The option pricing formula above may be used to obtain the value of a call option on a pure discount bond. However, these are not readily traded in all markets and one may be forced to use options on coupon paying bonds . Jamshidian [ 31 ] developed the following methodology allowing options on coupon paying bonds to be valued as portfolios of options on discount bonds.
Let P c ( r, t, s ) be the time t price of a coupon paying bond maturing at time s, t, s ˆˆ [0, T *]. P c ( r, t, s ) consists of n payments c i , i = 1, , n at times s i , i = 1, , n where s i ˆˆ [ t, s ]. The time t price of such a bond may be expressed as:
At option expiry time T , the bond has m payments remaining. Let r * be the instantaneous continuously compounded short- term interest rate at time T , such that the price of the coupon paying bond equals the option strike price, i.e.:
The time T payoff of the European call option on such a coupon paying bond is:
where X i = P ( r *, T,s i ), i = 1, , m . Hence the payoff of the i th option is:
and the time t price of this option is:
with
From equations (13.5) and (13.1) we may write the option strike price as:
where (refer to equations (7.15) and (7.16) Chapter 7):
and
where
Since B ( T,s i ) and ‚( T,s i ) are fully specified by the initial term structure, we may apply a numerical search technique such as Newton-Raphson to solve for r * such that (13.8) holds.