Quantitative Analysis.
Trading Platform.
Python for Excel.
Author.

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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.
Bibliography.
Forum Notation Index Contents

LIBOR market model.

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

MATH(Libor2)
The $Q_{k}$ is the probability measure 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 is that the process MATH is lognormalMATH under $Q_{k}$ for any $k$, where the MATH are some deterministic functions. The MATH, $j\neq k$, has a drift under $Q_{k}$MATH The calculation of MATH is our next task. According to the useful formula ( Change of drift recipe 1)MATH




By the ( Libor definition)MATH Hence, for math,MATHMATH Therefore,MATHMATH where the $\rho_{sj}$ are correlations and $\sigma_{j}$ are volatilities of the forward ratesMATH Similarly, for math,MATH

LIBOR market model introduces a curve--dependent drift. Hence, it has a state variable (:=description of filtration) that increases dimensionality if maturity and frequency of observations increase. For this reason this model has limited applicability.





Forum Notation Index Contents


















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