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BGM Model The Brace-Gatarek-Musiela (BGM) model, also called Libor Market Model, is a multi-factor log-normal model. This model applies to both currencies. Option values are discounted risk-neutral expectations of their pay-off. In a stochastic interest rate environment, the discounting should be taken as the accumulation of the spot rate The explicit modeling of market forward rates allows for a natural formula for interest rate option volatility that is consistent with the market practice of using the formula of Black for caps. It is generally considered to have more desirable theoretical calibration properties than short rate or instantaneous forward rate models. In terms of volatility surface, the Vvol introduces a positive smile, the correlation induces a skew and the mean reversion makes the smile decrease with maturity. The expected spot volatility drives the term structure. In general, it is believed that Monte Carlo simulation is the only viable numerical method available for the LMM. Diffusion dates are set in such a way that all calibration dates are diffusion dates. Then the diffusion step is specified on each slice. Diffusion dates are business days, approximately spaced by the diffusion step Generally speaking, all diffusions will be considered between diffusion dates, whereas discounting will be performed between spot dates. When computed at a diffusion date other than , these rates and discount factors are random variables that are materialized as Monte-Carlo arrays. Each index corresponds to a path and expectations will be replaced by averages over all paths. At each diffusion date, only a certain number of discount factors are provided. For the purpose of rate diffusion, one may need other discount factors. An interpolation procedure is thus necessary. Currently, we have implemented a linear interpolation of the logarithm of encompassing available maturities.
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