Hi again bluesteel!
These are the answers to you questions
Question 1
First of all, you can find a very good explanation of the two-period
binomial model at the following link.
The Binomial Model for Pricing Options
http://www2.sjsu.edu/faculty/watkins/binomial.htm
The site will walk you through the steps needed to arrive to the
pricing formula in this model. I think the explanation and steps are
clear enough so that I don't need to copy them here, but if you have
any trouble understanding any of them, please don't hesitate to
request clarification.
The formula for the valuation of a call option is thus:
C = [p*Cu + (1-p)*Cd]/(1+r)
where
r is the risk free rate
Cu is the value of the call if price increases
Cd is the value of the call if price decreases
and p = (r-d)/(u-d)
where
u is the possible price increase
d is the possible price decrease
So, we have in your case that
r = 0.04
u = 0.25
d = -0.15
Therefore,
p = (0.04+0.15)/(0.25+0.15) = 0.475
1-p = 0.525
Next we need to know Cu and Cd. The value of a call option at expiration is
Cu = max (Su-E, 0)
Cd = max (Sd-E, 0)
where Su is the price of the underlying if it rises, Sd is the price
of the underlying if it decreases and E is the strike price of the
option ($50 in your case). Since the underlying increases 25% when it
goes up, then
Su - E = 46.25*(1.25) - 50 = 57.8125 - 50 = 7.8125
Therefore, Cu = 7.8125. Now, since it decreases 15% when it falls, then
Sd - E = 46.25*(0.85) - 50 = 37.925 - 50 = -12.075
Therefore, Cd = 0 (there is no reason to exercise the option in this
case, so we let it expire worthless).
Now, we use the first formula to get the call option price:
C = (0.475*7.8215 + 0.525*0)/(1.04)
C = 3.57
So the call option should be valued at $3.57.
Question 1a and 2
I'm answering these two questions together because I'll give you a
link that gives the formula for the put-call parity (which can be used
in this case to calculate the price of a put option, as asked in
question 1a) and also explains why there can't be any arbitrage
opportunities when the put and call options are priced that way
(question 2). The link is
Put-Call Parity
http://www.riskglossary.com/articles/put_call_parity.htm
The link explains clearly (again, please request clarification if
there's anything you don't understand) why a portfolio consisting of
the call option and a cash position equal to the present value of the
strike price, and a portfolio consisting of the put option and the
underlying asset, have exactly the same return and thus should have
the same market value.
So we have that
C + 50/1.04 = P + 46.25
The lefthand side of this equation is value of the first portfolio,
while the right hand side is the value of the second one. We already
know that C=3.57 fromn question 1, so:
P = 3.57 + 50/1.04 - 46.25
P = 5.39
The put option should then be valued at $5.39. This is the only price
at which there are no arbitrage opportunities. If the put option where
priced such that one of the portfolio costs more than the other one, a
risk-free profit could be realized by going long the cheap portfolio
and short the expensive one, because at expiration date they will have
exactly the same value.
Google search terms
two period binomial model
http://www2.sjsu.edu/faculty/watkins/binomial.htm
put call parity
://www.google.com/search?hl=en&lr=&q=put+call+parity
I hope this helps! If you have any doubts regarding my answer, please
don't hesitate to request clarification before rating it; otherwise I
await your rating and final comments.
Best wishes!
elmarto |
Request for Answer Clarification by
bluesteel101-ga
on
06 Mar 2005 14:52 PST
elmarto -
A clarification on question 1. As I understand, the formula included
is one for finding the price of the call option. But where is it
taken into consideration the second period of a two-model?
thanks,
bluesteel
|
Clarification of Answer by
elmarto-ga
on
06 Mar 2005 16:22 PST
Hi bluesteel,
It appears that you're right, I gave the answer using the formula for
a one-period model. I apologize for misunderstanding the question. I
can't redo the answer right now, but rest assured that you will have
it by tomorrow. I hope this does not cause you any inconvenience.
Best regards,
elmarto
|
Clarification of Answer by
elmarto-ga
on
07 Mar 2005 04:59 PST
Hi bluesteel!
Fortunately, this same formula can be used to derive the call option
price when there are two periods instead of one like I thought before.
Here's how to do it.
This is a tree showing what might happen to the underlying's price:
|----- Suu
|--- Su --- |
| |----- Sud
|
S--
|
| |----- Sdu
|--- Sd --- |
|----- Sdd
This means that in the first period, the stock can either go up to Su
or down to Sd. In the next period, if it went up in the first it can
either increase again, going to Suu, or it can decrease, going to Sud.
A similar thing happens if the price goes to Sd in the first period.
Now, calculating the call option price at period 0 is just a matter of
iterating backwards through this tree.
First we find the values of Suu, Sud, Sdu and Sdd. These are simply:
Suu = S*(1+u)*(1+u) = 46.25*1.25*1.25 = 72.26
Sud = Sdu = 46.25*(1.25)*(0.85) = 49.14
Sdd = 46.25*0.85*0.85 = 33.41
Now, we find the value of the call option at each of these four
terminal nodes, which represent expiration date. Recall that
C=max(S-E,0). So
Cuu = max(Suu-E,0) = max(72.26-50,0) = 22.26
Cud = Cdu = 0
Cdd = 0
Using this information, we can now calculate, using the formula I
supplied in my answer, the possible values of the option at period 1.
There are 2 possibilities here, either the stock went up to SU or it
went down to Sd. So me must find the value of the option in each of
these cases:
Cu = [p*Cuu + (1-p)*Cud]/(1+r)
Cd = [p*Cdu + (1-p)*Cdd]/(1+r)
Plugging the values we found (p is always the same), we get:
Cu = 10.16
Cd = 0
Finally, now that we have the possible values for the option at period
1, we can calculate the value at period 0 (today) again using the
formula:
C = [p*Cu + (1-p)*Cd]/(1+r)
C = 4.64
So the correct answer is that the option should be valued at $4.64.
Also, notice that the same process of iterating backwards can be used
for calculating the option price of options expiring in any number of
periods, although the number of calculations required grows quickly
(with three perdios, we need first to find Cuuu, Cuud, Cudu, Cudd,
etc.)
Since these two periods represent two years, question 1a is also
wrong, because I discounted the strike price as if the option lasted
one year. The actual put-call parity equation should be:
C + 50/(1.04)^2 = P + 46.25
So P = 4.61
The put option should be priced at $4.61.
I ope this clarified your doubts. Please do request clarification
again if you need further assistance.
Best wishes!
elmarto
|
Request for Answer Clarification by
bluesteel101-ga
on
07 Mar 2005 08:31 PST
elmarto -
Thanks a bunch. Great additions in the clarficiation, specifically
with the chart and the two-period model.
I have two other questions out there - hopefully they get assigned to
you - you've been a great help.
bluesteel
|
Clarification of Answer by
elmarto-ga
on
07 Mar 2005 10:00 PST
Hi bluesteel,
I'm very happy that you liked the answer and that I've been of some
help to you. I may not have time to address your other questions, but
I'll do my best.
Best wishes,
elmarto
|
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