# Options pricing exercise - American call option on a futures contract

I am confused by a particular exercise I am doing right now, I am hopeful that someone can walk me through as to how to solve it. I further hope the question is not considered too basic for this forum.

Build a 15-period binomial model whose parameters should be calibrated to a Black-Scholes geometric Brownian motion model with: T=.25 years, S0=100, r=2%, σ=30% and a dividend yield of c=1%. Compute the fair value of an American call option with strike K=110 and maturity n=10 periods where the option is written on a futures contract that expires after 15 periods. The futures contract is on the same underlying security as described in the previous questions. What is the earliest time period in which you might want to exercise the American?

So what I would normally do to compute the value is to build a lattice and then work backwards in order to see what the value is. Yet the part with "where the option is written on a futures contract that expires after 15 periods", leaves me awfully confused as to what I should be doing as well as what implications that has for the exercise.

Thank you for any feedback!

[at this point I found the solution, big thanks to all the contributors, I deleted the lattices in order not to misguide anyone as they clearly did produce the wrong result]

• The relationship between the Spot and the Futures price is F = S exp(r-q) T. Oct 16 '15 at 18:03
• This is not answer. But I am not allowed to comment since I don't have 50 reputation points. Noir, would you mind sharing how you solved this? Jun 2 '16 at 6:44
• @DaleAngus could you tell me the chapter and the number of exercise. I will try to find it in my notes and tell you the individual steps (Its some time since I was doing the course).
– Noir
Jun 7 '16 at 7:29

You can build the binomial tree for the stock. After ten periods, the option expires and you enter in the future contract at a certain future price (noob2 gave a big help on this): you compute the future price at period 10 and then you work backwards for the option valuation.

• Thanks I will commit to the problem today and give feedback.
– Noir
Oct 17 '15 at 10:15
• Please see the update I made (Off topic: I am new to stack exchange, do you get automatically notified if I make an edit to the initial question?)
– Noir
Oct 17 '15 at 13:31
• At this point I can confirm that I obtained the correct result I had to change the way through which I calculated the futures lattice. The only thing I am yet to find out why it is exactly the period 7 at which it would make sense to early exercise the option.
– Noir
Oct 18 '15 at 9:32
• What about the price of an american put option with the same characteristics? which is the best time to exercise it? tkx
– user18105
Oct 31 '15 at 14:23

Noir : I'm studying the same coursera course as you and wonder why you used a multiplier "q" = 0.7483 in calculating the futures lattice eg first row of second column from left 175.42 = 0.7483 * 178.77 + (1-0.7483) * 165.45 and not the calculated value of q equal to 0.4925.

• Hi NDC, welcome to quant stack exchange. This would be better as a comment to the original post rather than an answer as it doesn't answer the question. Nov 10 '15 at 15:37
• @NDC the lattices which I posted are wrong (they don't produce the correct result as it is noted on the bottom). As such 175.42 is also incorrect and q of 0.4925 is correct. I will delete the lattices as they are misguiding.
– Noir
Nov 11 '15 at 10:57
• Hi NDC. Could you please share the Coursera course ID for this? Seems like an interesting course.
– AK.
Dec 5 '15 at 4:21
• It's a two part course from Columbia University. Both are well done. www.coursera.org/learn/financial-engineering-1 and www.coursera.org/learn/financial-engineering-2
– NDC
Dec 6 '15 at 9:52