Compare the graphs ofcs(t) in this case with thegraphs obtained in Problems 4 and 5(a). Canyou draw any conclusions about the possibleeffectiveness of source remediation? If so, whatare they?

Project 4Monte Carlo Option Pricing: Pricing FinancialOptions by Flipping a CoinA discrete model for change in price of a stock over a time interval [0,T] isSn+1=Sn+μSnt+σSnεn+1√t,S0=s(1)whereSn=S(tn) is the stock price at timetn=nt,n=0,...,N−1,t=T/N,μisthe annual growth rate of the stock, andσis a measure of the stock’s annual price volatilityor tendency to fluctuate. Highly volatile stocks have large values forσ, for example, valuesranging from 0.2 to 0.4. Each term in the sequenceε1,ε2,...takes on the value 1 or−1dependingonwhethertheoutcomeofacointossingexperimentisheadsortails,respectively.Thus, for eachn=1,2, . . .εn=1with probability=1/2−1with probability=1/2.(2)A sequence of such numbers can easily be created by using one of the random numbergenerators available in most mathematical computer software applications. Given such asequence, the difference equation (1) can then be used to simulate asample pathortrajec-toryof stock prices,{s,S1,S2,...SN}. The “random” termsσSnεn+1√ton the right-handside of (1) can be thought of as “shocks” or “disturbances” that model fluctuations in thestock price. By repeatedly simulating stock price trajectories and computing appropriateaverages, it is possible to obtain estimates of the price of aEuropean call option, a type offinancial derivative. A statistical simulation algorithm of this type is called aMonte Carlomethod.A European call option is a contract between two parties, a holder and a writer, whereby,for a premium paid to the writer, the holder acquires the right (but not the obligation) topurchase the stock at a future dateT(theexpiration date) at a priceK(thestrike price)agreed upon in the contract. If the buyer elects to exercise the option on the expiration date,the writer is obligated to sell the underlying stock to the buyer at the priceK. Thus, theoption has, associated with it, apayoff functionf(S)=max (S−K,0)(3)whereS=S(T) is the price of the underlying stock at the timeTwhen the option expires(see Figure 2.P.3).CSKFIGURE 2.P.3The value of a call option at expiration isC=max(S−K,0) whereKisthe strike price of the option andS=S(T) is the stock price at expiration.

Eq. (3) is the value of the option at timeTsince, ifS(T)>K, the holder can purchase,at priceK, stock with market valueS(T) and thereby make a profit equal toS(T)−Knotcounting the option premium. IfS(T)<K, the holder will simply let the option expire sinceit would be irrational to purchase stock at a price that exceeds the market value. The optionvaluation problem is to determine the correct and fair price of the option at the time thatthe holder and writer enter into the contract.15To estimate the price of a call option using a Monte Carlo method, an ensembleS(k)N=S(k)(T),k=1,...,M

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- Monte Carlo method, Stability theory, Monte Carlo methods in finance