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#1
04-09-2010, 04:15 PM
 Benjamen Member Join Date: Sep 2009 College: Penn State Posts: 44
MFE sample question 46

You are to price options on a futures contract. The movements of the futures price are modeled by a binomial tree. You are given:
(i) Each period is 6 months.
(ii) u/d = 4/3, where u is one plus the rate of gain on the futures price if it goes up, and d is one plus the rate of loss if it goes down.
(iii) The risk-neutral probability of an up move is 1/3.
(iv) The initial futures price is 80.
(v) The continuously compounded risk-free interest rate is 5%.
Let CI be the price of a 1-year 85-strike European call option on the futures contract, and CII be the price of an otherwise identical American call option. Determine CII - CI.

To find u and d, the online solution used the fact that Since u/d = 4/3 =>
u=4d/3,
p* = 1/3 = (1-d)/(u-d) => (1-d)/(4d/3 - d)
solving, u = 1.2 and d = 0.9 and p* = 1/3

My method to find u and d was to use the fact that for Binomial trees for futures: u = exp(sigma*sqrt(h)) , d = exp(-sigma*sqrt(h)) = 1/u
if, d = 1/u and we were given u/d = 4/3
then, 4/3 = u/(1/u) = u^2
therefor, u = sqrt(4/3) = (4/3)^0.5 and d = 1/u = (4/3)^(-0.5)

Obviously, I am doing something wrong in calculating u and d. Someone please enlighten me.
#2
04-09-2010, 04:32 PM
 optimizer Member Join Date: Mar 2009 Studying for VEE & FAP Posts: 352

Quote:
 Originally Posted by Benjamen You are to price options on a futures contract. The movements of the futures price are modeled by a binomial tree. You are given: (i) Each period is 6 months. (ii) u/d = 4/3, where u is one plus the rate of gain on the futures price if it goes up, and d is one plus the rate of loss if it goes down. (iii) The risk-neutral probability of an up move is 1/3. (iv) The initial futures price is 80. (v) The continuously compounded risk-free interest rate is 5%. Let CI be the price of a 1-year 85-strike European call option on the futures contract, and CII be the price of an otherwise identical American call option. Determine CII - CI. To find u and d, the online solution used the fact that Since u/d = 4/3 => u=4d/3, p* = 1/3 = (1-d)/(u-d) => (1-d)/(4d/3 - d) solving, u = 1.2 and d = 0.9 and p* = 1/3 My method to find u and d was to use the fact that for Binomial trees for futures: u = exp(sigma*sqrt(h)) , d = exp(-sigma*sqrt(h)) = 1/u if, d = 1/u and we were given u/d = 4/3 then, 4/3 = u/(1/u) = u^2 therefor, u = sqrt(4/3) = (4/3)^0.5 and d = 1/u = (4/3)^(-0.5) Obviously, I am doing something wrong in calculating u and d. Someone please enlighten me.
The problem does not say the binomial tree is based on forward prices, so you may not use the bolded formulas.
#3
04-09-2010, 04:33 PM
 sooner82 Member Join Date: Jul 2008 Location: Right behind you. Seriously, turn around. Studying for Fap..plus the Modules College: University of Oklahoma Posts: 2,790

According to ASM, you only use volatility to find u and d if it says 'the tree is built using forward rates'. I believe the rule of thumb is if u and d are given, don't use volatility.

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