By Umberto Cherubini
Copula tools in Finance is the 1st publication to deal with the math of copula services illustrated with finance applications. It explains copulas via functions to significant themes in by-product pricing and credits chance analysis. Examples contain pricing of the most unique derivatives (barrier, basket, rainbow techniques) in addition to probability administration issues. specific concentration is given to the pricing of asset-backed securities and basket credits spinoff items and the evaluate of counterparty threat in spinoff transactions.
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The second is that the risk factor represented by volatility must be provided with a market price, something that makes the model harder to calibrate. A model that is particularly easy to handle, and reminds us of the Hull and White (1987) model, could be based on the assumption that volatility risk is not priced in the market, and volatility is orthogonal to the price of the underlying. The idea is that conditional on a given volatility parameter taking value s, the stochastic process followed by the underlying asset follows a geometric brownian motion.
Moreover, another face of credit risk has become increasingly relevant along with the progressive shift from the classical standard intermediation business toward structured ﬁnance products, and the need to resort to over-the-counter (OTC) transactions to hedge the corresponding exposure. Contrary to what happens in derivatives transactions operated in futures-style organized markets, OTC deals involve some credit risk, as the counterparty in the contract may default by the time it has to honor its obligations.
This product is called a cap, and the price is obtained, assuming a strike rate LCAP , from N CAP(t, t1 , tN ) = δi EQ [D(t, tj ) max(L(tj −1 , tj ) − LCAP ), 0 | t] j =2 N = j =2 δi D(t, tj )EQtj [max(L(tj −1 , tj ) − LCAP ), 0 | t] and each call option is called caplet. By the same token, a stream of put options are called ﬂoor, and are evaluated as N FLOOR(t, t1 , tN ) = j =1 δ2 D(t, tj )EQtj [max(LFLOOR − L(tj −1 , tj )), 0 | t] where LFLOOR is the strike rate. The names cap and ﬂoor derive from the results, which may be easily veriﬁed L(tj , tj −1 ) − CAPLET(tj , tj −1 ) = min(L(tj −1 , tj ), LCAP ) L(tj , tj −1 ) + FLOORLET(tj , tj −1 ) = max(L(tj −1 , tj ), LFLOOR ) Setting a cap and a ﬂoor amounts to building a collar, that is a band in which the coupon is allowed to ﬂoat according to the interest rate.
Copula Methods in Finance by Umberto Cherubini