exercise: call option valuation

[fullofquestions]

Consider a non-dividend paying stock currently priced at $37. It is known with
certainty that over the next two 3-month periods, the price will either rise by 5%
or fall by 5%. The continuously compounded risk free rate is 7%. Calculate the
value of a six-month European call option with a strike price at $38.
a. $1.065
b. $1.234(ANS)
c. $1.856
d. $2.710

S0 = 37,
First 3 month period: Su = 37e(.05/4) = 37.465, Sd = 37e(-.05/4) = 36.540
Second 3 month period: Suu = 37e(.05/2) = 37.937, Sdd = 37e(-.05/2) = 36.086, Sud = Sdu = 37e(.05/4)e(-.05/2) = 37

(37.937 + 37 + 36.086) / 3 = 37.01, PV = 37.01e(-.07/2) = 35.727
Call = S – Ke(-rt) = 38 – 35.727 = 2.27 (we haven't used the standard normal distributions)

Anyway, this is incorrect. Could someone give me a hint?

 

[cash king]

I think we should figure out the probability of upward-move and downward-move first (risk neutral probabability) of the binomial tree, then we can discount the expected payoff (based on that probabability) using the risk-free rate and derive the option price.

 

[Roshan Ramdas]

Consider a non-dividend paying stock currently priced at $37. It is known with
certainty that over the next two 3-month periods, the price will either rise by 5%
or fall by 5%. The continuously compounded risk free rate is 7%. Calculate the
value of a six-month European call option with a strike price at $38.
a. $1.065
b. $1.234(ANS)
c. $1.856
d. $2.710

S0 = 37,
First 3 month period: Su = 37e(.05/4) = 37.465, Sd = 37e(-.05/4) = 36.540
Second 3 month period: Suu = 37e(.05/2) = 37.937, Sdd = 37e(-.05/2) = 36.086, Sud = Sdu = 37e(.05/4)e(-.05/2) = 37

(37.937 + 37 + 36.086) / 3 = 37.01, PV = 37.01e(-.07/2) = 35.727
Call = S – Ke(-rt) = 38 – 35.727 = 2.27 (we haven't used the standard normal distributions)

Anyway, this is incorrect. Could someone give me a hint?

Hello,
I've worked out the problem in the attached file.
Hope this helps.

 

[David Harper CFA FRM]

I got the same exact answer as @Roshan Ramdas (PV = $1.234), attached, but I would like to acknowledge that I was not 100% sure how to treat the assumption "It is known with
certainty that over the next two 3-month periods, the price will either rise by 5% or fall by 5%." I think it's instructive as an imprecise question. I think it's fair to interpret this as u^2 = 1.05, such that u = sqrt(1.05); i.e., 5% over the six-month period. Also understandable is the temptation to assume continuous compounding per u = exp(5%) or, consistent with the previous point, even u^2 = exp(5%) such that u = sqrt[exp(5%)]. I don't think your u = exp(.05/4) is a good candidate simply because it does not say "the price will either rise by 5% or fall by 5% per annum;" i.e., yours would be natural if the question included "per annum."

The weakness of the question is that if our $1.234 is correct, then in my opinion the question should read something like: "Consider a non-dividend paying stock currently priced at $37. It is known with certainty that over the next two 3-month periods, the price will either rise by 5% or fall by 5% in each of the periods."

 

[David Harper CFA FRM]

I was curious how Hull handles it and I found Hull question 12.5 (emphasis mine) "A stock price is currently $100. Over each of the next two 6-month periods it is expected to go up by 10% or down by 10% ..." and he treats in the solution, as you'd maybe expect (just like we did above), with u = 1.10 and d = 0.90. So, his "up by 10%" translates in a simple return over the one-step period. That's interesting because it is a simple (arithmetic) return, not a continuous (lognormal) without explicitly saying so ... it's sort of an understandable "exception" to the rule which states that rate assumptions should generally be per annum (a simple return is not per annum, it is only per annum is the special case of an effective annual return).

Volatility as an determinant of (u) and (d) in the binomial doesn't suffer this issue because it's always a per annum input. So, i totally get why you used exp(0.05/4).

A good question can never be too precise in its assumptions!