Download Adobe Premiere Pro CS4 Le Guide Complet by Franck Chopinet PDF

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We will focus on the computational aspects of the fair price. In Section 2, we shall sketch the obtention of the risk-minimizing hedging price for the European option and in Section 3, we will present two ways to compute the fair price obtained in Section 2. 2 Risk Minimizing Hedging Price First let us derive some properties of the market. 3), it follows that 7„ = cov (Xi, Xn+i - Xn) = 0 . Thus the stock return process is not a long memory process! I fact this process is a semimartingale. However, the volatility process at is a long memory process, this means that J2™=i 7« — °°Since we introduce another independent random factors, we can expect that the market is now no longer complete.

The proof of (i) is complete. Now let us prove (ii), indeed, it is easy to check that (—yf + 1,—zf) is the solution of BSDE: Vt I(xT>a)+ where g(y, z,t) := -g(-y We now prove / g(ys,zs,s)ds- / +l,-z,t). e. a G [ 0 , n Indeed, similar to the proof of (i), we can construct a C 3 function $„(x, a) such that $ n (XT,a) —> I(xT>a) as n —> oo and $„(x,a), $„(x, b) are comonotonic. 37 Let (yn'a>9iZn,a,gj be the solution of BSDE: yt = ¥ n ( X T , o ) + / g(y„z„s)ds- / zsdWs then by Lemma 2, zn'a'° -* -zA as n - • oo in L 2 ( 0 , T ) .

3) Then T is a non-negative definite, Hilbert-Schmidt operator on L2([0,T]). Thus there is an orthonormal basis {ei ,e2 ,• ••} of L 2 ([0,T]) and 0 < Ai < A2 < • • • such that 00 A E n<°° n= l 55 and l en — Anen , n — l, z, • • • Therefore K(t, s) = lim Km(t, s), m—*oo where m Km(t,s) = y^ X„en(t)en(s) n= l and the convergence is in L2([0,T]2) (in the operator's Hilbert-Schmidt norm). Introduce Qm(t) = Y/y/Ke„(t)Gn n=l where {Gi, G2 , • • •} is a sequence of independent standard normal variables.

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