By Umberto Cherubini

Copula equipment in Finance is the 1st booklet to handle the math of copula features illustrated with finance applications.  It explains copulas through functions to significant themes in by-product pricing and credits threat analysis.  Examples contain pricing of the most unique derivatives (barrier, basket, rainbow techniques) in addition to possibility administration issues.  specific concentration is given to the pricing of asset-backed securities and basket credits spinoff items and the assessment of counterparty probability in spinoff transactions.

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Apart from such possible complications, it is clear that option theory could be applied to recover both the value of debt and equity, and to decompose debt into the default-free part and the credit risk premium. Assume that the asset side of the firm V (t) follows a geometric brownian motion, so that under the risk-neutral measure we have dV (t) = rV (t) dt + σV V (t) dz (t) Then, the standard Black–Scholes formula can be applied to yield the value of equity C C (t) = V (t) (d1 ) − exp (−r (T − t)) DD (d2 ) ln(V (t) /DD) + (r + σV2 /2) (T − t) √ σV T − t √ d2 = d1 − σV T − t d1 = and the value of debt DD is recovered as DD(t) = V (t) − [V (t) (d1 ) − exp(−r(T − t))DD (d2 )] = (−d1 )V (t) + exp(−r(T − t))DD (d2 ) Notice that, by adding and subtracting exp (−r (T − t)) DD we can rewrite the value as DD (t) = exp (−r (T − t)) DD − −V (t) (−d1 ) + exp (−r (T − t)) DD (−d2 ) and we recognize the short position in the put option representing credit risk.

As g (t) /D (t, τ ) is a martingale, we have immediately g (D (t, τ, T ) , t) = P (t, τ ) EQτ [G (D (τ, T ) , τ ) | t] and the factorization of the discount factor and expected pay-off is now correct. To conclude, the cookbook recipe emerging from the forward martingale approach is that the forward price must be considered as the underlying asset of the derivative contract, instead of the spot. 3 LIBOR market model While the standard classical interest rate pricing models are based on the dynamics of instantaneous spot and forward rates, the market practice is to refer to observed interest rates for investment over discrete time periods.

We have then, under the risk-neutral measure dS (t) = rS (t) dt + σ (S, t) S (t) dz (t) The function σ (S, t) is called the local volatility surface and should then be calibrated in such a way as to produce the smile and volatility term structure effects actually observed on the market. A long-dated proposal is represented by the so-called constant elasticity of variance (CEV) models, in which dS (t) = rS (t) dt + σ S (t)α dz (t) Alternative local volatility specifications were proposed to comply with techniques that are commonly used by practitioners in the market to fit the smile.

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