Interest Rate Derivatives: One Factor Spot Rate Models

Abstract

In this chapter we survey models of interest rate derivatives which take the instantaneous spot interest rate as the underlying factor. The continuous hedging argument is extended so as to model the term structure of interest rates and other interest rate derivative securities. This basic approach is due to Vasicek (J Financ Econ 5:177–188, 1977) and hence we shall often refer to it as the Vasicek approach. By specifying different functional forms for the drift, the diffusion and the market price of risk, we develop three well known spot rate models, namely the Vasicek model, the Hull–White model and the Cox–Ingersoll–Ross model. Then we present a general framework for pricing bond options and we apply this framework to obtain closed form solutions for bond options under the specifications of the Hull–White and the Cox–Ingersoll–Ross model. Finally we discuss the calibration of the Hull–White model to the currently observed yield curve.

Appendices

Appendix

Appendix 23.1 Solution of the Ordinary Differential Equations (23.59) and (23.60)

Consider the ordinary differential equation

$$\displaystyle\begin{array}{rcl} \frac{db} {\mathit{dt}} =\alpha _{0}b^{2} +\alpha _{ 1}b - 1 =\alpha _{0}\left [b^{2} + \frac{\alpha _{1}} {\alpha _{0}}b -\frac{1} {\alpha _{0}} \right ].& &{}\end{array}$$

(23.125)

where α ₀ and α ₁ are constants. The quadratic in the brackets on the RHS can be factorised as

$$\displaystyle{b^{2} + \frac{\alpha _{1}} {\alpha _{0}}b -\frac{1} {\alpha _{0}} = (b -\phi _{1})(b -\phi _{2}),}$$

where

$$\displaystyle\begin{array}{rcl} \phi _{1}& =& -\frac{\alpha _{1}} {2\alpha _{0}} + \frac{\beta } {2\alpha _{0}}, \\ \phi _{2}& =& -\frac{\alpha _{1}} {2\alpha _{0}} - \frac{\beta } {2\alpha _{0}}, \\ \beta & =& \sqrt{\alpha _{1 }^{2 } + 4\alpha _{0}}.{}\end{array}$$

(23.126)

Thus the ordinary differential equation (23.125) can be written

$$\displaystyle{ \frac{db} {\mathit{dt}} =\alpha _{0}(b -\phi _{1})(b -\phi _{2}), }$$

(23.127)

or as

$$\displaystyle{ \frac{db} {(b -\phi _{1})(b -\phi _{2})} =\alpha _{0}\mathit{dt}.}$$

With some slight re-arrangement the last equation can be written

$$\displaystyle{ \left [ \frac{1} {b -\phi _{1}} - \frac{1} {b -\phi _{2}}\right ]db =\alpha _{0}(\phi _{1} -\phi _{2})\mathit{dt} =\beta \mathit{dt}. }$$

(23.128)

Integrating the last equation from t to T we obtain

$$\displaystyle{ \left [\ln \left (\frac{b -\phi _{1}} {b -\phi _{2}}\right ))\right ]_{t}^{T} =\beta (T - t), }$$

(23.129)

i.e.

$$\displaystyle{\ln \left (\frac{b(T,T) -\phi _{1}} {b(T,T) -\phi _{2}}\right ) -\ln \left (\frac{b(t,T) -\phi _{1}} {b(t,T) -\phi _{2}}\right ) =\beta (T - t),}$$

which on making use of b(T, T) = 0 becomes

$$\displaystyle{\ln \left (\frac{b(t,T) -\phi _{1}} {b(t,T) -\phi _{2}}\right ) =\ln \left (\frac{\phi _{1}} {\phi _{2}}\right ) -\beta (T - t),}$$

i.e.

