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4106-08-Notes-BLM

# 4106-08-Notes-BLM - Copyright c 2007 by Karl Sigman...

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Copyright c 2007 by Karl Sigman 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 0 , 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 bought \$ x worth of the risk-free asset (a bond for example, or placed money in a savings account at that fixed rate) at time n = 0). Given the value of S n , S n +1 = uS n , w.p. p ; dS n , w.p. 1 - p , n 0 , 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 0 × Y 1 × · · · × Y n , n 1 , (1) where S 0 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 0 for some i ∈ { 0 , . . . 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 0 ) = n i p i (1 - p ) n - i , 0 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 : 0 i n < ∞} , which is the state space for this Markov chain. Note that this lattice depends on the initial price S 0 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 that costs \$1 . 00 per share; x shares bought 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. Selling this asset is borrowing money (shorting this asset). 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 0 (1+ r ) S 0 u at time t = 1 by buying S 0 shares of the 1 This model is meant to approximate the continuous-time geometric Brownian motion (GBM) S ( t ) = S 0 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 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 Y (typically done by fitting the first two moments of H with those of Y ). As h 0 the approximation becomes exact.

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4106-08-Notes-BLM - Copyright c 2007 by Karl Sigman...

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