Question

You are offered a European Call option. This means you will have the option, but not the
obligation, to buy the stock at the strike price K of $100.

The price of the stock today is
$90. Your time discount rate is Beta=0.98, the risk-less rate of interest is 3%.

The price of the stock follows the following process over two periods: with probability
75% the price will not change from period 0 to period 1, but with probability 25% will go
up to $130. Then from period 1 to period 2, with probability 25% the price will stay the
same, and with probability 75% the price will go down by 20%.
How much would you be willing to pay as of period 0 for this option

Show work please!

Answer 1

Please see the stock price tree along with joint probability of each path.

Payoff from call option at the end of period 2: max (S2 - K, 0)

K = strike price = $ 100

If C_2i is the payoff from call option under path i at the end of period 2 then:

C₂₁ = max (90 - 100, 0) = 0 with probability p₂₁ = 18.75%

C₂₂ = max (72 - 100, 0) = 0 with probability p₂₂ = 56.25%

C₂₃ = max (130 - 100, 0) = 30 with probability p₂₃ = 6.25%

C₂₄ = max (104 - 100, 0) = 4 with probability p₂₄ = 18.75%

Hence expected payoff from call option at the end of period 2 = C₂ = 18.75% x 0 + 56.25% x 0 + 6.25% x 30 + 18.75% x 4 = 2.625

Hence, price today = C₀ = C₂ x Beta^{no. of period} = 2.625 x 0.98² = $  2.5211

Hence, I will be willing to pay as of period 0 for this option an amount = C₀ = $ 2.5211