By Umberto Cherubini
Copula tools in Finance is the 1st booklet to handle the math of copula capabilities illustrated with finance applications. It explains copulas via functions to significant themes in spinoff pricing and credits probability analysis. Examples contain pricing of the most unique derivatives (barrier, basket, rainbow thoughts) in addition to hazard administration issues. specific concentration is given to the pricing of asset-backed securities and basket credits by-product items and the overview of counterparty hazard in spinoff transactions.
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In order to compare the price of our bivariate product with that of the univariate ones, it would be great if we could write the price as DP = exp [−r (T − t)] Q (KNKY , KSP ) = exp [−r (T − t)] C (QNKY , QSP ) with C (x, y) a bivariate function. Without getting involved in heavy mathematics, we can also discover the general requirements that the function C (x, y) must satisfy in order to be able to represent a joint probability distribution. Beyond the basic requirement that the output of the function must be in the unit interval, as it must represent a probability, three requirements immediately come to mind.
Using the dynamics above one can compute or simulate the price from DD (t, T ; RR) = D (t, T ) RR + (1 − RR) D (t, T ) EQT exp − T t γ (u) du A ﬁnal comment is in order concerning the recovery rate. Extensions of the model refer to a stochastic recovery rate. Of course, the extension is immediate as long as one is willing to assume that the recovery rate is independent of the default intensity and interest rate. In this case the expected value is simply substituted for the deterministic value assumed in the analysis above.
Furthermore, let us assume that the recovery rate RR is equal to zero, so that the whole principal is lost in case of default. Under these assumptions, the price of a defaultable zero-coupon bond maturing at time T is simply DD (t, T ; RR = 0) = D (t, T ) EQ [1 − 1DEF ] = D (t, T ) exp −γ (T − t) and the credit spread is obtained as r ∗ (t, T ; RR = 0) − r (t, T ) ≡ − ln DD (t, T ; RR = 0) T −t − − ln D (t, T ) T −t =γ In this special case the credit spread curve is ﬂat and equal to the intensity ﬁgure of the default process.