$$\displaystyle\begin{array}{rcl} \frac{b(t,T) -\phi _{1}} {b(t,T) -\phi _{2}}& =& \exp \left [\ln (\frac{\phi _{1}} {\phi _{2}}) -\beta (T - t)\right ] = \frac{\phi _{1}} {\phi _{2}}e^{-\beta (T-t)}. {}\\ \end{array}$$

Solving the last equation for b(t, T) we obtain

$$\displaystyle{ b(t,T) = \frac{\phi _{1}\phi _{2}(1 - e^{-\beta (T-t)})} {\phi _{2} -\phi _{1}e^{-\beta (T-t)}}. }$$

(23.130)

Using the fact that $\phi _{1}\phi _{2} = -1/\alpha _{0}$ this simplifies slightly to

$$\displaystyle{ b(t,T) = \frac{1} {\alpha _{0}} \frac{[1 - e^{-\beta (T-t)}]} {[\phi _{1}e^{-\beta (T-t)} -\phi _{2}]}. }$$

(23.131)

To solve (23.59), we set $\alpha _{0} =\sigma ^{2}/2,\alpha _{1} =\alpha +\lambda$, so that we obtain

$$\displaystyle{ b(t,T) = \frac{2} {\sigma ^{2}} \frac{[1 - e^{-\beta (T-t)}]} {[\phi _{1}e^{-\beta (T-t)} -\phi _{2}]}, }$$

(23.132)

where

$$\displaystyle\begin{array}{rcl} \phi _{1}& =& -\frac{(\alpha +\lambda )} {\sigma ^{2}} + \frac{\beta } {\sigma ^{2}}, \\ \phi _{2}& =& -\frac{(\alpha +\lambda )} {\sigma ^{2}} - \frac{\beta } {\sigma ^{2}}, \\ \beta & =& \sqrt{(\alpha +\lambda )^{2 } + 2\sigma ^{2}}.{}\end{array}$$

(23.133)

Next from Eq. (23.62) of Sect. 23.4.3

$$\displaystyle{a(t,T) =\alpha \gamma \int _{ t}^{T}b(s,T)\mathit{ds}.}$$

Making the transformation $u =\beta (T - s)$ we see that

$$\displaystyle{a(t,T) = \frac{+\alpha \gamma } {\beta } \int _{0}^{\beta (T-t)}b(T -\frac{u} {\beta },T)\mathit{du}.}$$

Substituting the expression (23.132) for b(t, T) (and setting $\tau = T - t$) we obtain

$$\displaystyle{a(t,T) = +\frac{\alpha \gamma } {\beta } \frac{2} {\sigma ^{2}} \int _{0}^{\beta \tau } \frac{(1 - e^{-u})} {(\phi _{1}e^{-u} -\phi _{2})}\mathit{du}.}$$

Consider the integral

$$\displaystyle\begin{array}{rcl} I& =& \int _{0}^{\beta \tau }\left ( \frac{1 - e^{-u}} {\phi _{1}e^{-u} -\phi _{2}}\right )\mathit{du} {}\\ & =& \int _{0}^{\beta \tau } \frac{\mathit{du}} {\phi _{1}e^{-u} -\phi _{2}} -\int _{0}^{\beta \tau } \frac{e^{-u}\mathit{du}} {\phi _{1}e^{-u} -\phi _{2}} {}\\ & =& \int _{0}^{\beta \tau } \frac{e^{u}\mathit{du}} {\phi _{1} -\phi _{2}e^{u}} -\int _{0}^{\beta \tau } \frac{e^{-u}\mathit{du}} {\phi _{1}e^{-u} -\phi _{2}} {}\\ & =& \left [\frac{-1} {\phi _{2}} \ln (\phi _{1} -\phi _{2}e^{u})\right ]_{ 0}^{\beta \tau } + \left [\frac{1} {\phi _{1}} \ln (\phi _{1}e^{-u} -\phi _{ 2})\right ]_{0}^{\beta \tau } {}\\ & =& \frac{(\phi _{1} -\phi _{2})} {\phi _{1}\phi _{2}} \ln (\phi _{1} -\phi _{2}) -\frac{1} {\phi _{2}} \ln (\phi _{1} -\phi _{2}e^{\beta \tau }) + \frac{1} {\phi _{1}} \ln (\phi _{1}e^{-\beta \tau }-\phi _{ 2}) {}\\ & =& \frac{(\phi _{1} -\phi _{2})} {\phi _{1}\phi _{2}} \ln (\phi _{1} -\phi _{2}) -\frac{1} {\phi _{2}} \ln (\phi _{1} -\phi _{2}e^{\beta \tau }) + \frac{1} {\phi _{1}} \ln [e^{-\beta \tau }/(\phi _{ 1} -\phi _{2}e^{\beta \tau })] {}\\ & =& -\frac{\beta \tau } {\phi _{1}} + \frac{(\phi _{1} -\phi _{2})} {\phi _{1}\phi _{2}} \left \{\ln (\phi _{1} -\phi _{2}) -\ln (\phi _{1} -\phi _{2}e^{\beta \tau })\right \}. {}\\ \end{array}$$

