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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

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

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

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

  4. 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.

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