Modeling a Credit Spread Exit Option with Vasicek Dynamics

# Pricing with Vasicek model on basket of credit spreads


# Pricing with Vasicek model on basket of credit spreads












I would appreciate help with a valuation of a fixed income derivative, with an embedded exit option.

Summary: Goal is to provide valuation of a fixed schedule of quarterly cash flows with an option to exit at any quarter (3 years from today) for a lump sum exit payment (to dispose by refinancing). The lump sum payment is calculated for each quarter based on the spot level of a basket of spreads, and there are no cash flows after the option is executed.

INPUTS:

- $CF_T$ - array of quarterly cash flows assuming no exit,

- $r_{k \times 12}$ - matrix containing 12 month history of $k$ discount margins,

- $\operatorname{ExitCF}(r[1..k][\![t]\!],t)$ - function for calculation of the exit cash flow which depends on k spot-spreads at time t when the exit option is executed,

- $\operatorname{DiscRate}$ - being the discount rate for the valuation.

Main Questions

- Assuming there is only one spread in the basket, how to model the value of the option?

- Does it make sense to use Vasicek model for modelling spreads as interest rates?

- Does it make sense to normalize all spreads and model their Z-score through Vasicek model?

- How to calibrate Vasicek model on normalized Z-scores of a basket of spreads?

Spreads: The basket of spreads is dominated by the first item which makes some 75%, first two are 85%. It is also highly correlated where last item has 60% correlation to first, second is some 82% to first. Having said that, I would normalize all spreads and look at their Z-score:

$$ z_{k,t+1} = \frac{r_{k,t+1}-\operatorname{Mean}(r_{k,1 \cdots t+1})}{\operatorname{StDev}(r_{k,1 \cdots t+1}) \cdot \sqrt{4} /\sqrt{t+1}}, $$ where standard deviation is annualized by multiplying by $\sqrt{4}$ and divided to adjust for sample size, as the sample grows in the future.

Modelling Spreads: Given that the first and most important spread in the basket is not a measure of credit risk (a very senior credit instrument), I would like to model it as an interest rate. I also don't want to apply any credit migration provisions. My long term reversion rate should be the present rate at $(t+1)$. I would like to model $z$ variable and derive all rates from there. I initially thought about Cox-Ingersoll-Ross model, but given that the standard deviation factor (containing $\sqrt{r_t}$ was making issues, I think it is more appropriate to use Vasicek model instead.

Assume $z$ is a Vasicek process ($k$ assumed to be $1$, and omitted):

$$ d z_t = a(b - z_t) d t + \sigma_z d W_t, $$

then by replacing $z_t=\frac{r_t-\bar{r_t}}{\sigma_t}$ and $\sigma_z=1$, this is equivalent to $r_t$ being also a Vasicek process:

$$ d r_t = a([b \sigma_r + \bar{r} ( 1 + \frac{1}{\sigma_r}) ] - r_t ) d t + \sigma_r d W_t . $$

I would then fit the Vasicek model on spread $k=1$ (the most important one), and solve for $a$ and $b$, based on the terms for $d r_t$ expression.