Finally

$$\displaystyle{I = -\frac{\beta } {\phi _{1}}\tau -\frac{(\phi _{1} -\phi _{2})} {\phi _{1}\phi _{2}} \ln \left (\frac{\phi _{1} -\phi _{2}e^{\beta \tau }} {\phi _{1} -\phi _{2}} \right ),}$$

and so

$$\displaystyle{a(t,T) = \frac{2\alpha \gamma } {\beta \sigma ^{2}} \left [\frac{-\beta (T - t)} {\phi _{1}} -\frac{(\phi _{1} -\phi _{2})} {\phi _{1}\phi _{2}} \ln (\frac{\phi _{1} -\phi _{2}e^{\beta (T-t)}} {\phi _{1} -\phi _{2}} )\right ].}$$

Using the fact that $\phi _{1}\phi _{2} = -2/\sigma ^{2}$ and $\phi _{1} -\phi _{2} = 2\beta /\sigma ^{2}$ we finally obtain

$$\displaystyle{ a(t,T) = \frac{2\alpha \gamma } {\sigma ^{2}} \left [-\frac{(T - t)} {\phi _{1}} +\ln \left (\frac{\phi _{1} -\phi _{2}e^{\beta (T-t)}} {\phi _{1} -\phi _{2}} \right )\right ]. }$$

(23.134)

When allowing for time-varying coefficients, the steps leading to (23.128) remain the same as in the constant coefficients case, only now ϕ ₁, ϕ ₂, σ and β become functions of time. Thus in order to use the same functional form for the solution we need to define

$$\displaystyle{\bar{\beta }(t,T) = \frac{1} {T - t}\int _{t}^{T}\beta (s)\mathit{ds}.}$$

Thus integration of (23.128) will yield (23.132) with β replaced by $\bar{\beta }(t,T)$.

Appendix 23.2 Calculating θ(T) in the Calibration of the Hull–White Model

From Eq. (23.110) we note that

$$\displaystyle{ \frac{\partial b} {\partial T} = e^{\mathcal{K}(t)-\mathcal{K}(T)}.}$$

Differentiating Eq. (23.117) with respect to T yields

$$\displaystyle\begin{array}{rcl} \frac{\partial } {\partial T}a(0,T)& =& b(T,T)\theta (T)+\!\int _{0}^{T}e^{\mathcal{K}(s)-\mathcal{K}(T)}\theta (s)\mathit{ds}- \frac{\partial } {\partial T}\left (\frac{1} {2}\int _{0}^{T}\!\!\sigma ^{2}(s)b^{2}(s,T)\mathit{ds}\right ) {}\\ & =& e^{-\mathcal{K}(T)}\int _{ 0}^{T}e^{\mathcal{K}(s)}\theta (s)\mathit{ds} - \frac{\partial } {\partial T}\left (\frac{1} {2}\int _{0}^{T}\sigma ^{2}(s)b^{2}(s,T)\mathit{ds}\right ). {}\\ \end{array}$$

