Term Structure Derivatives

Vasicek bond option pricing

If the term structure model is Vasicek's model there is a solution for the price of an option on a zero coupon bond, due to Jamshidan (1989).

Under Vacisek's model the process for the short rate is assumed to follow.

\begin{displaymath}dr = a(b-r)dt + \sigma dZ \end{displaymath}

where $a$, $b$ and $\sigma$ are constants. We have seen earlier how to calculate the discount factor in this case. We now want to consider an European Call option in this setting.

Let $P(t,s)$ be the time $t$ price of a zero coupon bond with a payment of $1 at time $s$ (the discount factor). The price at time $t$ of a European call option maturing at time $T$ on on a discount bond maturing at time $s$ is( See Jamshidan (1989) and Hull (1993))

\begin{displaymath}P(t,s)N(h) - XP(t,T)N(h-\sigma_P) \end{displaymath}


\begin{displaymath}h=\frac{1}{\sigma_P}\ln \frac{P(t,s)}{P(t,T)X} + \frac{1}{2}\sigma_P \end{displaymath}

\begin{displaymath}\sigma_P = v(t,T)B(T,s) \end{displaymath}

\begin{displaymath}B(t,T) = \frac{1-e^{-a(T-t)}}{a} \end{displaymath}

\begin{displaymath}v(t,T)^2 = \frac{\sigma^2({1-e^{-a(T-t)}})}{2a} \end{displaymath}

In the case of $a=0$,

\begin{displaymath}v(t,T)=\sigma\sqrt{T-t} \end{displaymath}

\begin{displaymath}\sigma_P = \sigma(s-T)\sqrt{T-t} \end{displaymath}

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