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3.5 Hedging a bond with another bond (Vasicek) $ \spadesuit $

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In contrast to the price of a share (which can never drop below zero because of the regulations of bankruptcies), it is in principle possible to live with negative interest rates: these have even been observed in Switzerland in the 1960s, albeit only for a short period. This motivates the development of stochastic models for long term interest rates where the random walk of the spot rate increments $ dr$ is based on (3.3.1#eq.1) and the distribution chosen somewhere between $ \kappa=0$ (normal walk assumed by Vasicek [23]) and $ \kappa=1/2$ (assumed by Cox, Ingersoll and Ross [5]).
Here we follow the classical derivation from Vasicek using $ \kappa=0$ and apply Itô's lemma to calculate the stochastic price increment for a bond $ P(t,T)$ of maturity $ T$

$\displaystyle dP(t,T)= \underbrace{ \left[ \frac{\partial P}{\partial t} +\frac...
...erbrace{\left[\sigma_s\frac{\partial P}{\partial r}\right]}_{\sigma(t,T)} dW(t)$ (3.5#eq.1)

The drift and volatility of a bond $ (\mu,\sigma)$ depend on the spot rate $ r(t)$ and can be parametrized using market data $ (\mu_s,\sigma_s)$ . Having no anti-correlated underlying as for the case of stock options, the trick here is to create here a portfolio that is long one bond $ P(t,T_1)$ and short a number $ (-\Delta)$ of bonds $ P(t,T_2)$ with a different maturity $ T_1<T_2$ . The portfolio value and its incremental change per time step become

$\displaystyle \Pi(t)=P(t,T_1) -\Delta P(t,T_2)$ (3.5#eq.2)

$\displaystyle d\Pi(t)= \left[ \mu(t,T_1) -\Delta \mu(t,T_2)\right] dt +\left[ \sigma(t,T_1) -\Delta \sigma(t,T_2)\right] dW(t)$ (3.5#eq.3)

Choosing $ \Delta=\sigma(t,T_1)/\sigma(t,T_2)$ to eliminate the random component, the portfolio becomes deterministic and, using no-arbitrage arguments, earns exactly the risk-free spot rate

$\displaystyle d\Pi(t) = dP(t,T_1)- \Delta dP(t,T_2) = r(t)\Pi(t)dt = r(t)dt\left[ P(t,T_1)- \Delta P(t,T_2)\right]$ (3.5#eq.4)

Substitute the value for $ \Delta$ , insert (3.5#eq.1) for the increments and move all the terms with the same maturity on the same side of the equation to obtain

$\displaystyle \frac{\mu(t,T_1) -r(t)P(t,T_1)}{\sigma(t,T_1)} = \frac{\mu(t,T_2) -r(t)P(t,T_2)}{\sigma(t,T_2)} \equiv \lambda(t,r), \hspace{1cm} \forall T_1, T_2$ (3.5#eq.5)

This shows that the so-called market price of risk $ \lambda(t,r)$ is independent of the maturity and can therefore be used to parameterize the market. Rewriting the bond drift $ \mu(t,T)$ in (3.5#eq.1) in terms of the market price of risk (3.5#eq.5), the properly hedged portfolio (3.5#eq.2) finally yields the bond pricing equation

$\displaystyle \frac{\partial P}{\partial t} +\frac{1}{2}\sigma_s^2 \frac{\parti...
...}{\partial r^2} +(\mu_s -\lambda\sigma_s)\frac{\partial P}{\partial r} - rP = 0$ (3.5#eq.6)

which can be solved using the normalized terminal payoff at maturity $ P(T,T)\equiv 1, \forall r$ as the terminal condition. Boundary conditions depend on the model for the spot rate, e.g.
$\displaystyle \mu_s(r,t)$ $\displaystyle =$ $\displaystyle a(t)r -b(t)$  
$\displaystyle \sigma_s(r,t)$ $\displaystyle =$ $\displaystyle \sqrt{c(t)r-e(t)}$ (3.5#eq.7)

which can be used with $ P(r,t)\rightarrow 0, \forall t$ when $ r\rightarrow \infty$ and keeping $ P(r,t)$ finite for small $ r$ .
Using (3.5#eq.6) to re-write the deterministic component of a single bond (3.5#eq.1), Itô's lemma can be put into another form

$\displaystyle dP = \left[ \lambda\sigma_s \frac{\partial P}{\partial r} + rP \right] dt + \sigma_s\frac{\partial P}{\partial r} dW(t)$ (3.5#eq.8)


$\displaystyle dP -rPdt = \sigma_s\frac{\partial P}{\partial r} \left[ \lambda dt + dW(t) \right]$ (3.5#eq.9)

showing on the left hand side that a higher return can be earned exceeding the risk-free interest rates, provided that the investor accepts a certain level of risk $ dW(t)$ . Indeed, the portfolio grows by an extra $ \lambda dt$ per unit of risk $ dW$ . This justifies the interpretation of $ \lambda$ as the market price of risk, with investors that are either risk seeking or risk averse depending whether $ \lambda$ is positive or negative.

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