Finance: ‘Risk-Free’ Trading and Option Pricing

Abstract

Financial investment reached new levels of sophistication some decades ago with the development of ‘futures’ markets. In these the seller of a contract accepts payment now against a guarantee to provide a specified service at some time in the future. An example is the European call option concerning a stock whose unit value at time t is S _t . At time t = 0 (say) the seller offers the buyer the right (but not the obligation) to buy unit amount of the stock at time T at a fixed price K, the striking price. The gain to the buyer at time T is thus max (S_T - K, 0) = (ST - K)+, and this is the amount that the seller must undertake (at time t = 0) to produce at time T. The transaction can be regarded as a device to share uncertainty between buyer and seller. The buyer is offered a fixed price but is not certain whether he will wish to buy at that price when the time comes. The seller does not know the future course of events but knows that he has the opportunity of trading himself over the time period 0 < t < T. The question is then: what is the fair price for the option? That is, what sum should the seller demand from the buyer at time t = 0 to finance the service he has undertaken to provide at time T?