# Stochastic Calculus for Finance I: The Binomial Asset Pricing Model (Springer Finance)

Language: English

Pages: 187

ISBN: 0387249680

Format: PDF / Kindle (mobi) / ePub

Developed for the professional Master's program in Computational Finance at Carnegie Mellon, the leading financial engineering program in the U.S.

Has been tested in the classroom and revised over a period of several years

Exercises conclude every chapter; some of these extend the theory while others are drawn from practical problems in quantitative finance

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payments Cn, . . . ,CN. It is just the sum of the value E,1 of each of the payments C�c to be made at times k = n, k = n + 1, . . . , k = N . Note that the payment at time n is included. This payment Cn depends on £nly the first n tosses and so can be taken outside the conditional expectation En, i.e., [(I+r�tn-k)] Vn = Cn + E n [k=n+l t +�)le-n ] (1 In the case of n = N, (2.4.13) reduces , n = 0, 1, . . . , N - 1 . (2.4.18) to (2.4.19) vN = eN . Consider an agent who is short the cash

contmct that at time N if and only if w occurs. The state price density 0z_;��N tells us the time-zero price of this contract per unit of actual probability. For this reason, we call it a density. Of course, most contracts make payoffs for several different values of and these payoffs are not all necessarily 1. w, Such a contract can be regarded as a portfolio of simple contracts, each of which pays off 1 if and only if some particular w occurs, and their prices can be computed by

pricing formula. Theorem 3.2. 7. Consider an N -period binomial model with 0

it below. Let m be a. b�· posit.iv<' integPr. The risk-neut ral value of the put if the owner usrs t.hP Pxercise policy · to 4 2 - "' , is r- m , which exercises the first time the stock price fall s This is the risk-neutral expected payoff of the option at the time of exercise, discounted from the exercise time back to time zero. Because J\Jn is a sym metric random walk under the risk-neutral probabilities. we can compute the right.-hand side of be � · so that (5.4.2) using

prices) . sure on the space [l of all possible sequences of that every sequence w Ro, R1 , R2 , . . . , RN-l 1 • • • wN Let N P be a probability mea coin tosses, and assume has st7ictly positive probability under be an interest rate process, with each Rn P. Let depending on (6.2.1}. Define the discount 1n·ocess = 0, 1 , . . . , N. by (6.2.2). For 0 $ 11 ::; m ::; N. the price at time n of the zero-coupon bond maturing at time m is defined to be only the first n coin tosses and