Agricultural Finance: From Crops to Land, Water and Infrastructure

Chapter 4Plain Vanilla Options on Commodity Spot and Forward Prices. The Bachelier–Black–Scholes Formula, the Merton Formula, the Black Formula

‘The pricing formula will become a self-fulfilling prophecy.’

1973 – Robert Merton, Nobel Prize winner 1997

4.1 Introduction

Some technicalities in this chapter may be skipped. However, a few properties deserve to be emphasized in this Chapter because of their economic importance.

An option is a financial security granting its holder the right (but not the obligation) to buy (or sell) a given asset or commodity, called the underlying, at a predetermined price, called the ‘strike’ of the option, or at a given date, called the ‘maturity’ of the option. This unique exercise date can be chosen at any moment prior to maturity in the case of American options.

Hence, the characteristics of the option may be summarized as follows:

• Underlying asset: stock, stock index, currency, bond, commodity, etc. (denoted as S).
• Option type: call C or put P.
• Position type: buyer (long) or seller (short).
• Strike price: purchase or selling price guaranteed by the ownership of the option (denoted as k).
• Inception date: day on which the option is ‘written’ by the seller.
• Maturity date: or expiration date (denoted T in what follows); day on which the option may be exercised (European option) or last day of the exercise interval in the case of an American option.
• Option price: or premium, denoted C(0) for the call and P(0) for the put.

Indeed, two categories ...