MFE sample question 46

[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.

[optimizer]

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.

[sooner82]

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.