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Unformatted text preview: Copyright c 2010 by Karl Sigman 1 Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n +1 = S n Y n +1 , n ≥ , where the { Y i } are iid with distribution P ( Y = u ) = p, P ( Y = d ) = 1 p . Here 0 < d < 1 + r < u are constants with r the riskfree interest rate ((1 + r ) x is the payoff you would receive one unit of time later if you placed $ x in a bank account at fixed rate r at time n = 0 (or bought a bond that increases in value at fixed rate r ). Given the value S n , at any time n ≥ 0, S n +1 = ( uS n w.p. p , dS n w.p. 1 p. independent of the past. Thus the stock either goes up (“u”) or down (“d”) in each time period, and the randomness is due to iid Bernoulli ( p ) rvs (flips of a coin so to speak) where we can view “up=success”, and “down=failure”. Expanding the recursion yields S n = S × Y 1 × ··· × Y n , n ≥ 1 , (1) where S is the initial price per share and S n is the price per share at time n . 1 It follows from (1) that for a given n , S n = u i d n i S for some i ∈ { ,...n } , meaning that the stock went up i times and down n i times during the first n time periods ( i “successes” and n i “failures” out of n independent Bernoulli ( p ) trials). The corresponding probabilities are thus determined by the binomial( n,p ) distribution; P ( S n = u i d n i S ) = n i p i (1 p ) n i , ≤ i ≤ n, which is why we refer to this model as the binomial lattice model (BLM) . The lattice is the set of points { u i d n i S : 0 ≤ i ≤ n < ∞} , which is the state space for this Markov chain. Note that this lattice depends on the initial price S and the values of u,d . Portfolios of stock and a riskfree asset In addition to our stock there is a riskfree asset (money) with fixed interest rate 0 < r < 1: $ x now (at time t = 0) would be worth the deterministic amount x (1 + r ) n at time t = n, n ≥ 1 (interest is compounded each time unit). Buying this asset is lending money (putting money in the bank). Selling this asset is borrowing money (taking out a loan). We must have 1 + r < u for otherwise there would be no reason to invest in the stock: you could instead obtain a riskless payoff of S (1+ r ) ≥ S u at time t = 1 by buying S shares of the 1 This model is meant to approximate the continuoustime geometric Brownian motion (GBM) S ( t ) = S e X ( t ) model for stock, where X ( t ) = σB ( t )+ μt is Brownian motion (BM) with drift μ and variance term σ 2 . The idea is to break up the time interval (0 ,t ] into n small subintervals of length h = t/n , (0 ,h ] , ( h, 2 h ] , ... (( n 1) h,nh = t ], and rewrite S ( t ) = S (0) × H 1 × ··· × H n , where H i = S ( ih ) /S (( i 1) h ) , i ≥ 1 are the succesive price ratios, and are in fact iid (due to the stationary and independent increments of the BM X ( t )). Then we find an appropriate p , u , d so that the distribution of H is well approximated by the twopoint distribution of...
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This note was uploaded on 10/16/2010 for the course IEOR 4701 taught by Professor Karlsigma during the Summer '10 term at Columbia.
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