10.2 Models for the Interest Rate and Interest Rate Derivatives

As we saw in the previous chapter, important assumptions of the Black and Scholes application are

Both assumptions are violated for bond options and the longer time periods which are typically found in the analysis of these options.

A bond produces at the time of maturity $ T$ a fixed amount $ Z$, the nominal value, and if applicable at predetermined dates before $ T$ dividend payments (coupon). If there are no coupons, the bond is referred to as a zero coupon bond or zero bond for short.

When valuing a Bond Option coupons can be treated as discrete dividend payments when valuing stock options.

10.2.1 Bond Value with Known Time Dependent Interest Rate

To begin we calculate the bond value $ V(t)$ at time $ t$ with a time dependent but known interest rate $ r(t)$.

From the assumption of no arbitrage we conclude that a bond's change in value over the time period $ [t,t+dt]$ with possible coupon payments $ K(t)dt$ coincides with the change in value of a bank account with a value $ V(t)$ and with an interest of $ r(t):$

$\displaystyle V(t+dt) - V(t) = \left( \frac{dV}{dt} + K(t) \right) dt = r(t) V(t)\
dt $

Together with the boundary restrictions $ V(T) = Z $ it follows that:

$\displaystyle V(t) = e^ {I(t)} \{ Z + \int^ {T}_t K(s) \, e^ {-I(s)} ds \}$ (10.1)

with $ I(t) = - \int^ {T}_t r(s)ds$ , the antiderivative of $ r(t).$

For a zero bond this simplifies to: $ V(t) = Z \cdot e^ {I(t)} $

10.2.2 Stochastic Interest Rate Model

Due to the uncertainty associated with the future development of the interest rate, $ r(t)$ is modelled as a random variable. In order to have an unambiguous, fixed interest rate, one usually considers the interest rate of an investment over the shortest possible time period:

$ r(t) = $ spot rate = Interest rate for the shortest possible investment.

$ r(t)$ does not follow a geometric Brownian motion so that the Black-Scholes application cannot be used. There are a number of models for $ r(t)$, that are special cases of the following general Ansatz which models the interest rate as a general Itô Process:

$\displaystyle dr(t) = \mu (r(t),t) dt + \sigma(r(t),t) dW_t$ (10.2)

$ \{ W_t\} $ represents as usual a standard Wiener process.

Three of the most used models are simple, special cases and indeed are the coefficient functions of the models from

In general $ \mu(r,t)$ and $ \sigma(r,t)$ can be conveniently chosen, in order to replicate in the model empirically observed phenomena. In the following we write $ w(r,t)$ for $ \sigma(r,t)$, in order to clearly differentiate between the function $ w
\stackrel{\mathrm{def}}{=}\sigma$ and the constant $ \sigma$, which appears as a parameter in the three models mentioned above.

10.2.3 The Bond's Value Equation

A stock option can be hedged with stocks, and Black and Scholes use this in deriving the option pricing formula. Since there is no underlying financial instrument associated with a bond, bonds with varying life spans have to be used to mutually hedge each other, in order to derive the equation for valuing bonds.

Consider a portfolio made up of a zero bond with a remaining life time of $ \tau_1$ and $ - \Delta$ zero bonds (i.e., $ \Delta$ sold zero bonds) with a remaining life time of $ \tau_2$. The value of the portfolio at time $ t$ for the current interest rate $ r(t) = r$ is:

$\displaystyle \Pi (r,t) = V_1(r,t) - \Delta \cdot V_2 (r,t).$

where $ V_i, i=1,2$ stands for the value function of both bonds. We write $ \Pi _t = \Pi \{r(t),t\}, V_{it} = V_i\{r(t),t\}, i=1,2, \;
\mu_t = \mu\{r(t),t\}, \; w_t = w\{r(t),t\}$. Using Itôs Lemma it follows that

$\displaystyle d\Pi _t = \frac{\partial V_{1t}}{\partial t} dt + \frac{\partial
...
...r} dr(t)+ \frac{1}{2} w_t^ 2 \cdot \frac{\partial^ 2 V_{1t}}{\partial
r^ 2} dt $

$\displaystyle - \Delta \left( \frac{\partial V_{2t}}{\partial t} dt + \frac{\pa...
...r(t) + \frac{1}{2} w_t^ 2 \, \frac{\partial^ 2V_{2t}}{\partial r^2} dt \right) $

By hedging the risks the random component disappears. This is achieved by choosing

$\displaystyle \Delta = \frac{\partial V_{1t}}{\partial r} \bigg/ \frac{\partial
V_{2t}}{\partial r} .$

The insertion and comparison of the portfolio with a risk free investment and taking advantage of the no arbitrage assumption, that is the equality of the change in value of the portfolio and investment:

$\displaystyle d\Pi _t = r(t) \cdot \Pi _t \, dt ,$

produces altogether
$\displaystyle \left( \frac{\partial V_{1t}}{\partial t} + \frac{1}{2} w_t^ 2
\f...
...{\partial r^ 2} - r(t)\ V_{1t}\right) \bigg/
\frac{\partial V_{1t}}{\partial r}$      
$\displaystyle = \left( \frac{\partial V_{2t}}{\partial
t} + \frac{1}{2} w_t^ 2 ...
...partial r^ 2} - r(t)\ V_{2t}
\right) \bigg/ \frac{\partial V_{2t}}{\partial r}.$     (10.6)

This is only correct when both sides are independent of the remaining life times $ \tau_1, \tau_2 $. $ V_{1t}, V_{2t} $ therefore satisfy both of the following differential equations

