FAQ Exam Will the exam always provide N(d1) and N(d2) or do we need to know how to calculate them? (Are formula sheets provided? Answer: No)

Nicole Seaman

Director of FRM Operations
Staff member
Question: The BSM formula is actually simple except for N(d1) and N(d2) in the case of the call, and N(-d1) and N(-d2) in the case of the put. Will the exam always provide N(d1) and N(d2) or do we need to know how to calculate them?

Answer: That's a good point that BSM is simple except for the standard normal cumulative distribution functions; aka, normal CDFs. After all, the minimum value of the call on a non-dividend paying is given by min[call] = S(0) - K*exp(-rT) which can be understood intuitively. The minimum value is the option's value if the volatility is zero, so we can imagine the addition of the N(.) functions as increasing the option's value to account for non-zero volatility. It is helpful to have a bit of an intuitive understanding of why N(d2) is the probability that a call option will expire in the money (ie, will be exercised) in the risk-neutral world; see http://trtl.bz/merton-model-summary. While it is more difficult to get an intuition of N(d1), at least memorize that N(d1) is the Greek percentage delta of a non-dividend paying stock, and please don't forget that for a dividend-paying stock call option delta is given by exp(-qT)*N(d1). With that context, let's finally answer the question...

As of 2017, authorized exam calculators include the Texas Instruments (TI) BA II Plus and Hewlett Packard (HP) 12C which do not naturally return CDFs. Therefore, you cannot be expected to retrieve N(d1) or N(d2) from scratch. The most likely scenario is that these probability values will be provided as assumptions. For example, here is a snippet from GARP's 2015 Part 1 Practice Exam, Question 11:


Another possible scenario is for the question to require you to calculate d2 or d1 and then retrieve N(d2) or N(d1) by using a provided Z-lookup table. For example, here is the provided Z lookup table in GARP's 2017 Part 2 Practice Exam:

A final possibility--although this seems unlikely--is for the (d2) or (d1) value to conveniently calculate to a recognizable quantile such as -2.33, -1.65, +1.65, +2.33 or even +/- 1.96 because if d2 is conveniently equal to 1.65 then we do not need a calculator to realize that N(1.65) equals about 95.0%.

Finally, please remember that d2 = d1 - σ*sqrt(T). Thanks!

Nicole Seaman

Director of FRM Operations
Staff member
@David Harper CFA FRM A side question, what are the formulas that will be provided during the exam? eg BSM formula is so complex, do we have to memorise? thx

Please note that I moved your post to this thread, which discusses the BSM formula specifically. Also, there are many threads in the forum that have been posted over the years asking about the formulas that will be provided on the exam and the formulas that you need to memorize. If you use the search function and type in "formulas", "formulas on exam", "BSM formula", or anything similar, you will find that many threads come up with a great deal of discussion. I hope that helps!



@David Harper CFA FRM A side question, what are the formulas that will be provided during the exam? eg BSM formula is so complex, do we have to memorise? thx

The only information that is provided for the exam is the Z-table. Don't expect any (full) formula to be provided. But in the spirit of the top post, most formulas make sense once the individual terms are well understood... Given the volume though, it is a good practice to jot them down a couple dozen times in the weeks leading up to the exam to make sure you've committed them to memory.

Suky Dehal

New Member
Hi Guys quick question what formula pages can we expect in the FRM Part I exam, or do we need to memorise a set of key formulas from doing exam papers?

David Harper CFA FRM

David Harper CFA FRM
Staff member
HI @Suky Dehal i moved your question to this FAQ. The exam will not provide formulas; at most, you can expect the normal Z reference/lookup tables.

Also, see GARP FAQ https://www.garp.org/#!/frm/frequently-asked-questions (emphasis below is mine)
"Another consideration for potential candidates is the quantitative aspect of the subject matter. The level of mathematical difficulty of the Exam is consistent with an advanced undergraduate or introductory graduate level finance course at most universities. It should also be noted that while the FRM Exam is conceptual in nature, you will still need to know important formulas and calculations and how to apply them correctly. Formula sheets are not provided with the Exam. For guidance on which formulas to focus on, please refer to the Learning Objectives and look for any statements that include the words ‘calculate,’ ‘compute’ or ‘derive.’ These words will generally indicate an associated formula to commit to memory.

Z-Table Information
A standard table presenting the values for the cumulative distribution function (CDF) of the standard normal distribution (“z-table”) is provided, as is other relevant information pertaining to the CDFs of other probably distributions (F, chi-square, etc.) where necessary and applicable. Memorization of CDF values is therefore not expected. A list of common abbreviations used throughout the Exam is also provided."

I will just add: notice how GARP's 2018 Practice Paper provides a full normal Z reference table, but contains no formulas, and the following "special instructions and definitions:"
Special Instructions and Definitions
1. Unless otherwise indicated, interest rates are assumed to be continuously compounded.
2. Unless otherwise indicated, option contracts are assumed to be on one unit of the underlying asset.
3. VaR = value-at-risk
4. ES = expected shortfall
5. GARCH = generalized auto-regressive conditional heteroskedasticity
6. EWMA = exponentially weighted moving average
7. CAPM = capital asset pricing model
8. LIBOR = London interbank offer rate
9. OIS = overnight indexed swap
10. CDS = credit-default-swap(s)
11. CCP = central counterparty or central clearing counterparty
12. MBS = mortgage-backed-security(securities)
13. CDO = collateralized debt obligation(s)
14. ERM = enterprise risk management
15. RAROC = risk-adjusted return on capital
16. bp(s) = basis point(s)
17. The CEO, CFO, CIO, and CRO are the chief executive, financial, investment, and risk officers, respectively.
18. The following acronyms are used for selected currencies ...
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Active Member
Better you be ready to calculate d1 and d2. However, on GARP Mocks i cannot recall that I saw the question with no N(d1) and N(d2) inputs already given. And BSM formula is not more complex than some other formulas.
Agreed with David and other folks on the availability of formula in the exam. GARP only provided common reference list of the terminology used in question paper for better understanding of the question & to correlate with the concept, but not formulas. Regarding N(d1) & N(d2), we all just need to know d1 formula, b'coz d2 is d1-(denominator of d1).


Active Member
we all just need to know d1 formula, b'coz d2 is d1-(denominator of d1).

In addition, you could solve Put value if asked through Put-Call Parity. Thus you need to know exactly how to calculate Call value and d1 via BSM. And recall how to correctly define N(d) from Z-Table. Only 2 formulas.