A careful description of the credit process of the issuer is key to model the call feature present in AT1 CoCo bonds or non-cumulative preferred shares, and therefore also the effective maturity of the instrument. We use a regime switching model where each regime is a jump diffusion with its own volatility level. In each regime, a jump in the underlying share price may be due to multiple factors. It could result from a regulatory or a contractual bail-in, or it could be due to default which sends the share price to zero. Finally a switch between regimes is also associated with a jump in the underlying share price, thereby generating complex correlation patterns between volatility, share price, credit and bail-intensities.
In practice two regimes are usually sufficient, although more complex structures with three regimes may be possible in some complex cases, in particular to accommodate non monotonic CDS term structures. With two regimes, one of them can be labelled as the good state, while the other one is viewed as the bad state for the issuing bank.
The good state is typically the current situation, characterized by low volatility and a good credit rating, therefore with a low probability of default and low intensty of bail-in. With some probability the bank may switch at any time to the bad state (think of it as a downgrade) characterized by a high risk of default, a higher volatility level, and larger bail-in intensities. At the time of the switch or downgrade one would expect to see a significant negative jump on the price of the underlying share price.
The optimal decision to call the bond depends on the future regime, which renders both the timing of the call and the actual maturity of the bond stochastic.
The parameters of this regime switching model are calibrated on all derivative quotes available for the issuing bank. In practice this often means a list of American vanilla options and CDS quotes, a framework known as the Equity-to-Credit universe.
We consider up to three relevant CDS term structure: the ISDA 2003 subordinated and senior CDS curve and the ISDA 2014 subordinated curve. In absence of CDS quotes, standard bond quotes of various maturities can be used to calibrate the credit component of the model.
The regime switching model has been shown to capture efficiently a large number of derivative instruments in a consistent way, with the same model explaining both the short term and the long term behaviour of the options and the credit default swaps.
The conversion of a CoCo bond may be triggered mechanically by a particular CET1 level or it may be forced by the regulator at the point of non viability. We consider a reduced form model where the bail-in of the CoCo bond is triggered by two Poisson processes whose intensities depends on the credit regime. One Poisson process corresponds to the contracual bail-in which depends on the CET1 level while the other describes the regulatory bail-in. In a simple two regime setting, the bail-in intensities in the current good credit state is low while they becomes significant in the bad regime.
For the perpetual non-cumulative preferred shares issued by North American banks, we only consider the regulatory bail-in since these instruments do not have a CoCo feature.