Suppose a binomial tree model for a stock where price can go up (u = 1.2) or dow

Question

Suppose a binomial tree model for a stock where price can go up (u = 1.2) or down (d = 0.7). The riskless interest rate is r = 10%. Consider a European call option with a strike K = 105 and that matures at the end of one period. Take several stock spot prices: S=$80, S=$90, S=$100, S=$110, S=$120. For each spot price compute the option premium (c) and the replication portfolio (Delta, B). Plot your results in 3 function graphs: one with c and S, one with Delta and S, and one with B and S. Interpret your results.

Explanation / Answer

Ans) Spot Price = $                                          80 Strick Price = $105 Risk free Rate of Return = 10% d = 0.7 U = 1.2 Lower Price = SP0*d = 80*0.70 $                                                    56 upper Price = SP0*u = 80*1.20 $                                                    96 Probability of Upper price limit = F-d/u-d = (1.1052)-(0.7)/(1.2)-(0.7) FP1 = 0.81 Probability of Lower price limit = u-f/u-d = (1.2)-(1.1052)/(1.2)-(0.7) FP2 = 0.19 Prod(0.81) FP1 = 96, C=0 Out of Money SP0 = $80 Prod(0.19) FP2= 56, C=0 Out of Money Prod(0.81) FP1 = 108, C=3 In the money Computation of Option Delta SP0 = $90 C=$2.43 Prod(0.19) FP2= 63, C=0 Out of Money If the Price is raised to $90 = Changes valus of Option/ Changes in Spot Price (3*0.81) (2.43)/(90-80) 0.243 Prod (0.81) FP1=120, C=15 In the money SP0= $100 If the Price is raised to $100 = (12.15-2.43)/(100-90) C= $12.15 0.972 Prod (0.19) FP2 = 70, C=0 Out of Money Prod (0.81) FP1=132, C=27 In the money If the Price is raised to $110 = (21.87-12.15)/(110-100) SP0= $110 0.972 C= $21.87 (27*0.81) Prod (0.19) FP2 = 77, C=0 Out of Money Prod (0.81) FP1=144, C=39 In the money SP0= $120 If the Price is raised to $120 = (31.59-21.87)/(120-110) C= $31.59 0.972 (39*0.81) Prod (0.19) FP2 = 84, C=0 Out of Money Value of the present value 0.9048 Computation of Option Delta If the Price is raised to $120 = Changes valus of Option/ Changes in Spot Price (31.59-21.87)/(120-110) Present value of Call Present Value of the Call SP0 = $80 Call=0 SP0 = $90 Call=(2.43*0.972) $                                      2.36 C=$2.43 (3*0.81) SP0= $100 Call= (12.15*0.9720 $                                    11.81 C= $12.15 SP0= $110 Call=(21.87*0.972) $                                    21.26 C= $21.87 (27*0.81) SP0= $120 Call=(31.59*0.972) $                                    30.71 C= $31.59 (39*0.81)

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