Thursday, December 5, 2019

SABR/LIBOR Market Mode

For which class of instruments cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 SABR/LIBOR Market Model does perform better than cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 classical LIBOR Market Model?

The LIBOR Market Model

The LIBOR Market Model — also known as Brace, Gatarek, Musiela model — is an interest rate model capable of reproducing cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 correlation structure of forward rates. One-factor models are unable to reproduce this structure and cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365refore cannot price accurately derivatives whose prices reflect cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365se correlations. A typical example of such derivatives are swaps paying a non-linear function of cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 difference two swap rates for two different maturities.
The model is constructed by using a family of LIBOR rates: L0(t),,Ln(t)" id="MathJax-Element-2-Frame" role="presentation" style="position: relative;" tabindex="0">
, where Li(t)" id="MathJax-Element-3-Frame" role="presentation" style="position: relative;" tabindex="0"> is LIBOR forward rate starting at ti" id="MathJax-Element-4-Frame" role="presentation" style="position: relative;" tabindex="0"> and ending at ti+1" id="MathJax-Element-5-Frame" role="presentation" style="position: relative;" tabindex="0">, following

dLi(t)=σi(t)Li(t)dWi+1(t)." id="MathJax-Element-6-Frame" role="presentation" style="position: relative; text-align: center;" tabindex="0">

The SABR LIBOR-Market Model

An important flaw of cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 LMM is known as sticky volatilities: if cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 model is calibrated in a highly volatile market it assumes that this high volatility lasts forever, which leads to inaccurate results.
The SABR LMM attempts to address this issue. In this model, each LIBOR rate is assumed to follow a log-normal dynamic having stochastic volatility:

dLi(t)=σi(t)Liβi(t)dWi+1(t)dσi(t)=αiσi(t)dZ(t)<dW,dZ>=ρdt" id="MathJax-Element-7-Frame" role="presentation" style="position: relative;" tabindex="0">
dLi(t)=σi(t)Liβi(t)dWi+1(t)dσi(t)=αiσi(t)dZ(t)<dW,dZ>=ρdt" id="MathJax-Element-7-Frame" role="presentation" style="position: relative;" tabindex="0">