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The Actuary’s Free Study Guide for Exam 3F / Exam MFE – Second Edition – G. Stolyarov IIA point system helps one establish a common denominator by which to measure one's efforts attasks that can often vary considerably. Just like money provides a convenient way to measureeconomic value while dealing with millions of goods, so does a point system allow one tocompare disparate studying approaches and have some rough measure of accomplishment on aparticular day. Like money, any point system is also highly imperfect at measuring the desiredobjectives - effort and learning. Develop a point system that you consider to be the most accurateand the most relevant to the approaches that work best for you.The incentives provided by my point system have shown to be effective in my own experience.In preparing for Exam 1/P, I maintained an average of 114.95 points per day over 98 days. Inpreparing for Exam 2/FM, I maintained an average of 128.24 points per day over 105 days. I amdoing even better thus far in preparing for Exam 3F/MFE; I have a current average of 143.35over 79 days. My studying for the prior two exams seems to have been sufficient; I passed bothon the first try, with scores of 10/10 and 9/10, respectively. And I still have detailed records ofthe studying I did on every day that I prepared for each exam.3. Do as many practice problems as possible. This includes writing your own! The way totruly learn each concept relevant for the exam is to immerse oneself in it and repeatedly exposeoneself to ways in which it might be applied. For me personally, a mathematical formula or ideadoes not stay in my mind if I just read about it. If I write down the relevant formulas, mymemory functions somewhat better, but it is in the use of the formulas that I truly memorizethem; after I have used a formula five or six times, I no longer have any difficulty recalling it.I am a highly frugal individual and pride myself on only spending money when it is absolutelynecessary for the improvement of my well-being. When I began to study for this exam, I noticedthat there was a dearth of freely available practice problems for it - problems that would helpstudents to make sense of the theoretical concepts described in McDonald's Derivatives Marketsand to see how those concepts apply in situations relevant to them - i.e., in exam situations.Granted, some admirable efforts have been made by parties like Dr. Ewa Kubicka, Dr. SamBroverman, Dr. Bill Cross, Dr. Abraham Weishaus, and Actuarial Brew to offer free practiceproblems - but even those combined were not enough to enable students to learn the entiresyllabus from them alone. This is why I wrote The Actuary's Free Study Guide for Exam 3F /Exam MFE, which, in May 2008, accounted for 418 of the 625 free Exam 3F/MFE practiceproblems with free available solutions of which I was aware.I encourage you to write any problems you think might be helpful in teaching actuarial studentsimportant exam-related concepts or skills. It is even possible to earn some small passive incomefrom these problems if you render them available online on sites like Associated Content, whichwill pay you 1.50 per 1000 page views to your articles. While this is not a living, it might earnyou more than the interest in your bank account and may even compensate for a significantfraction of your exam fees. If you choose to publish your practice problems via AssociatedContent, submit them under the "Display-Only" terms, which will enable you to edit yourmaterial, albeit with substantial time delays of about a week for the edits to get implemented. Asis virtually inevitable with problems of this level of difficulty, any author will sometimes makesubstantial errors or oversights - and indeed, virtually all textbooks and practice manuals for thisexam are accompanied by substantial sections of errata. The advantage of publishing problems5

The Actuary’s Free Study Guide for Exam 3F / Exam MFE – Second Edition – G. Stolyarov IIonline is that user feedback regarding any errors is extremely fast and contributes to anultimately better product. Freely available online practice problems and solutions can be assuredto have high quality in a manner similar to open-source software, which also gets continuallyimproved on the basis of user feedback.Exam-Specific Approaches4. Assure yourself some points by mastering the easier material. Exam 3F/MFE is comprisedof material of highly varying difficulty and open-endedness. Fortunately, several major topics onthe syllabus - mostly represented in the earlier chapters of McDonald's Derivatives Markets - arequite systematic and straightforward in their application. Answering questions regarding themproperly only requires you to thoroughly memorize a few formulas and problem-solvingtechniques. The easier topics on this exam include put-call parity, binomial option pricing,and the Black-Scholes formula. Questions related to these three topics combined have in thepast accounted for over 50% of both the Casualty Actuarial Society's and the Society ofActuaries' exams - and there is no reason why you should get any of these questions wrong.5. Learn as much as possible about the intermediate-level material. Approximately another25% of the exam should be composed of material pertaining to delta-hedging and exoticoptions. All of this material is also manageable, although the variety of problems that could beasked is somewhat greater.Much of your success at hedging questions will be assured if you memorize the formula for thedelta-gamma-theta approximation: C new C old єΔ (1/2)є2Γ hθ. The delta-gammaapproximation is just the delta-gamma-theta approximation without the hθ term. You should alsobe able to answer questions regarding what it takes to delta- and gamma-neutralize a portfolio.In those questions, you will be given delta and gamma values for different kinds of options.Always work with gamma-neutralization first, because once you have rendered a portfolio'sgamma zero, you can always compensate for the leftover delta by adding or subtracting shares ofstock (as each stock has a delta of 1). The mathematics behind these problems is simplearithmetic, and again, there is no reason to get any of them wrong. Sometimes, you might begiven a delta in disguise; remember that Δ e-δTN(d 1 ). If you are given δ, T, and N(d 1 ), you canfind Δ using this formula.Also remember the formulas for option elasticity (Ω SΔ/C) and option volatility (σ option σ stock * Ω ). These are easy to memorize and apply.Less likely but quite possible are questions regarding the return and variance of the return to adelta-hedged market-maker. To be able to answer those questions, memorize the formulasR h,i (1/2)S2σ2Γ(x i 2-1)h for the return and Var(R h,i ) (1/2)(S2σ2Γh)2 for the variance of thereturn. Also remember that you can model the stock price movement in any time period h asσS t (h). You might be asked to determine said stock price movement, given h, S t , and somefunction of σ - probably N(d 1 ) - from which you will be able to figure out σ.6

