Binomial Option Pricing Model

Question

This isn't homework. I'm going through sample questions for an exam. They include the answer, but no explanation. I've studied this model, but I don't know how to setup this tree to get any of the vales they are showing.

I'm confused because they say "2-step tree" and "current price is 100". However, using 100 as the start of the tree doesn't get me to any of the answers. Also, if 100 were at the end of the tree, which node would it be?

While the tree (starting with 100 at node-0) is not recombining, it turns out that ud = du = 96. (Where u and d indicate going up or down for a particular step)

I think this question has some flaws to it, but can someone work out what the question was trying to do, and thus get one of the answers?

The correct answer is supposed to be C, 12.49

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I had something like $K*(1.02)^{-step}*(1-0.55) - S*(1.02)^{-step}*0.55 = Put~value~$
(if positive, else 0)

Plugging in 104 and 100 doesn't work. Also, I tried using no discounting on the strike price since there's no dividend, but sill not getting an answer in the list.

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Question No : 1

A 2-step binomial tree is used to value an American put option with strike 104, 
given that the underlying price is currently 100. 
At each step the underlying price can move up by 20% or down by 20% 
and the risk-neutral probability of an up move is 0.55. 
There are no dividends paid on the underlying and the discretely compounded 
risk free interest rate over each time step is 2%. 

What is the value of the option in this model?
A. 11.82
B. 12.33
C. 12.49
D. 12.78

Answer 1

Note that the tree is recombining. You have $u=1.2$ and $d=0.8$ with $ud=0.96$. Your tree for the asset price reads as

• At time zero: 100
• At time one: 80 or 120
• At time two: 64 or 96 or 144

The transition probabilities are $q_u=0.55$ and $q_d=0.45$. For your put option with strike price $K=104$, you thus obtain by backward induction

• At time two: 40 or 8 or 0.
• At time one: 24 or 3.53.
• At time zero: 12.49.

Hence, answer c) is indeed correct.

You firstly build the tree for the asset price. These values are given by $S_0$ at time zero, $S_0u$ and $S_0d$ at time one and by $S_0u^2$, $S_0ud$ and $S_0d^2$ at time two. Then, you start with backward induction, i.e. you build a second tree starting at its end. You firstly compute the option payoff for time point 2 via $\max\{K-S_2,0\}$ for all three cases and work your way backwards through the tree. The final value at time zero is then your current option price.

Since your option is American and may be exercised at any node, you have to compare the immediate payoff with the ``discounted expected value'' (continuation value if you do not exercise the option). So, to compute the value 24 at time step one, you evaluate $$\max\left\{K-80,\frac{1}{1+2\%}\big(q_u\cdot 8 + q_d\cdot 40\big)\right\}\approx\max\{24,21.96\}=24.$$

Note that your discount factor is $\frac{1}{1.02}$ and not $e^{-0.02}$ since you ought to use discrete compounding.