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section7 - Derivative Securities Section 7 Fall 2004 Notes...

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Derivative Securities – Section 7 – Fall 2004 Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. Topics in this section: (a) further discussion of SDE’s, including some examples and appli- cations; (b) reduction of Black-Scholes PDE to the linear heat equation; and (c) discussion of what happens when you hedge discretely rather than continuously in time. When I taught this class in Fall 2000 I discussed barrier options at this point. This time around I prefer to postpone that discussion. But you now have enough background to read about barrier options if you like; see Section 7 of my Fall 2000 notes, or the discussion in the “student guide” by Dewynne, Howison, and Wilmott. ******************** Further discussion of stochastic differential equations . Several students requested more information on examples of SDEs’ and how they can be used. Therefore the discussion that follows goes somewhat beyond the bare minimum we’ll be using in this class. (Every- thing here is, however, relevant to financial applications.) For simplicity, we restricted the discussion to problems with a “single source of randomness,” i.e. scalar SDE’s of the form dy = f ( y ( t ) , t ) dt + g ( y ( t ) , t ) dw (1) where w is a scalar-valued Brownian motion. The main things we will use about stochastic integrals and SDE’s are the following: (1) Ito’s lemma. We discussed in the Section 6 notes the fact that if A is a smooth function of two variables and y solves (1) then z = A ( t, y ( t )) solves the SDE dz = A t dt + A y dy + 1 2 A yy dy dy = ( A t + A y f + 1 2 A yy g 2 ) dt + A y g dw. We’ll also sometimes use this generalization: if y 1 and y 2 solve SDE’s using the same Brownian motion w , say dy 1 = f 1 dt + g 1 dw and dy 2 = f 2 dt + g 2 dw, and A ( t, y 1 , y 2 ) is a smooth function of three variables, then z = A ( t, y 1 ( t ) , y 2 ( t )) solves the SDE dz = A t dt + A 1 dy 1 + A 2 dy 2 + 1 2 A 11 dy 1 dy 1 + A 12 dy 1 dy 2 + 1 2 A 22 dy 2 dy 2 with the understanding that A ij = 2 A/∂y i ∂y j and dy i dy j = g i g j dt. The heuristic Taylor-expansion-based explanation is exactly parallel to the one sketched in Section 6. 1
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(2) A stochastic integral b a F dw has mean value zero. We used (and explained) this assertion at the end of Section 6, but perhaps we didn’t emphasize it enough. The explanation is easy. The integrand F = F ( t, y ( t )) can be any function of t and y ( t ). (The key point: its value at time t should depend only on information available at time t .) The stochastic integral is the limit of the Riemann sums F ( t j , y ( t j ))[ w ( t j +1 ) - w ( t j )] and each term of this sum has mean value zero, since the increment w ( t j +1 ) - w ( t j ) has mean value 0 and is independent of F ( t j , y ( t j )). (3) Calculating the variance of a dw integral. We just showed that b a F dw has mean value 0. What about its variance? The answer is simple: E b a F ( s, y ( s )) dw 2 = b a E [ F 2 ( s, y ( s ))] ds. (2) The justification is easy. Just approximate the stochastic integral as a sum. The square of the stochastic integral is approximately i,j F ( s i , y ( s i ))[ w ( s i +1 ) - w ( s i )] i,j F ( s j , y ( s j ))[ w ( s j +1 ) - w ( s j )] = i,j F ( s i , y ( s i )) F ( s j , y ( s j ))[ w (
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