This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ) = 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 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...
View Full
Document
 Summer '10
 KarlSigma

Click to edit the document details