section7 - Derivative Securities – Fall 2007– Section 7...

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Unformatted text preview: Derivative Securities – Fall 2007– Section 7 Notes by Robert V. Kohn, extended and improved by Steve Allen. Courant Institute of Mathematical Sciences. Further discussion of the continuous time framework. Topics in this section: (a) more stochastic calculus; (b) what are the consequences of hedging only at discrete times? (c) the link between risk-neutral expectation and PDE’s; and (d) martingales and their importance for option pricing. ******************** More stochastic calculus . You’ll need the following for HW4. Consider a stochastic integral of the form R b a g ( s ) dw ( s ) where g is a deterministic function of s . It has mean value zero – we explained this in Section 6. What about its variance? The answer is simple: E Z b a g ( s ) dw ! 2 = Z b a g 2 ( s ) ds. Here is why. Approximating the stochastic integral by a sum, we see that the square of the stochastic integral is approximately X i g ( s i )[ w ( s i +1 )- w ( s i )] ! X j g ( s j )[ w ( s j +1 )- w ( s j )] = X i,j g ( s i ) g ( s j )[ w ( s i +1 )- w ( s i )][ w ( s j +1 )- w ( s j )] . For i 6 = j the expected value of the i, j th term is 0 since [ w ( s j +1 )- w ( s j )] and [ w ( s i +1 )- w ( s i )] are independent Gaussians, each with mean value 0. For i = j the expected value of the i, j th term is g 2 ( s i )( s i +1- s i ). So the expected value of the squared stochastic integral is approximately X i g 2 ( s i )( s i +1- s i ) , which is a Riemann sum for R b a g 2 ( s ) ds . By the way: since we are assuming that g is deterministic, R b a g ( s ) dw ( s ) is a Gaussian random variable. (Proof: recall that a sum of Gaussians is Gaussian; therefore ∑ i g ( s i )[ w ( s i +1 )- w ( s i )] is Gaussian. Now use the fact that a limit of Gaussians is Gaussian.) Since we know its mean and variance, we have completely characterized this random variable. We are in the habit of focusing on lognormal dynamics, because this is the most basic model for the price of a stock (or the forward price of a stock). Another exactly-solvable SDE is the Ornstein-Uhlenbeck process, which solves dy =- cydt + σdw, y (0) = y 1 with c and σ constant. (This is not a lognormal process, because the coefficient of dw is not proportional to y .) Ito’s lemma gives d ( e ct y ) = ce ct ydt + e ct dy = e ct σdw so e ct y ( t )- y = σ Z t e cs dw, or in other words y ( t ) = e- ct y + σ Z t e c ( s- t ) dw ( s ) . We see (using the discussion at the beginning of this section) that y ( t ) is a Gaussian random variable. So it is entirely characterized by its mean and variance. They are easy to compute: the mean is clearly E [ y ( t )] = e- ct y since the “dw” integral has expected value 0, and the variance is E h ( y ( t )- E [ y ( t )]) 2 i = σ 2 E " Z t e c ( s- t ) dw ( s ) 2 # = σ 2 Z t e 2 c ( s- t ) ds = σ 2 1- e- 2 ct 2 c ....
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This note was uploaded on 01/02/2012 for the course FINANCE 347 taught by Professor Bayou during the Fall '11 term at NYU.

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section7 - Derivative Securities – Fall 2007– Section 7...

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