# CoCo-bond modeling & Vasicek calibratrion

#### hamudenben

Dear folks,

I'm trying to do a case-study on the CoCo-bonds issued by Credit Suisse, and I want to calibrate the model from Pennacchi's article: A Structural Model Of Contingent Bank Capital.

So, basically I know the following about the deal:
Issue date: February 17, 2011
Maturity: February 24, 2041
Call date: Call-able at par from August 24, 2016
Rating: BBB+
Coupon: 7.875%

The time to maturity is easy: 30 yrs

But then I need to find or calibrate the numbers for a 30 yrs interest-curve, modeled as:

$$dr_t = \kappa (\bar{r} - r_t)dt + \sigma_r \sqrt{r_t} d \zeta, \, d\zeta dz = \rho dt$$

rho is the corr between the yield and a given capital-ratio (let's just say asset value for now).

And on top of that, I need to use the knowledge that rating from Fitch is BBB+ and make reason for why I choose such and such in regards to volatility (sigma in equation), jump-intensity and jump-size (both captured in Y in equation), in this equation:

$$d \ln(x_t) = \left[(r_t-\lambda k)-\frac{r_t+h_t+c_t b_t}{x_t} - g(x_t - \hat{x}) - \frac{\sigma^2}{2} \right]dt + \sigma dz + \ln(Y_{q_{t-}}) dq$$

Approximation:

$$r_{t+\Delta t} = r_t + \kappa (\bar{r} - r_t) \Delta t + \sigma_r \sqrt{r_t} \sqrt{\Delta t} \eta_{t+\Delta t}, \, \eta_{t+\Delta t} \sim N(0,1)$$

Approximation:
$\ln(x_{t+\Delta t}) = \ln x_t + \left[ (r_t-\lambda_t k_t) - \frac{r_t+h_t+c b_t} {x_t} - g(x_t - \hat{x}) - \frac{\sigma^2}{2} \right] \Delta t + \sigma \sqrt{\Delta t} \epsilon_{t+\Delta t} + \ln(Y_{t+\Delta t}) \omega_{t+\Delta t}, \epsilon_{t+\Delta t} \sim N(0,1), E^Q_t[\epsilon_{t+\Delta t} \eta_{t+\Delta t}]$
$$=\rho,\ln(Y_{t+\Delta t}) \sim N(\mu_y, \sigma_y^2),\omega_{t+\Delta t} = 1$$ with prop
$$\Delta t \lambda_t$$ and $$\omega_{t+\Delta t} = 0$$ with prop. $$1- \Delta t \lambda_t$$

So, what I'm trying to figure out is how to calibrate the approximation for dr_t to a 30-yrs yield curve. I need:
- $$r_0$$, starting number for yield curve (I think this could be chosen fairly arbitrarily, but lower than $$\bar{r}$$, they use: 3.5%)
- $$\bar{r}$$, long-term mean level (they use: 0.069, but this is for 5yrs)
- $$\kappa$$, speed of reversion (they use: 0.114, 5yrs, also)

And for the dln(x)-approximation, I need:
- $$\sigma$$ (IMO, this can stay the same)v
- $$mu_y$$, mean for jump-size (they use: -1%)
- $$\sigma_y$$, std.dev for jump-size (they use: -2%)
- $$\lambda$$, jump-intensity (they use: 1)

All inputs are very welcome - also if you just read the article and want to share some thoughts on it.

Thanks,

J