Option pricing in continuous time

The method of pricing financial securities in discrete time, studied in the previous chapter, makes use of an arbitrary concept: that of the unitary holding period of financial securities. In fact, we considered two successive instants t and t + 1, put into place an arbitrage portfolio at instant t, then examined the rebalancing of this operation at instant t + 1, the interval of time between t and t + 1 being fixed and given. A procedure of this type excludes the possibility of rebalancing at an intermediate instant. In practice, however, investors are free to operate on the financial market at any time. 1 A more natural procedure therefore involves using a continuous timescale. It is in this spirit that we showed in the previous chapter what the price limit of an option would be if we indefinitely reduced the interval of time between two successive instants. This procedure is, however, indirect. We shall now study, and apply to option pricing, a mathematical technique allowing us to reason directly in continuous time: the technique of stochastic differential calculus.

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