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Create a four-period binominal price tree and find the fair value of an European call and...

Create a four-period binominal price tree and find the fair value of an European call and put options and an American put option on a nondividend-paying stock if the initial stock price is 82 PLN, the strike price of 80 PLN is expiring at the end of the fourth month, the compound risk-free interest rate is 12% per annum, and σ= 0.1 . Please solve in details.

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Valuation of European call option: Call option gives option buyer the right to buy the Stock at a srike price Payoff of call option Max(S-X,0) where S is stock price and X is exercise price Given the following data: Risk free rate (r) Current Price, So Exercise Price, X Standard Deviation Period (Month) Step Period (h) 12.00% 10 12 10% 4 0.081-month 14 15 16 17 Step-1: Calculation of upside and downside change ratio and probability to rise: u -exp(o*sqrt(h)) 1.03 EXP(D12 SQRT(D14)) 97%)-1/D18 19 Uu Probability of rise, p (Exp(rf h)-d)/(u-d 0.666833-(EXP(D9*D14)-D19)/(D18-D19) 0.333167 1-D20 Step-2: Building Binomiial tree using u and d: 24 First period stock price will either go up Sou or go down Sod as shown below: 26 27 $84.4 -C29 D18 $82.00 31 $79.7 -C29 D19 32 Simillary for next period price will move up and down as each node of period one Final binomial tree of prices is as follows: 35 36 1-Month Period 0$82.00 $84.40 $86.87 $89.42 $92.04G37*SD$18 S79.67S82.00S84.40S86.87IF(HS36> SC38,G37*SDS19,- 77.40 $79.67 $82.00F(H$36-$C39,638*$D$19,- $75.20 $77.40 IFH$36> SC40,G39*SD$19,) $73.06IF(H$36>-$C41,G40*SD$19,-) 38 39 41 42

Step-3: Calculation of Payoff of Put Option at the maturity of the lattice Payoff of Call Option at the maturity of the lattice is calculated using following formula: Payoff of Call Option Max(S-X,0) where S is stock price and X is exercise price 43 $12.04MAX(H37-SD$11,0) $6.87MAX(H38-SD$11,0) $2.00MAX(H39-SD$11,0) $0.00MAX(H40-SD$11,0) $0.00MAX(H41-SD$11,0) 47 0 49 2 51 52 53 4 Step-4: Calculation Payoff at the rest of the lattice The payoff at the period before the maturity is calculated by calculating the present value of expected payoff at the maturity as follows Payoff of Call Option at each node is calculated backward using following formula: Value of Call Option 57 (p*Payoff in case of rise+(1-p) Payoff in case of fall in Price)*EP-r*h) =IF(G$60>=$C61, SUMPRODUCT($D$20:SD$21,H61:H62)*EXP(-$DS9*SD$14),- 59 0$5.47 $6.86 $8.46 $10.21 $2.84$3.87 $5.20 $0.87 $1.32 $0.00 12.0-H47 6.9H48 2.0-H49 0.0 H50 0.01 =H51 61 63 2 4 Similarly payoff for all other periods is calculated by moving backward. The payoff at the period zero is the value of the option Hence Value of Call Option is $5.47-D61

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