Question

3. A stock price is currently $30. Over each of the next two three-month periods it is expected to go up by 20% or down by 20%. The risk-free interest rate is 4% per annum. What is the value of a six-month American put option with a strike price of $32? (15 marks)

Answer 1

- Binomial Tree Pricing Step 1 - High price=Current price*up move=30*1.2=36 Low price=Current price*down move=30*0.8=24 Risk neutral probability for up move -q=(e^(risk free rate*time)-D)/(U-D) = =(e^(0.04*0.25)-0.8)/(1.2-0.8)=0.52513 Step 5 Scenario price=Spot price*(U)=30*1.2=36 Payoff H=Max(Strike price-Scenario price,0) =Max(32-36,0) =Max(-4,0)=0 Put option value at node H=e^(-r*t)*(q*Payoff HH+(1-) *Payoff HL) =e^(-0.04"0.25)*(0.525125*0+(1-0.525125)*3.2) =1.5045 Final value at node H=Max(Payoff H,Option value) =Max(0,1.5045)=1.5045 Step 2 Scenario price=Spot price*(U)^2=30*1.2^2=43.2 Payoff HH=Max(Strike price-Scenario price,0) =Max(32-43.2,0) =Max(-11.2,0) =0 Node HH Node H Step 3 Scenario price=Spot price*(U)*(D)=30*1.2*0.8=28.8 Payoff HL=Max(Strike price-Scenario price,0) =Max(32-28.8,0) =Max(3.2,0) =3.2 T=0 Today T=0.25 T=0.5 Node HL Step 7 - Put price = e(-r*t)*(q*Node H value+(1-0) "Node L value) =e^(-0.04*0.25)*(0.525125*1.5045+(1-0.525125)*8) - -4.5434 Node L Step 6 Scenario price=Spot price*(D)=30*0.8=24 Payoff L=Max(Strike price-Scenario price, 0) =Max(32-24,0)=Max(8,0)=8 Put option value at node L=e^(-r*t)*(q*Payoff HL+(1-9) *Payoff LL) =e^(-0.04*0.25)*(0.525125*3.2+(1-0.525125)*12.8) =7.6816 Final value at node L=Max(Payoff L,Option value) =Max(8,7.6816)=8 Step 4 Node LL Scenario price=Spot price*(U)^2=30*0.8^2=19.2 Payoff LL-Max(Strike price-Scenario price,0) =Max(32-19.2,0) =Max(12.8,0) =12.8 A m st in in