ITOLEMMA - EC3070 FINANCIAL DERIVATIVES ITO’S LEMMA...

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Unformatted text preview: EC3070 FINANCIAL DERIVATIVES ITO’S LEMMA Preliminaries Ito’s lemma enables us to deduce the properties of a wide vari- ety of continuous-time processes that are driven by a standard Wiener process w ( t ). We may begin an account of the lemma by summarising the properties of a Wiener process under six points. First, we may note that (i) E { dw ( t ) } = 0 , (ii) E { dw ( t ) dt } = E { dw ( t ) } dt = 0 , (iii) E { [ dw ( t )] 2 } = dt. (1) The first of these is true by construction, since the increments of a Wiener process over any finite interval have expectations of zero. The second result follows from the fact that the expectation of the product of a random variable dw ( t ) and a constant dt is the constant times the expectation of the variable. The third result is also part of the construction, since it is the variance of the mean-zero increment dw ( t ) of the Wiener process. Next, are some derived properties: (iv) V { [ dw ( t )] 2 } = E { [ dw ( t )] 4 } − ( E { [ dw ( t...
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This note was uploaded on 03/02/2012 for the course EC 3070 taught by Professor D.s.g.pollock during the Spring '12 term at Queen Mary, University of London.

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ITOLEMMA - EC3070 FINANCIAL DERIVATIVES ITO’S LEMMA...

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