Rearranging

$$\displaystyle{e^{\mathcal{K}(T)} \frac{\partial } {\partial T}a(0,T) =\int _{ 0}^{T}e^{\mathcal{K}(s)}\theta (s)\mathit{ds} - e^{\mathcal{K}(T)} \frac{\partial } {\partial T}\left (\frac{1} {2}\int _{0}^{T}\sigma ^{2}(s)b^{2}(s,T)\mathit{ds}\right ).}$$

Differentiating again with respect to T we obtain

$$\displaystyle\begin{array}{rcl} \frac{\partial } {\partial T}\left (e^{\mathcal{K}(T)} \frac{\partial } {\partial T}a(0,T)\right )& =& e^{\mathcal{K}(T)}\theta (T) {}\\ & & \quad - \frac{\partial } {\partial T}\left (e^{\mathcal{K}(T)} \frac{\partial } {\partial T}\left (\frac{1} {2}\int _{0}^{T}\sigma ^{2}(s)b^{2}(s,T)\mathit{ds}\right )\right ). {}\\ \end{array}$$

Rearranging this last equation we obtain

$$\displaystyle\begin{array}{rcl} \theta (T)& =& e^{-\mathcal{K}(T)} \frac{\partial } {\partial T}\left (e^{\mathcal{K}(T)} \frac{\partial } {\partial T}a(0,T)\right ) {}\\ & & \qquad + e^{-\mathcal{K}(T)} \frac{\partial } {\partial T}\left (e^{\mathcal{K}(T)} \frac{\partial } {\partial T}\left (\frac{1} {2}\int _{0}^{T}\sigma ^{2}(s)b^{2}(s,T)\mathit{ds}\right )\right ). {}\\ \end{array}$$

Problems

Problem 23.1

Equation (23.15) shows that under the equivalent martingale measure the bond price dynamics are given by

$$\displaystyle{\frac{\mathit{dP}(t,T)} {P(t,T)} = r(t)\mathit{dt} +\sigma _{P}(r,t,T)d\tilde{z}(t)}$$

with σ _P (r, t, T) defined by (23.5). Use Ito’s lemma to show that the yield to maturity

$$\displaystyle{\rho (t,T) = - \frac{1} {(T - t)}\ln P(t,T)}$$

satisfies

$$\displaystyle{d\rho (t,T) = \frac{1} {(T - t)}\left [\left (\rho (t,T) + \frac{1} {2}\sigma _{P}^{2}(r,t,T) - r(t)\right )\mathit{dt} -\sigma _{ P}(r,t,T)d\tilde{z}(t)\right ].}$$

Problem 23.2

Recall the relation (22.9) between the bond price P(t, T) and the instantaneous forward rate f(t, T). By considering the quantity

$$\displaystyle{B(t,T) =\ln P(t,T)}$$

and assuming that

$$\displaystyle{d\left (\frac{\partial B(t,T)} {\partial T} \right ) = \frac{\partial } {\partial T}\mathit{dB}(t,T),}$$

show that f(t, T) satisfies the stochastic differential equations

$$\displaystyle{df(t,T) = -\sigma (t,T)\sigma _{P}(t,T)\mathit{dt} +\sigma (t,T)\mathit{d}\tilde{z}}$$

where

$$\displaystyle{\sigma (t,T) = - \frac{\partial } {\partial T}\sigma _{P}(t,T).}$$

Problem 23.3

Consider the Hull–White model of Sect. 23.4.2, for which the bond price P(t, T) is given by (23.25) with a(t, T) and b(t, T) given by Eqs. (23.47) and (23.48). Obtain the corresponding expression for the forward rate f(t, T). Show what this expression reduces to in the special case of the Vasicek model.