Mathematical Techniques in Finance

CHAPTER 5Risk‐Neutral Valuation

While forward contracts separate the agreement date and the forward transaction date, they require both counterparties to abide by the terms of the forward contract, regardless of any profit or loss consideration. For example, the buyer in a forward contract has to buy the asset for the previously agreed upon price, even if the market price of the asset is below the contract price. An option contract, on the other hand, provides the right, but not the obligation, to transact an asset at some future date for predetermined terms. While the cash‐and‐carry argument allowed us to price a variety of forward contracts, the pricing of options requires more advanced techniques, and their pricing falls under the modern pricing paradigm of risk‐neutral valuation.

5.1 CONTINGENT CLAIMS

Option contracts are examples of contingent claims that allow the owner to transact at the option owner's sole discretion. We will focus on the economic value of the contract at transaction time and assume that an option owner will transact if and only if the economic value of the underlying transaction is positive.

The prime example of an option is a European‐style exercise option, which has a specified payoff at a specific exercise/expiration date in the future. For example, a European‐style call option, , with strike on an asset gives the owner the right—but not the ...