14.4 Pricing contingent claims - Backward Induction

14.4 Pricing contingent claims - Backward Induction

Once the short-term interest rate tree has been constructed such that the short- term interest rate and associated discount factor are known at each node of the tree, any interest rate contingent claim may be priced by a simple backward induction procedure.

14.4.1 Pricing Coupon Bonds

Consider a coupon bond maturing at time ^[3] T = N _T ” t and paying coupons at discrete time steps { t ₁ , t ₂ , , t _m } where m is the number of coupons due until maturity. If c is the amount payable at each coupon time and the last coupon payment coincides with bond maturity, i.e. t _m = N _T , the maturity value of the bond may be initialised at time i = N _T as:

where P ^c _i,j is the value of the coupon paying bond at node ( i, j ).

Now for i < N _T , the value of the coupon bond is equal to the discounted expected value of the bond at the next time step ( i + 1). Since the risk-neutral probability associated with each branch of the binomial tree is ½, the value of the coupon bond for all i < N _T is determined as ^[4] follows .

If i ˆˆ { t ₁ , t ₂ , , t _m } then

else

14.4.2 Pricing European Options on Coupon Bonds

Once we have determined the value of the coupon paying bond at each node of the binomial tree, we may price claims contingent on this coupon bond. Consider a European call option on the above coupon paying bond with:

• expiry date s = N _s ” t ,

• strike price X .

Knowing the value of the coupon bond in all states at option expiry time i = N _s , we determine the option payoff as:

where C _i,j is the value of the European call option at node ( i, j ). For each i < N _s the value of the European call is determined as the discounted expectation of option values at time ( i + 1); hence:

More directly, utilise the discounted probabilities or state prices Q _i,j to determine the time i = 0 European option value directly from the expiry condition (14.6) as:

^[3] Hence, this bond may be seen to mature N _T time steps from initial time i = 0.

^[4] Here j represents each node at time step i , hence j = ˆ’ i, ˆ’ i +2, , i ˆ’ 2, i .