Download e-book for iPad: Avions de papier by Didier Boursin

By Didier Boursin

ISBN-10: 204720142X

ISBN-13: 9782047201428

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18) we conclude that only one boundary condition and one initial condition are needed in order to specify a unique solution (see Friedrichs, 1958; Duffy, 1977). 16) should we choose? , 1979). 19) The lines have positive or negative slope depending on whether μ has negative or positive values. 16) is superfluous. 3), the so-called fundamental solution plays an important role. In general the fundamental solution has a singularity of a certain type – for example, a Dirac delta function δ(x). This function is defined on the real ∞ line (−∞, ∞) and is zero there, except at x = 0.

Many of the diffusion processes originate in the physical sciences and we shall attempt to apply some of the results in this book. In particular, we need to show the role of the heat equation in the context of the Black–Scholes equation. In fact, the original Black-Scholes equation with constant volatility and risk-free interest rate can be reduced to the heat equation by a suitable change of variables (Wilmott, 1998). Other reasons for studying the heat equation are: r It is an essential component in more general convection–diffusion equations.

5) + 12 σ 2 S 2 2 + (r − D)S ∂t ∂S ∂S where V is the derivative type, S is the underlying asset (or stock), σ is the constant volatility, r is the interest rate and D is a dividend. 6) (see Bhansali, 1998). This equation models a multi-asset environment. In this case σi is the volatility of the ith asset and ρi j is the correlation between assets i and j. 1) we note that it has an infinite number of solutions in general. In order to reduce this number to 1, we need to define some extra constraints.

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Avions de papier by Didier Boursin

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