forward rates MATH Similarly, for $j<k$ , MATH

LIBOR market model introduces a curve-dependent drift. Hence, it has a state variable (=description of filtration) of high dimensionality.





Notation. Index. Contents.


















Copyright 2007