The Black-Scholes Model is about one risky asset, (such as stock), and one riskless asset (such as cash). It also depends on asset returns to be normally distributed.
I don't know what model LTCM was using in their Fixed Income Arbitrage, but it was unlikely to be the Black-Scholes Model, and anyway also a big part of the LTCM downturn was when Russian government defaulted on their domestic local currency bonds, more of a "Black Swan" than a "Normal Distribution."
It has been written "Despite the presence of Nobel laureates closely identified with option theory it seems LTCM relied too much on theoretical market-risk models and not enough on stress-testing, gap risk and liquidity risk. There was an assumption that the portfolio was sufficiently diversified across world markets to produce low correlation. But in most markets LTCM was replicating basically the same credit spread trade. In August and September 1998 credit spreads widened in practically every market at the same time."