The Black model

We started this chapter by defining interest rate derivatives as assets with interest-rate-dependent cash flows. It is worth noting that the value of financial products is almost always dependent on some interest rates because of the need to discount the future cash flows. However, in the case of interest rate derivatives, not only the discounted value but the payoff itself depends on the interest rates. This is the main reason why interest rate derivatives are more complicated to price than stock or FX derivatives (Hull, 2009 discusses these difficulties in detail).

The Black model (Black, 1976) was developed to price options on futures contracts. Futures options grant the holder the right to enter into a futures contract at a predetermined futures price (strike price or exercise price, X) on a specified date (maturity, T). In this model, we keep the assumptions of the Black-Scholes model, except that the underlying is the futures price instead of the spot price. Hence...