We introduce a general continuous stochastic time-change model on Brownian motion in a first-passage-time setting and derive analytical formulas for the first-passage-time distribution in one and several dimensions. Thereby the multi-dimensional model introduces a dependence structure via the time transformation. The two-dimensional model allows, in addition, for correlation between the Brownian motions.
The model is an extension of the classic Merton model and of the deterministic time-change model by Overbeck and Schmidt.
The model is applied to derive an explicit formula for credit-spread and corresponding credit-spread dynamics, especially for the credit-spread volatility.
The model can also be used across other assets. In the last chapter we give a note on barrier option pricing.
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