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The Handbook of Hybrid Securities: Convertible Bonds, CoCo Bonds and Bail-In by Jan De Spiegeleer, Wim Schoutens, Cynthia Van Hulle

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9 Modeling Hybrids: Advanced Issues

9.1 TAIL RISK IN HYBRIDS

So far we have been putting the emphasis on Brownian motion as the main engine governing the random walk of financial assets. This is a Gaussian setting because the increments of a Brownian motion W is normally distributed: WtWsN(0, ts). This assumption forces us indeed to accept that the log return of share prices between t and s is normally distributed according to the following equation:

(9.1)image

This idea has suited us for a lot of purposes. It allowed us to develop closed-form solutions for a lot of derivative securities. Accepting the normal density function also simplified the risk management of a trading desk. The concept of Value-at-Risk, for example, initially had the normal distribution as its backbone. This led, in the beginning, to a situation where the modeled asset prices and risk factors were very distant from the financial reality. As early as 1963, Mandelbrot already recognized in [145] the heavy-tailed nature of financial time series. This was elaborated further in [146]. A 3-sigma event is much more likely to be encountered in reality than suggested by the perfect bell-shaped Gaussian distribution. The market meltdown that took place when the subprime bubble burst in 2007 led to extreme movements in asset-backed portfolios. Or as one CFO of a major investment bank told a journalist of the Financial ...

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