Problem on binomial tree

[indra]

Hi, I came across the following problem in Hull.

Q. Calculate the price of a three-month European call option on the spot value of silver. The three-month futures price is $12, the strike price is $13, the risk-free rate is 4% and the volatility of the price of silver is 25%.

This is my approach at solving it.

u = exp(0.25*sqrt(3/12)) = 1.133148 d= 1/u i.e 1/1.133148 = 0.882497

risk neutral factor = (exp (0.04*3/12) - d) / (u - d) = 0.508887

So there is a 50.8887% chance of the futures price being $12 * u = 13.59778 and a 49.1113% chance of the futures price being $12*d = 10.58996

If futures price turns out to be $13.59778, call payoff = $13.59778 - $13 (strike price) = 0.59778
If futures price turns out to be $10.58996, call payoff = 0

So call price = (0.508887 * 0.59778) + (0.491113 * 0) = 0.304203
Present call price = 0.304203 * exp (-0.04*3/12) = 0.301176

However asper the solution handbook the answer is 0.244. Please let me know where have I gone wrong. Thanks!!

 

[ShaktiRathore]

You have done the error :So there is a 50.8887% chance of the futures price being $12 * u = 13.59778 and a 49.1113% chance of the futures price being $12*d = 10.58996
You should have taken the spot price of the silver instead of its future price, we are calculating three-month European call option on the spot value of silver not the future price. so that SPot price=12*Exp(-0.04*0.25)=11.90 ,So there is a 50.8887% chance of the spot price $11.90* u = 11.90*1.1331=13.48389 and a 49.1113% chance of the spot price being $11.90*d= 11.90*0.8825=10.50175
If spot price turns out to be $13.48389 , call payoff = $13.48389 - $13 (strike price) = .48389
If spot price turns out to be $10.50175, call payoff = 0
So call price = (0.508887 * .48389) + (0.491113 * 0) = 0.24624533043
Present call price = 0.24624533043 * exp (-0.04*3/12) = 0.244
my solution:
upJump=u=EXP(sigma*SQRT(Δt))=u=EXP(0.25*SQRT(0.25))=1.1331
downJump=1/u=0.8825
probabilityofUpjump=p=(EXP((r-D)*Δt)-d)/(u-d)=(EXP((0.04-0)*0.25)-0.8825)/(1.1331-0.8825)=0.5090
InitialStockPrice=S=11.8806=11.90
payoff if upjump=max(11.90*1.1331-13,0)=0.48389
payoff if downjump=max(11.90*0.8825-13,0)=0
value=0.48389*Exp(-0.04*0.25)*0.5090=0.244
thanks

 

[indra]

Thanks for the clarification Shakti. Much appreciated.