The Actuary’s Free Study Guide for Exam 3F / Exam MFE – Second Edition – G. Stolyarov IIIt also would not hurt to memorize the Black-Scholes partial differential equation: rC(S t ) (1/2)σ2S t 2Γ t rS t t θ.For exotic options, each type of option is not difficult to understand conceptually, but there are alot of types to keep in mind. For Asian options, make sure you remember that they are pathdependent and that you know how to take a geometric average. Pay attention to the differencebetween geometric average price options and geometric average strike options. For barrieroptions, remember the parity relationship Knock-in option Knock-out option Ordinaryoption. Know about rebate options as well; the concept is not difficult. For compound options,remember the parity relationship CallOnCall - PutOnCall xe-rt 1 BSCall. Gap options arepriced just like ordinary options via the Black-Scholes formula, with the exception that thetrigger price rather than the strike price is used in the formula for d 1 , while the strike price is stillused in the formula for the overall option price. For pricing exchange options, the Black-Scholesformula holds as well, with two significant differences. Sigma in the formula for exchangeoptions is σ [σ S 2 σ K 2 - 2ρσ S σ K ]; you will be given the individual volatilities of theunderlying asset and the strike asset, as well as their correlation ρ. Also, S is your underlyingasset price, K is your strike asset price, and you will be given the dividend yields δ S and δ Kpertaining to these assets. Then your Black-Scholes exchange call price will beC Se-(δ S)TN(d 1 ) - Ke-(δ K)TN(d 2 ).6. Memorize shortcut approaches to Brownian motion and interest rate models. The mostdifficult - and most open-ended - topics on the exam will be based on Chapters 20 and 24 ofMcDonald's Derivative Markets - dealing respectively with Brownian motion and interest ratemodels. The best way to approach these topics is not from the vantage point of theory orderivation from first principles. Rather, you will save yourself a lot of time and stress bymemorizing the general form of results obtained by doing specific kinds of problems.You do need to know what arithmetic, geometric, and mean-reversion (Ornstein-Uhlenbeck)Brownian processes look like. You should also be familiar with terminology pertaining to them,including drift, volatility, and martingale (a martingale is a process for which E[Z(t s) Z(t)] Z(t)).Ito's Lemma can be easily memorized in the formdC(S, t) C S dS (1/2)C SS (dS)2 C t dt, which is applicable to geometric Brownian motion. It isdoubtful that you will be asked to apply Ito's Lemma to non-geometric kinds of Brownianmotion. When you apply Ito's Lemma, remember the multiplication rules, which state that (dZ)2 dt, and any other product of multiple dZ and dt is 0. If you have two correlated Brownianmotions Z and Z', then dZ * dZ' ρ.If you memorize some results derived using Ito's Lemma, then you will not have to go throughthe derivation on the test. For example, if dX(t)/X(t) αdt σdZ(t) (i.e., X follows a geometricBrownian motion, then d[ln(X)] (α - 0.5σ2)dt σdZ(t) (i.e., ln(X) follows an arithmeticBrownian motion with the exact same volatility.) Just memorizing the latter formula can makeseveral currently known prior exam questions extremely easy to solve.7

