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It has been shown that today digital modems are mainly based on specific or programmable gate arrays, data and signal processors. The technology of digital integrated circuits shows continuous progress. This should soon permit the concentration of the digital part, as a first step, in a group of gate arrays and processors and eventually into only two or three devices. Beside this progress in digital technology which permits the concentration or addition of more digital processing, the analogue parts may benefit from the new integrated circuits designed for terrestrial radiocommunications and mobile phone technology. Significant results of the technical progress should also provide: more compact equipment at reduced price and ensured reliability; improved customer facilities, ease of use, higher synchronization speed and also compatibility with powerful network management means. However, it should be noted that progress in modems is not only related to the product itself but also to the definition of the whole communication system.

is the P/L from all money market account trades (this is a standard integral over time since the price of one share of the money market account is not stochastic). We now want to nd such a self- nancing dynamic trading strategy {( t , Nt ), 0 t T} such that it exactly replicates an option with the given payoff F(ST ). If such a strategy does exist, we call it a replicating strategy. DEFINITION. If such a self- nancing strategy exists using a nite number of securities then the contingent claim (i.e., option) is said to be attainable. We wish to nd such a trading strategy that its P/L at time T always exactly matches the option payoff F(ST) for every possible terminal stock price ST (0, ). If such a trading strategy does exist that matches the option payoff in all possible states of the world, the no-arbitrage principle requires that the fair value of the option ft at any time t, 0 t T, should equal the value of the replicating portfolio ( t St Nt At ) at time t: tSt Nt A t ft = 0 From Ito s lemma we can express the process for the option s price: df = mf fdt + f fdz (1.49) (1.48)

2

We can extend the model to price American-style options on foreign currencies (FX options), stock indexes, and options on futures by letting r r rf where rf is the foreign risk-free rate, r r q where q is the dividend yield on the index, and r q (r = q), respectively. Thus, for an FX option, the up probability is e

3.400 - 3.700 GHz 3.400 - 3.600 3.400 - 3.500 3.500 - 3.700 3.600 - 4.200