Question

The current price of a non-dividend-paying stock is 30. The volatility of the stock is 0.3 per annum. The risk free rate is 0.05 for all maturities. Using the Cox-Ross-Rubinstein binomial tree model with two time steps to do the valuation, what is the value of a European call option with a strike price of 32 that expires in 6 months?

Answer 1

uret dre ovi = 1

u = e^(0.3*(0.25)^(1/2) = 1.1618

d = 1/u = 1/1.1618 = 0.8607

Step 5 Binomial Tree Pricing Step 1 Step 2 Scenario price=Spot price*(U)^2=30*1.1618^2=40.49 Payoff HH=Max(Scenario price-Strike price,0) =Max(40.4933772-32,0) =Max(8.4933772,0) =8.4933772 Call option value at node H=e^(-r*t)*(q*Payoff HH+(1-9) *Payoff H =e^(-0.05*0.25)*(0.504412*8.4933772+(1-0.504412)*o) -4.2309 Risk neutral probability for up move q=(e^(risk free rate*time)-D)/(U-D) 1=(e^(0.05 0.25)-0.8607)/(1.1618-0.8607)=0.50441 Node HH =Max(2.854,4.2309) 4.2309 Node H Step 3 Scenario price-Spot price*(U)*(D)=30*1.1618*0.8607=30 Payoff HL=Max(Scenario price-Strike price, 0) =Max(29.9988378-32,0) Node HL =Max(-2.0011622,0) T=0 Today T=0.25 T=0.5 Node L Step 6 Step 7 Call price = e^(-r*t)*(q*Node H value+(1-9)*Node L value) =e^(-0.05*0.25)*(0.504412*4.2309+(1-0.504412) *0) =2.1076 Call option value at node L=e^(-r*t)*(q*Payoff HL+(1-)*Payoff LL =e^(-0.05*0.25)*(0.504412*0+(1-0.504412) *0) Step 4 Node LL Scenario price=Spot price*(U)^2=30*0.8607^2=22.22 Payoff LL-Max(Scenario price-Strike price, 0) =Max(22.2241347-32,0) =Max(-9.7758653,0) =Max(0,0)=0