The Actuary’s Free Study Guide for Exam 3F / Exam MFE – Second Edition – G. Stolyarov IIDo not forget about the formula for the Sharpe ratio (φ (α - r)/σ) and the fact that the Sharperatios of two perfectly correlated Brownian motions are equal.To figure out whether a particular Brownian motion has zero drift (or what kind of drift factor ithas), you will need to apply Ito's Lemma to the given Brownian motion and see what theresultant dt term is.Also recall that the quadratic variation of a Brownian process is expressible asn2n lim i 1 Σ(Z[ih] - Z[(i -1)h]) T from time 0 to time T. Here, Z is the Standard BrownianMotion. In X is some other Brownian motion with volatility factor σ, thenn lim i 1Σ(Y[ih] - Y[(i -1)h])2 σ2T.nOther useful facts to know are as follows.Var[σdZ(t) Z(t)] σ2Var[dZ(t) Z(t)] σ2dt.For any arithmetic Brownian motion X(t), the random variable [X(t h) - X(t)] is normallydistributed for all t 0, h 0, and has a mean of X(t) αh and a variance of σ2h.The Black-Scholes option pricing framework is based on the assumption that the underlyingasset follows a geometric Brownian motion:dS(t)/S(t) αdt σdZ(t).When X(t) follows an arithmetic Brownian motion, the following equation holds:X(t) X(a) α(t - a) σ (t-a)ξ.When X(t) follows a geometric Brownian motion, the following equation holds:X(t) X(a)exp[(α - 0.5σ2)(t - a) σ (t-a)ξ].If given either of those two equations, you should be able to recognize arithmetic and geometricBrownian motion.Be sure to know how to value claims on power derivatives (of the formS(T)a). This is a newlyadded topic and is thus likely to be tested on at least one question. Memorize the followingformulas.d(Sa)/Sa (a(α - δ) 0.5a(a-1)σ2)dt aσdZ(t)γ a(α - r) r, where r is the annual continuously compounded risk-free interest rate.δ* r - a(r - δ) - 0.5a(a-1)σ2F 0,T [S(T)a] S(0)aexp((a(r - δ) 0.5a(a-1)σ2)T)FP 0,T [S(T)a] e-rTS(0)aexp((a(r - δ) 0.5a(a-1)σ2)T)8

The Actuary’s Free Study Guide for Exam 3F / Exam MFE – Second Edition – G. Stolyarov IIWhile many actuarial students might think that Brownian motion is the most difficult topic onthe exam, I believe that the interest-rate models covered in Chapter 24 are in fact harder,because a virtually endless variety of practically impossible questions can be asked regardingthem. I hope that the SOA and CAS will be reasonable in what they choose to test. For instance,asking students to use the explicit bond-price formulas for the Vasicek and Cox-Ingersoll-Ross(CIR) interest rate models would be uncalled for, as these formulas take tremendous effort tomemorize. The past exam questions I have seen have not asked for these formulas. Instead, theytended to involve various "auxiliary" formulas for these models.Of course, it is essential to know the Brownian motions associated with the Vasicek and CIRmodels. For the Vasicek model, dr a(b - r)dt σdZ. For the CIR model, dr a(b - r)dt σ (r)dZ. Note that the only difference is a (r) factor in the dZ term. This has importantimplications, however, as discussed in Section 72 of my study guide. Make sure you read thissection to find out about the essential similarities and differences between these models as wellas why the time-zero yield curve for either of these models cannot be exogenously prescribed (alearning objective on the exam syllabus).For both the Vasicek model, the following "auxiliary" equations are important to know, asquestions involving them have appeared on prior exams.P[t, T, r(t)] exp(-[α(T - t) β(T - t)r]), where α(T - t) and β(T - t) are constants that stay thesame whenever the difference between T and t is the same, even if the values of T and t aredifferent.Now say you have a Vasicek model wheredP[r(t), t, T]/P[r(t), t, T] α[r(t), t, T]dt q[r(t), t, T]dZ(t). You are given some α(r 1 , t 1 , T 1 ) andare asked to find α(r 2 , t 2 , T 2 ). All you need to do is to solve the following equation:[α(r 1 , t 1 , T 1 ) - r 1 ]/(1 - exp[-a(T 1 - t 1 )]) [α(r 2 , t 2 , T 2 ) - r 2 ]/(1 - exp[-a(T 2 - t 2 )])The derivation of this formula is quite involved and is provided in Section 71. You would bewell advised to simply memorize the end result (and read the derivation, if you are interested inthe reasoning, but do not repeat the derivation each time unless you enjoy pain and suffering).(It would not hurt to also memorize that q[r(t), t, T] -σB(t, T) in the case above. A questionusing this formula is not likely to be asked, and if it is, I hope that σ and B(t, T) will be given.)The Black-Derman-Toy (BDT) interest rate model has easy and difficult applications. The easyapplications involve pricing caplets and interest rate caps, provided that you have had practicewith the BDT discounting procedure. Remember that the risk-neutral probability for the BDTmodel is always 0.5 and the interest rates at the nodes in any given time period differ by aconstant multiple of exp[2σ i (h)]. To discount the expected value of a payment at any node,divide it by 1 plus the interest rate at that node. Remember that, for interest rate caps, you willneed to account for payments made each time period when the interest rate exceeds the cap strikerate.9

