Lecture7_BinomialOptionPricing

15192 13269 100115192 for i 1 to n n2 for s1

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Unformatted text preview: 132.69 100 75.36 0 0 10.72 0 24.64 15-17 Details in the code * In the program “binomial_euro”, we initialise asset prices at In maturity step n maturity S(0) = Stock price of all ups S(0) = 100*(1.1519^2) = 132.69 100*(1.1519^2) For I = 1 To n (n=2), For S(1) = (S(1 - 1) / u) * d = (132.69/1.1519)*0.8681 = 100 S(1) S(2) = (S(2 - 1) / u) * d = (100/1.1519)*0.8681= 75.36 Now, we can easily compute the value of the option at the maturity (n) = MAX( OptionType*(S(I) - K), 0 ) We now have the all option prices in step 2: price(0)=0, price(1)=0, price(2)=24.64 price(0)=0, 15-18 Details in the code ** (1) In the program “binomial_euro”, we step back the tree For j = n - 1 To 0 Step -1 For For I = 0 To j For price(I) = pv * (Pu * price(I) + Pd * price(I + 1)) price(I) Next I Next Next j For j = n – 1 = 1 For I = 0 to 1, we will have the following option price in j=1: For price(0) = pv*(Pu*price(0)+Pd*price(1)) = 0.9753*(0.5539*0 + (1-0.5539)*0) = 0 (Note that price(0) was step 2 option price but is now replaced by the updated value!! price(0) is now the option 15-19 price in step 1) price Details in the code ** (2) price(1) = pv*(Pu*price(1)+Pd*price(2)) = 0.9753*(0.5539*0 + (1-0.5539)*24.64) = 10.72 (Note that price(1) is now replaced by the option price in step 1) in For j = n – 2 = 0 For For I = 0 to 0, we will have the following option price in j=0: For price(0) = pv*(Pu*price(0)+Pd*price(1)) = 0.9753*(0.5539*0 + (1-0.5539)*10.72) = 4.66 (Note that price(0) is now replaced by the updated value!! So, price(0) was step 1 option price, but is now step 0 option price) step At the end, price(0) gives us the option price for today (step At 0) 0) 15-20 Cox, Ross, Rubinstein’s (CRR) binomial trees – American options 15-21 Using binomial tress to price American options (1) In the final step, the payoffs of American In options are the same as the payoffs of same European options European So, pricing American options is similar to what So, we have done for European options as we all start from the final step’s payoffs final In any step before the final step, the risk-neutral In investor makes the decision between (a) exercise the option now and (b) wait for the next step next So, investor compare (a) the payoff from So, exercise and (b) the expected payoff for waiting exercise 15-22 Using binomial tress to price American options (2) Check the “option” tab for American options Take the bottom triangle of step 8 (J57) as the Take example, we find the following example, p=0.5177 $14.05 1-p= 0.4823 $17.29 $? As shown earlier, the claim in this node is As 15.54, i.e., if the investor choose not to exercise the option now, the expected payoff is 15.54 (present value) 15.54 15-23 Using...
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