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# Difference in GARP vs BT Binomial calculation of "u" and "d"

#### nicholasjalonso

##### New Member
Subscriber
Based on the binomial trees video you provided, we calculate the u and d with volatility as:

However, in the 2019 GARP Practice Exam question #87 when calculation risk neutral probability it is simply 1+volatility an 1-volatility for u and d, respectively:

Is there a reason for the logic differences between this coursework and the GARP exam?

Thank you!

#### Nicole Seaman

Staff member
Subscriber
Based on the binomial trees video you provided, we calculate the u and d with volatility as:
View attachment 2895
However, in the 2019 GARP Practice Exam question #87 when calculation risk neutral probability it is simply 1+volatility an 1-volatility for u and d, respectively:View attachment 2896

Is there a reason for the logic differences between this coursework and the GARP exam?

Thank you!
Hello @nicholasjalonso

Can you please tell me which video you are referring to so I can move your post to the correct thread? If it is a YouTube video that we have posted, please provide me with the link (forum link or YouTube link if we don't have it posted here in the forum). If you are referring to an instructional video in the study planner, then I will need to move the post out of our YouTube section and into our regular Valuation & Risk Models section of the forum. Thank you.

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#### nicholasjalonso

##### New Member
Subscriber
Just want to bump this as new, hoping to gain some clarity as this discrepancies is causing issues in my risk-neutral probability switching between BT and GARP material..thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
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Hi @nicholasjalonso GARP's practice exams, um, they aren't always the best. In the practice exam, GARP is explicitly providing (u) and (d) which is available as an approach but the better (more sophisticated) approach is to match volatility with (u) and (d) as even GARP's new materials state (emphasis mine) in their Chapter 14:
"Before moving on to show how these equations are used in multi-step trees, we will explain how u and d are determined. The parameters u and d should be chosen to reflect the volatility of the stock price. [footnote 3: See Section 3.3 for a discussion of the measurement of volatility.] If we denote the volatility per year by s, then appropriate values for the parameters are

u = exp[σ *sqrt(Δt)]
d = exp[-σ *sqrt(Δt)]

where ∆t is measured in years. The appendix to this chapter explains why these parameters match the volatility."
-- May, B. (2019). 2020 Financial Risk Management Part I: Valuation and Risk Models, 10th Edition. [[VitalSource Bookshelf version]].

If you are given explicitly u = 1.2 and d = 0.8, then of course use them; as such, they implicitly assume normally distributed arithmetic returns. However, the exam should be more likely to ask u and d to be determined by (i.e., matched to) volatility per the text above; this approach implicitly assumes normally distributed geometric returns and this binomial converges on the BSM. Either approach is okay, but matching volatility is better. The question will tell you which approach. I hope that's helpful,

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#### nicholasjalonso

##### New Member
Subscriber
Thank you, David! Will take that approach going forward.