Here is the data on the stock and call option
CFAI says the value of the call option is $14.60
Using the one period binomial model results in a value of $11.9 instead. Why is this the wrong approach?
The one period binomial method is the correct approach. What’s incorrect is your implementation.
Don’t forget that stock prices are assumed to follow a LOGNORMAL random walk. And the drift term is usually taken to be the risk free rate.
If the initial stock price is S_{0}=100,
you say that the stock prices at the end of the first period are
S_{1}^{(u)}=100\times1.25 and S_{1}^{(d)}=100\times0.85.
Because it is lognormal, there needs to be a drift term, so CFAI says that the stock prices at the end of the first period are
S_{1}^{(u)}=100\times1.25\times1.05 and S_{1}^{(d)}=100\times0.85\times1.05.
The payoff from the option is \max(S_{1}-100,0), and you have to evaluate it along each branch of the tree, and then discount it back to the present time. You have discounted it back at the risk-free rate while CFAI has discounted it at the Lafayette bond rate. I assume that Lafayette is this company.
So you say the option value is
V=\frac{1}{2}\left[\max(S_{1}^{(u)}-100,0)+\max(S_{1}^{(d)}-100,0)\right]/1.05
=\frac{1}{2}\left[\max(100\times1.25-100,0)+\max(100\times0.85-100,0)\right]/1.05=11.90476190
CFAI says the value of the option is
V=\frac{1}{2}\left[\max(S_{1}^{(u)}-100,0)+\max(S_{1}^{(d)}-100,0)\right]/1.07
=\frac{1}{2}\left[\max(100\times1.25\times 1.05-100,0)+\max(100\times0.85\times 1.05-100,0)\right]/1.07
=14.60280374
Thank you so much @guest
May I clarify two points:
- How do we know which is the correct discount rate to use? The risk-free rate or the company’s bond rate?
- How do we know when the drift term (i.e., multiplying by 1.05 in the example) needs to be accounted for?
I ask because when the question previously asked us to value a two-year European call option on Lafayette Corporation, I arrived at the correct answer of 15.59 with my ‘wrong’ method of implementation, i.e.:
- I used the risk-free rate as the discount rate
- I did not multiply by 1.05 to account for the drift
You need to post the ENTIRE question and the ENTIRE answer, as a screenshot or scan, verbatim so we know exactly WHAT was said in the question and EXACTLY what was said in the ANSWER.
There’s something you haven’t told us.
I’ve told you how to get the value of 14.60.
I also found this online which supposedly derives 14.60 by arbitrage free pricing but is wrong:
If you look at the last line of math, they have
(-.625\times 100) which gives -62.5 but instead they replace it with -65.2 with the 2 and 5 interchanged.
If that is where the value of 14.60 came from in the official answer then the official answer is wrong and the entire premise of your original question was wrong (i.e. 14.60 is not the correct answer)
The screenshot I posted was a risk neutral valuation. They got the wrong answer, but the idea is to value the option by constructing a portfolio which has the same value whether the stock goes up or down.
If you have a portfolio consisting of long a call option and short a quantity h of stock S,
portfolio = C - hS
If the stock goes down, you have S=85 and C=0
If the stock goes up, you have S=125 and C=25
the value of the portfolio needs to be the same at time t=1 whether the stock goes up or down
-85h=25-125h (left hand is value when goes down, right is value when goes up)
so 40h=25 or h=0.625. That’s where the value of 0.625 comes from in the screenshot and h is the hedge ratio, because the portfolio is hedged.
Next, you discount that back to the present day, when it has to be equal to C(0)-100 h, which is the value of the portfolio at time 0
C(0)-100h=-85h/1.05
C(0)=100h-85h/1.05=11.9048, which is the value they would have got in the screenshot if they hadn’t swapped the 2 and the 5.
Here you use 1.05 because it’s the risk-free rate at which the investors can borrow,
Again, different from what you got with a binomial tree, but they are different things, one being the risk-neutral value, the other the expected value of the discounted cash flows.
Why is that value (11.9048) important? Because if the option price differs to that, there are arbitrage opportunities you can exploit by constructing a portfolio consisting of the option and an amount h of the stock.
You need to post the entire question and the entire answer.
Dear @guest , my bad, I was trying to minimize information overload for the reader.
Here is the entire section of the vignette that is relevant for this particular question.
And here is CFAI’s full answer:
It does seem indeed that CFAI’s answer has a typo
May I confirm that 11.9 is the correct value of the option based on the info provided by the question? Thanks so much
1 Like
Exactly, 11.90 is correct.