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that in two months, the stock will trade for either $18 per share or $29 pershare. Use the one-period binomial option pricing to find the today’s priceof the call.Problem 15.10‡A nondividend-paying stock S is modeled by the tree shown below.
15 THE REPLICATING PORTFOLIO METHOD119A European call option on S expires att= 1 with strike priceK= 12.Calculate the number of shares of stock in the replicating portfolio for thisoption.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 strikeprice of 105.Problem 15.12Which of the following binomial models with the given parameters representan 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%.
120OPTION PRICING IN BINOMIAL MODELS16 Binomial Trees and VolatilityThe goal of a binomial tree is to characterize future uncertainty about thestock price movement. In the absence of uncertainty (i.e. stock’s return iscertain at the end of the period), a stock must appreciate at the risk-freerate less the dividend yield. Thus, from timetto timet+hwe must haveSt+h=Ft,t+h=Ste(r-δ)h.In other words, under certainty, the price next period is just the forwardprice.What happens in the presence of uncertainty (i.e. stock’s return at the endof the period is uncertain)? First, a measure of uncertainty about a stock’sreturn is itsvolatilitywhich is defined as the annualized standard deviationof the return of the stock when the return is expressed using continuouscompounding. Thus, few facts about continuously compounded returns arein place.LetStandSt+hbe the stock prices at timestandt+h.Thecontinuouslycompounded rate of returnin the interval [t, t+h] is defined byrt,t+h= lnSt+hSt.Example 16.1Suppose 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 areln103100= 0.02956 and ln97103=-0.06002Now, if we are givenStandrt,t+hwe can findSt+husing the formulaSt+h=Stert,t+h.Example 16.2Suppose 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 theyear.
16 BINOMIAL TREES AND VOLATILITY121Solution.The answer isS1= 100e-5= $0.6738Now, supposert+(i-1)h,t+ih,1≤i≤n,is the continuously compounded rateof return over the time interval [t+ (i-1)h, t+ih].Then the continuouslycompounded return over the interval [t, t+nh] isrt,t+nh=nXi=1rt+(i-1)h,t+ih.(16.1)Example 16.3Suppose that the stock price on three consecutive days are $100, $103, $97.