The Actuary’s Free Study Guide for Exam 3F / Exam MFE – Second Edition – G. Stolyarov IIThe difficult parts of using the BDT model involve finding yield volatilities and actuallyconstructing BDT binomial trees. I am aware of a scant few exam-style questions on the formerand of none on the latter - which I hope will be indicative of the composition of future MFEexams. There is no way around memorizing the formula for yield volatility in the BDT model:Yield volatility 0.5ln(y[h, T, r u ]/y[h, T, r d ]) 0.5ln([P[h, T, r u ]-1/(T-h) - 1]/[P[h, T, r d ]-1/(T-h) - 1])You will likely be given a multi-period BDT tree and asked to calculate the volatility in timeperiod 1 of the bond used in constructing the tree. I am hard-pressed to see how the formulaabove might be used in finding the volatilities for time periods greater than 1, since the formularelies on the presence of two and only two nodes in the binomial tree.For constructing binomial trees, I hope that, if a question is asked, it will involve constructing atree for a one-period model. To do this, learn the following formulas:If P h is the price of an h-year bond, where h is one time period in the BDT model, thenP h 1/(1 r 0 ) and thus r 0 1/P h - 1. If P 2h is the price of an 2h-year bond, then R h r d and σ hmeet the following conditions: r 0 σ h and P 2h 0.5P h (1/(1 R h exp[2r 0 ]) 1/(1 R h )).Anything beyond this is intense algebraic busy work, and I hope that the SOA and CAS are kindenough not to embroil students in it.Also remember to review the Black formula for pricing options on futures contracts (Section 37)and caplets (Section 75).7. Do not forget the "miscellaneous" topics! I can think of two "miscellaneous" topics thatappear seldom or not at all in McDonald's text but are highly likely to be tested. One of these iscomputing historical volatility. Fortunately, this is a straightforward and systematic procedure,and you should have no difficulty with it after working through Section 81. The other topic,equity-linked insurance contracts, can involve applications of virtually anything else on thesyllabus. Nonetheless, you are likely to be asked to use one or both of the following formulas todiscover option payoffs in disguise:Formula 80.1: max(AB, C) B*max(A, C/B)Formula 80.2: max(A, B) A max(0, B - A)Section 80 gives some equity-linked insurance problems that you might find beneficial.8. Take practice tests under exam conditions several days before the actual exam. Takingpractice exams will give you an idea of your current knowledge level as well as areas to whichyou might need to devote additional attention. If you have studied well and you do well on thepractice exams, this will give you a lot of confidence coming into the actual test. I expect thepassing threshold for future sessions of Exam 3F/MFE to be around 13 or 14 out of 20, since theFall 2007 exam - a particularly difficult test - had a passing threshold of 12. You can compilemany practice exams from the exam-style questions offered in various sections of my studyguide.10

The Actuary’s Free Study Guide for Exam 3F / Exam MFE – Second Edition – G. Stolyarov IIThis is as much essential studying advice as immediately comes to mind. Remember, however,that the more you know, the better your chances are on this exam, and that the information I haveprovided here does not substitute for comprehensive months-long preparation. Use all theresources at your disposal and learn everything you can. I hope that the free resources providedin this study guide will aid you in passing the exam and achieving a high mark.11

The Actuary’s Free Study Guide for Exam 3F / Exam MFE – Second Edition – G. Stolyarov IISection 1Put-Call ParityPut-call parity for European options with the same strike price and time to expiration isCall - put present value of (forward price - strike price)Equation for put-call parity:C(K, T) - P(K, T) PV 0,T (F 0,T - K) e-rT(F 0,T - K)Meaning of variables:K strike price of the optionsT time to expiration of the optionsC(K, T) price of a European call with strike price K and time to expiration T.P(K, T) price of a European put with strike price K and time to expiration T.F 0,T forward price for the underlying asset.PV 0,T the present value over the life of the options.e-rT*F 0,T prepaid forward price for the asset.e-rT*K prepaid forward price for the strike.r the continuously compounded interest rate.Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 9,p. 282.Original Practice Problems and Solutions from the Actuary's Free Study Guide:Problem PCP1. The European call option on Asset Q that expires in one year has strike price 32and option price 4.

Study Methods for Actuarial Exam 3F / Exam MFE . To accompany . The Actuary's Free Study Guide for Exam 3F / Exam MFE, I offer a list of general studying methods, techniques, and insights that have guided my own preparation for the financial economics half of the third actua