Question

21. The current stock price of firm AAa= $20. It is expected that this firm’s stock price will go up by 20%, or it might go down by 20%. No dividends. The one year risk free rate = 5%. A call option’s strike price is also $20. Using the binomial pricing model , calculate that to set up a risk free portfolio, for each call option, how many stocks (or portion of a stock) is needed.

22. Using the binomial pricing model, calculate the price of a one-year call option on this stock with a strike price of $20 is:

Answer 1

21) Note: we use 1 period binomial tree model since not explicitly mentioned.

S₀=20

S₊=20*1.2=24

u=24/20=1.2

S_-=20*0.8=16

d=16/20=0.8

p=(e^r-d)/(u-d)=(e^0.05-0.8)/(1.2-0.8)=62.8177741%

(1-p)=1-62.8177741%=37.1822259%

Call possibilities at maturity,

_C+ = MAX(S₍₊₎ - K,0) = MAX(24-20,0) = 4

_C- = MAX(S_(-) - K,0) = MAX(16-20,0) = 0

We will need Delta stocks to set up the risk free portfolio.

Delta = [C₍₊₎ - C_(-)]/[S₍₊₎ - S_(-)] = (4-0)/(24-16) = -4/8 = 0.5.

Thus we need 0.5 stock for each call to set up aformentioned portfolio.

22) Continuing from (21)-

C_t=0 = {[p * C₊] + [(1-p) * C_-] }*e^-r = [(62.8177741%*4)+(37.1822259%*0)]*e^-5%=$ 2.390164604

Excel Calculations for reference-

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