The Mathematics of Derivatives: Tools for Designing Numerical Algorithms

CHAPTER 11

Specific Models

11.1. STOCHASTIC RATES AND DEFAULT

The combination of the static default model described in Chapter 10 and a stochastic rates model like the Hull-White discussed in Chapter 8 is now straightforward. The spot rate r—that is, instantaneous riskless borrow rate—follows a risk-neutral process consistent with arbitrage-free bond pricing in the HJM framework,

dr = (v²(t) − k(r − f(t)) + f′ (t)) dt − σ dz(t)

for an initial (riskless) forward curve f(t); model specifications σ and k (both constants; the spot rate standard deviation and the bond volatility/mean reversion decay rate respectively) and

According to this model, a riskless zero-coupon bond, Z_T(r, t), has value

where

and it satisfies the pricing equation implied by the short rate process,

Now a simple model combining rates and default might be: the spot rate continues the above process independently of default; default is a jump-probability event (i.e., it is not stochastic and a known constant probability curve for default exists) and the probability of default is not correlated to the riskless rates ...