Binomial tree

[darshang3]

Hi david

In calculating the size of up move factor U wat does square root of t mean is it time to maturity or the length of the step in the binomial model.

for example to calculate the value of a 6 month american call option using a 2 step binomial model wat will be the t value taken here to calculate U.

THANKS.

 

[David Harper CFA FRM]

darshang - it's the SQRT (delta T): just the length of the step - David

 

[darshang3]

i m still not getting it...
can u tell me wat will be the t value in my above example.

is it SQRT of 2....correct me if m wrong.


thanks.

 

[David Harper CFA FRM]

sorry - read too fast:
re: 6 month american call option using a 2 step binomial model
each step is 3 months and the usage is: SQRT(3/12) because typically the vol input is annual
e.g, if annual volatility = 30%, (volatility connotes annualized standard deviation)
then up (u) = exp(30%*SQRT(3/12)) for the 1st 3-month step
i.e., you can see the "square root rule" at work: SQRT(3/12) scales an annual volatility into a 3 month volatility

David

 

[surojitpalb]

@David Harper CFA FRM @Nicole Seaman ;
In the study notes, term (a) has been calculated. I am not getting what is it. Please explain what is it and how is it calculated.

 

[brian.field]

"a" sometimes refers to alpha = the actual expected return from the asset but I can't see the entire question here. It may be irrelevant.

 

[surojitpalb]

yes @brian.field , it is irrelevant in the question which is why it is confusing

 

[brian.field]

yes @brian.field , it is irrelevant in the question which is why it is confusing

that is true - and that is exactly why it is included in the question. The exam will sometimes include irrelevant info to confuse the test taker....

 

[David Harper CFA FRM]

Hi @surojitpalb Actually (a) is used, but it's not Brian's fault, the notes could be better help here (we are updating this note this semester) In case you want to see it "in action," I pulled out the above example (from the learning XLS) to a single sheet here at https://www.dropbox.com/s/6q5vxl1lmyrqvvv/0110-Hulls-2step-binomial.xlsx?dl=0 (snapshot below). This is Hull's own example.

Here a = exp(Rf - q)*Δt = exp(0.05 - 0.02)*0.25 = 1.007528. Then (a) does inform (p) via p = (a-d)/(u-d) = (1.007528 - 0.9048)/(1.1052-0.9048) = where u = exp[σ*sqrt(Δt)] = exp[0.20*sqrt(0.25)] = exp(and d = 1/u = exp(-σ*sqrt Δt). I hope that clarifies!

 

[brian.field]

ahhh, so a is the forward price (without the S component)

 

[surojitpalb]

I got it, thanks a ton !!! But I am not being able to derive the value of 53.39 above. Am I missing something? @David Harper CFA FRM @brian.field

 

[David Harper CFA FRM]

Hi @surojitpalb It's the expected discounted price based on the two forward nodes:

• Expected future value (at time 0.25) is weighted by p and (1-): (100.66 * 51.26% + 5.06 * 48.74%) = $54.07; this is the "expected" part
• Discounted to current value at risk-free rate = $54.07 *exp(-5%*0.25 years) = $53.39; this is the "discounted" part
• Together $53.39 = (100.66 * 51.26% + 5.06 * 48.74%) / exp(-5%*0.25 years) which is rightly called by Tuckman the expected discounted value.

Each of 100.66, which is called node[1,1] for node[state 1 in time, row 1 up from the bottom at 0], and 5.06 which is called node[1,0] for node[state 1, row 0] are themselves each discounted expected values. But the final option node values (e,g, $189.24) are just intrinsic values at maturity. I hope that helps!

 

[surojitpalb]

how are u getting the values of 100.66 and 5.06?

 

[brian.field]

The figures in blue are the option values - I can't tell if they are American or European but you should be able to simply calculate the option values using the data that is provided. Take a look at the referenced reading as well.

 

[David Harper CFA FRM]

@surojitpalb I trust you noticed that I shared the XLS? To get node[1,0] and node [1,1] is it the same "backward induction;" eg, for node [1,1]

• Expected future value (at time 0.50) is weighted by p and (1-): ($189.3362 * 51.26% + $10.00 * 48.74%) = $101.9276; ie, conditional at node [1,1] the expected future value is probability-weighed average of node[2,1] and node[2,2]
• Discounted to current value at risk-free rate = $101.9276 *exp(-5%*0.25 years) = $100.6614; ie, discounted from t=0.5 to t=0.25
• Together $100.6614 = ($189.3362 * 51.26% + $1.00 * 48.74%)/exp(-5%*0.25 years); i.e., "discounted expected value"

@brian.field re the (a) I just noticed for the first time that Hull calls it "growth factor per step" consistent with your label for it I just happen to be currently revising these binomial exhibits, I think they look better (eg, including some formulas in the exhibit):

 

[surojitpalb]

it did answer my doubts. Thanks @David Harper CFA FRM @brian.field

 

[flawless21]

Small correction above @David Harper CFA FRM - should be "$10 + 48.74%" - you have $1.

I understand from your two examples how you discount to the present, but there is a slide that you have in your lecture video that I can't get your values when using the method above. If you look at the image I attached, to get to 2.0256, I did: (3.2*.652) + (0*0.348) = 2.1056. Now if I discount that: 2.1056 * e(-.25*.12), I get 2.024738. You have 2.0256. Is this a rounding issue? If so, how do I approach these issues on the exam?

 

[ShaktiRathore]

Hi,
p=(exp(.12*.25)-0.90)/(.20)
[3.20*(exp(.12*.25)-0.90)/(.20)+0*(1-(exp(.12*.25)-0.90)/(.20))]*exp(-.12*.25) = 2.0256 (checked this formula in excel its coming exactly 2.0256) yes its due to rounding.
thanks

 

[David Harper CFA FRM]

@omar72787 thanks for the correction mentioned (fixed above). I agree with @ShaktiRathore if i use p = 0.6520 then i get c[1,1] = 2.024738, but the true value of p = p=(exp(.12*.25)-0.90)/(.20). Re: If so, how do I approach these issues on the exam? Over the years, GARP has gotten lots of feedback about this sort of thing: the modern exam should never expose the candidate to such rounding error trade-offs. The choices given should not be near enough for it to matter. I hope that helps!