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">L0(t),…,Ln(t)
, where
Li(t)" id="MathJax-Element-3-Frame" role="presentation" style="position: relative;" tabindex="0">Li(t) is LIBOR forward rate starting at
ti" id="MathJax-Element-4-Frame" role="presentation" style="position: relative;" tabindex="0">ti and ending at
ti+1" id="MathJax-Element-5-Frame" role="presentation" style="position: relative;" tabindex="0">ti+1, 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">dLi(t)=σi(t)Li(t)dWi+1(t).
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)Lβii(t)dWi+1(t)dσi(t)=αiσi(t)dZ(t)<dW,dZ>=ρdt
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)Lβii(t)dWi+1(t)dσi(t)=αiσi(t)dZ(t)<dW,dZ>=ρdt
