Hull-White 2-factor model : 1) Introduction

This post introduces Hull-White 2-factor model and derives integrations of some important stochastic process which are ingredients of short rate process.

Introduction

We are going to derive the Hull-White 2-factor model.

Given money market account \(B_t\) as a numeraire under the Q measure, short rate \(r(t)\) is assumed as follows. \[\begin{align} r(t) &= x(t) + y(t) + \varphi(t)\\ dx(t) &= -a(t)x(t)dt+\sigma(t)dW_x (t)\\ dy(t) &= -b(t)y(t)dt+\eta(t) dW_y (t)\\ dW_x & (t) dW_y (t) = \rho dt, \\ \\ x(0)&=0, y(0)=0, -1≤\rho≤1 \end{align}\] Here \(a(t)\), \(b(t)\) and \(\sigma (t)\), \(\eta (t)\) are mean-reversion and volatility parameters for each process respectively. \(W_x (t)\) and \(W_y (t)\) are correlated standard Wiener process and \(\varphi(t)\) is the deterministic process which is adapted to an initial term structure.

Like 1-factor model, \(\theta(t)\) and \(\varphi(t)\) are reflected in the process of derivation implicitly. Hence our focus is on \(x(t)+y(t)\).

Using \(W_1 (t)\) and \(W_2 (t)\) as independent Wiener processes, \(x(t)\) and \(y(t)\) can be rewrited. \[\begin{align} dx(t) &= -a(t)x(t)dt+\sigma(t)dW_1 (t)\\ dy(t) &= -b(t)y(t)dt+\eta(t) (\rho dW_1(t) + \sqrt{1-\rho^2}dW_2 (t) )\\ \end{align}\] For any \(s( < t)\), we can get the integrated from of \(r(t)\) from \(dr(t)\) as follows.
\[\begin{align} r(t) &= \varphi(t) + x(s)e^{- \int_{s}^{t} a(v)dv} + y(s)e^{- \int_{s}^{t} b(v)dv} \\ &+ \int_{s}^{t} \sigma(u) e^{- \int_{u}^{t} a(v)dv} dW_1 (u) \\ &+ \rho\int_{s}^{t} \eta(u) e^{- \int_{u}^{t} b(v)dv} dW_1 (u) \\ &+ \sqrt{1-\rho^2}\int_{s}^{t} \eta(u) e^{- \int_{u}^{t} b(v)dv} dW_2 (u) \end{align}\] The derivation of the above equation is skipped because that is the similar logic of the corresponding derivation of HW 1-factor model.

Using these results, we will derive a zero coupon bond price of Hull-White 2-factor model in the next post. \(\blacksquare\)