Rate Options and the Black Model

In 1976 Fisher Black presented a slightly modified version of the B-S model, using similar assumptions, for pricing forward contracts and interest rate options. Banks today employ this modified version, the Black model, to price swaptions and similar instruments in addition to bond and interest rate options, such as caps and floors. The bond options described in this section are options on bond futures contracts, just as the interest rate options are options on interest rate futures. 

The Black model refers to the underlying assets or commodity s spot price, S t). This is defined as the price at time t payable for immediate delivery, which, in practice, means delivery up to two days forward. The spot price is assumed to follow a geometric Brownian motion. The theoretical price, F t,T), of a futures contract on the underlying asset is the price agreed at time t for delivery of the asset at time T and payable on delivery. When t= T, the futures price equals the spot price. As explained in chapter 12, futures contracts are cash settled every day through a clearing mechanism, while forward contracts involve neither daily marking to market nor daily cash settlement. 

The values of forward, futures, and option contracts are all functions of the futures price F(t,T), as well as of additional variables. So the values at time r of a forward, a futures, and an option can be expressed, respectively, asf F,t), u F,t), and C F,i). Since the value of a forward contract is also a function of the price of the underlying asset S at time T, it can be represented hy f F,t,S,T). Note that the value of the forward contract is not the same as its price. As explained in chapter 12, a forward s price, at any given time, is the delivery price that would result in the contract having a zero present value. When the contract is transacted, the forward value is zero. Over time both the price and the value fluctuate. The futures price 

Calculate the price of a European call option with a strike price of 100 and a maturity of one year, written on a bond with the following characteristics  

Semiannual coupon Time to maturity Bond price volatility Coupon payments 

The Black model refers to the underlying asset s or commodity s spot price, S t). This is defined as the price at time r payable for immediate delivery, which, in practice, means delivery up to two days forward. The spot price 

Because futures contracts are repriced each day at the new forward price, their prices imply those of forward contracts. When Arises, so that F S,f is positive when F falls, / is negative. When the contract expires and delivery takes place, the forward contract value equals the spot price minus either the contract price or the spot price, futures price equals the spot price, and the value of the forward contract equals the spot price minus the contract price or the spot price. 

The value of a bond or commodity option at maturity is either the difference between the spot price of the underlying and the contract price or zero, whichever is larger. Since the futures price on the maturity date equals the spot price, the equivalence expressed in (8.31) holds. 

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