By Bradford Angier; Lamar Underwood

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**Sample text**

Then we have Z(t) = f (t,X(t)) where X = W and f is given by f(t,x) = x4. 28), means that μ = 0 and σ = 1. Furthermore we have and . Thus the Itô formula gives us Written in integral form this reads Now we take the expected values of both members of this expression. 4, the stochastic integral will vanish. 13 Compute E [eα W(t)]. Solution: Deﬁne Z by Z(t) = eα W(t). The Itô formula gives us so we see that Z satisﬁes the stochastic differential equation (SDE) 42 STOCHASTIC INTEGRALS In integral form this reads Taking expected values will make the stochastic integral vanish.

Thus, from a ﬁnancial point of view, the portfolio h and the claim X are equivalent so they should fetch the same price. The “reasonableness” of the pricing formula above can be expressed more formally as follows. The proof is left to the reader. 21 Suppose that X is reachable using the portfolio h. Suppose furthermore that, at some time t, it is possible to buy X at a price cheaper than (or to sell it at a price higher than) . Then it is possible to make an arbitrage proﬁt. We now turn to the completeness of the model.

As a corollary we obtain the following important fact. 8 For any process g ∈ £2, the process X, deﬁned by is an -martingale. In other words, modulo an integrability condition, every stochastic integral is a martingale . Proof Fix s and t with s < t. t. 5 we have STOCHASTIC CALCULUS AND THE ITÔ FORMULA 35 We have in fact the following stronger (and very useful) result. e. X has no dt-term. Proof We have already seen that if dX has no dt-term then X is a martingale. The reverse implication is much harder to prove, and the reader is referred to the literature cited in the notes below.