$\displaystyle \frac{\partial V_t}{\partial t} + \frac{1}{2} w_t^ 2
\frac{\parti...
...partial r^ 2} - r(t)\ V_t = -a\{r(t),t\}
\cdot
\frac{\partial V_t}{\partial r} $

for the function $ a(r,t)$ which is independent of one of the remaining life times. With the economically interpretable value

$\displaystyle \lambda (r,t) = \frac{ \mu (r,t) -a (r,t)}{w(r,t)} $

this produces with the abbreviation $ \lambda_t = \lambda
\{r(t),t\}$ the zero bond's value equation for $ V(r,t):$

$\displaystyle \frac{\partial V(r,t)}{\partial t} + \frac{1}{2} w_t^2 \frac{\par...
...- (\mu_t - \lambda_t w_t) \frac{\partial V(r,t)}{\partial r} - r(t)\ V(r,t) = 0$ (10.7)

with the boundary restrictions $ V(r,T) = Z$ at the time of maturity and with additional boundary restrictions dependent on $ \mu, w$. It should be noted that in the equation $ \mu_t, w_t,
\lambda_t$ stand for functions of $ r$ and $ t$.

The value $ \lambda (r,t)$ has the following interpretation. Consider a risky portfolio, that is not hedged, consisting of a bond with the value $ V_t = V\{r(t),t\}.$ For the change in value within the time period $ dt$ we obtain using Itôs Lemma and the zero bond's value equation:

$\displaystyle dV_t = r(t)\ V_t\ dt + w_t \cdot \frac{\partial V_t}{\partial r} (dW_t +
\lambda\{r(t),t\} \, dt) $

Since $ \mathop{\text{\rm\sf E}}[dW_t] = 0,$ the mean change in value $ \mathop{\text{\rm\sf E}}[dV_t] $ is

$\displaystyle \left( w_t \frac{\partial V_t}{\partial r}\right) \cdot \lambda\{r(t),t\} \, dt
$

above the increase in value $ r(t)\ V_t\ dt $ of a risk free investment. $ \lambda(r,t) \, dt$ is therefore the bonus on the increase in value, which one receives at time $ t$ with a current interest rate $ r(t) = r$ for taking on the associated risks. $ \lambda (r,t)$ is thus interpreted as the market price of risk.

10.2.4 Solving the Zero Bond's Value Equation

Consider the special case:

$\displaystyle w(r,t) = \sqrt{\alpha(t) r- \beta(t)}$

$\displaystyle \mu (r,t) = - \gamma(t) \cdot r + \eta (t) + \lambda (r,t) w(r,t).$

Inserting the solution assumption

$\displaystyle V(r,t) = Z \cdot e^ {A(t)- r B(t)}$

into the zero bond's value equation results in the two equations

$\displaystyle \frac{\partial A(t)}{\partial t} = \eta(t)\ B(t) +
\frac{1}{2} \, \beta(t)\ B^2(t) $

$\displaystyle \frac{\partial B(t)}{\partial t} = \frac{1}{2} \alpha (t)\ B^2(t) + \gamma
(t)\ B(t) - 1 $

with boundary restrictions $ A(T) = B(T) = 0 $ (since $ V(r,T) = Z). $

For the time independent $ \alpha, \beta, \gamma, \eta $ there is an explicit solution, which with a remaining life time of $ \tau = T - t$ has the form

$\displaystyle B(t) = \frac{2(e^ {\psi _1 \tau} - 1 ) }{(\gamma + \psi_1)(e^
{\psi_1 \tau} -1) + 2\psi_1} \, ,\ \, \, \psi_1 = \sqrt{\gamma^ 2 +
\alpha} $

$\displaystyle \frac{2}{\alpha} A(t) = b_2 \psi_2 \ln (b_2-B) + (\psi_2 -
\frac{\beta}{2} ) b_1 \ln (\frac{B}{b_1} + 1 ) $

$\displaystyle + \frac{1}{2} B \, \beta - b_2 \psi_2 \ln b_2 $

with

$\displaystyle b_{1/2} = \frac{\pm \gamma + \sqrt{\gamma^ 2 + 2\alpha}}{\alpha},\
\psi_2 = \frac{\eta + b_2 \beta/2}{b_1 + b_2} $

Choice of parameters:
1)
The spot rate volatility is $ \sqrt{\alpha r(t)
-\beta}.$ With this representation $ \alpha, \beta $ can be estimated from historical data, in a fashion similar to the historical volatility of stocks.
2)
Taking the yield curve (see Section 10.4.3) into consideration, the discussion of which goes beyond the scope of this section, estimators for $ \gamma$ and $ \eta$ can be derived.

10.2.5 Valuing Bond Options

As an example consider a European Call with a strike price $ K$ and a maturity $ T_C$ on a zero bond with a maturity of $ T_B
> T_C, $, i.e., the right is given to buy the bond at time $ T_C$ at a price $ K$.

$\displaystyle V_B(r,t) =$   Value of the bond at time $\displaystyle t$   with the current interest rate$\displaystyle \; r(t)=r$

$\displaystyle C_B(r,t) =$   Value of the Call at time $\displaystyle t$   with the current interest rate$\displaystyle \; r(t)=r$

$ C_B$ is only dependent on the random variable $ r(t)$ and time $ t$ and therefore itself also satisfies the zero bond's value equation, but with the boundary restrictions

$\displaystyle C_B (r,T_C) = \max \left( V_B (r,T_C) - K,\ 0\right) . $

This equation, analogous to the corresponding Black-Scholes equation can be numerically solved.