10 - Copyright c 2010 by Karl Sigman 1 Binomial lattice...

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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 risk-free 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 risk-free asset In addition to our stock there is a risk-free 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 continuous-time geometric Brownian motion (GBM) S ( t ) = S e X ( t ) model for stock, where X ( t ) = B ( t )+ t is Brownian motion (BM) with drift and variance term 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 re-write 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 two-point distribution of...
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10 - Copyright c 2010 by Karl Sigman 1 Binomial lattice...

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