Question

Using a binomial tree, what is the price of a $40 strike 6-month American put option, using 3-month intervals as the time period? Assume the following data: S=$37.90, r=5.0%, 5=35%, =0.

Answer 1

u = e^(std dev*(t)^(1/2))

=e^(0.35*(0.25)^(1/2))

=1.1912

d = 1/u

=1/1.1912

=0.8395

Step 5 Scenario price=Spot price*(U)=37.9*1.1912=45.14648 Payoff H=Max(Strike price-Scenario price, 0) =Max(40-45.14648,0)=Max(-5.14648,0)=0 Put option value at node H=e^(-r*t)*(q*Payoff HH+(1-9) *Payoff HL) =e^(-0.05*0.25)*(0.49212*0+(1-0.49212)*2.09953004) =1.0531 Final value at node H=Max(Payoff H, Option value) =Max(0,1.0531)=1.0531 Step 2 Scenario price=Spot price*(U)^2=37.9 1.1912^2=53.7 Payoff HH=Max(Strike price-Scenario price, 0) =Max(40-53.778486976,0) =Max(-13.778486976,0) Binomial Tree Pricing Step 1 High price=Current price*up move=37.9*1.1912=45.14648 Low price=Current price*down move=37.9*0.8395=31.81705 Risk neutral probability for up move q=(e^(risk free rate*time)-D)/(U-D) =(e^(0.05*0.25)-0.8395)/(1.1912-0.8395)=0.49212 Node HH Node H Step 3 Scenario price=Spot price*(U)*(D)=37.9*1.1912*0.8395=37.9 Payoff HL=Max(Strike price-Scenario price,0) =Max(40-37.90046996,0) Node HL -Max(2.09953004,0) =2.09953004 T=0 Today T=0.25 T=0.5 Step 7 Put price = e(-r*t)*(q*Node H value+(1-4) "Node L value) =e^(-0.05*0.25)*(0.49212*1.0531+(1-0.49212)*8.18295) =4.6161 Node L Step 6 Scenario price=Spot price*(D)=37.9*0.8395=31.81705 Payoff L=Max(Strike price-Scenario price,0) =Max(40-31.81705,0)=Max(8.18295,0)=8.18295 Put option value at node L=e^(-r*t)*(q*Payoff HL+(1-9) *Payoff LL) =e^(-0.05*0.25)*(0.49212*2.09953004+(1-0.49212) *13.289586525) =7.6861 Final value at node L=Max(Payoff L,Option value) =Max(8.18295,7.6861)=8.18295 Step 4 Node LL Scenario price=Spot price*(U)^2=37.9*0.8395^2=26.7 Payoff LL-Max(Strike price-Scenario price,0) =Max(40-26.710413475,0) =Max(13.289586525,0) =13.289586525