# That in two months the stock will trade for either 18

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that in two months, the stock will trade for either \$18 per share or \$29 per share. Use the one-period binomial option pricing to find the today’s price of the call. Problem 15.10 A nondividend-paying stock S is modeled by the tree shown below.
15 THE REPLICATING PORTFOLIO METHOD 119 A European call option on S expires at t = 1 with strike price K = 12 . Calculate the number of shares of stock in the replicating portfolio for this option. Problem 15.11 You are given the following information: A particular non-dividend paying stock is currently worth 100 In one year, the stock will be worth either 120 or 90 The annual continuously-compounded risk-free rate is 5% Calculate the delta for a call option that expires in one year and with strike price of 105. Problem 15.12 Which of the following binomial models with the given parameters represent an arbitrage? (A) u = 1 . 176 , d = 0 . 872 , h = 1 , r = 6 . 3% , δ = 5% (B) u = 1 . 230 , d = 0 . 805 , h = 1 , r = 8% , δ = 8 . 5% (C) u = 1 . 008 , d = 0 . 996 , h = 1 , r = 7% , δ = 6 . 8% (D) u = 1 . 278 , d = 0 . 783 , h = 1 , r = 5% , δ = 5% (E) u = 1 . 100 , d = 0 . 982 , h = 1 , r = 4% , δ = 6% .
120 OPTION PRICING IN BINOMIAL MODELS 16 Binomial Trees and Volatility The goal of a binomial tree is to characterize future uncertainty about the stock price movement. In the absence of uncertainty (i.e. stock’s return is certain at the end of the period), a stock must appreciate at the risk-free rate less the dividend yield. Thus, from time t to time t + h we must have S t + h = F t,t + h = S t e ( r - δ ) h . In other words, under certainty, the price next period is just the forward price. What happens in the presence of uncertainty (i.e. stock’s return at the end of the period is uncertain)? First, a measure of uncertainty about a stock’s return is its volatility which is defined as the annualized standard deviation of the return of the stock when the return is expressed using continuous compounding. Thus, few facts about continuously compounded returns are in place. Let S t and S t + h be the stock prices at times t and t + h. The continuously compounded rate of return in the interval [ t, t + h ] is defined by r t,t + h = ln S t + h S t . Example 16.1 Suppose that the stock price on three consecutive days are \$100, \$103, \$97. Find the daily continuously compounded returns on the stock. Solution. The daily continuously compounded returns on the stock are ln 103 100 = 0 . 02956 and ln 97 103 = - 0 . 06002 Now, if we are given S t and r t,t + h we can find S t + h using the formula S t + h = S t e r t,t + h . Example 16.2 Suppose that the stock price today is \$100 and that over 1 year the contin- uously compounded return is - 500% . Find the stock price at the end of the year.
16 BINOMIAL TREES AND VOLATILITY 121 Solution. The answer is S 1 = 100 e - 5 = \$0 . 6738 Now, suppose r t +( i - 1) h,t + ih , 1 i n, is the continuously compounded rate of return over the time interval [ t + ( i - 1) h, t + ih ] . Then the continuously compounded return over the interval [ t, t + nh ] is r t,t + nh = n X i =1 r t +( i - 1) h,t + ih . (16.1) Example 16.3 Suppose that the stock price on three consecutive days are \$100, \$103, \$97.
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