Option Pricing

A contingent claim (or option) is a contract giving the buyer of the contract (or simply the buyer) the right to buy from or sell to the contract writer (or simply writer) a share of an underlying stock at a predetermined price q > 0 (called the strike price) and at or prior to a prespecified time T > 0 (called the expiration date) in the future. The right to buy (respectively, to sell) a share of the stock is called a call (respectively, a put) option. The European (call or put) option can only be excised at the expiration date T, but the American (call or put) option can be excised at any time prior to or at the expiration date. The pricing problem of a contingent claim is, briefly, to determine the fee (called the rational price) that the writer should receive from the buyer for the rights of the contract and also to determine the trading strategy the writer should use to invest this fee in a (B, S)-market in such a way as to ensure that the writer will be able to cover the option if and when it is exercised. The fee should be large enough that the writer can, with riskless investing, cover the option but be small enough that the writer does not make an unfair (i.e., riskless) profit.