Exam MFE study method extract<br/><br/>1. Assure yourself some points by mastering the easier material. Exam 3F/MFE is comprised of material of highly varying difficulty and open-endedness. Fortunately, several major topics on the syllabus - mostly represented in the earlier chapters of McDonald's Derivatives Markets - are quite systematic and straightforward in their application. Answering questions regarding them properly only requires you to thoroughly memorize a few formulas and problem-solving techniques. 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 the past accounted for over 50% of both the Casualty Actuarial Society's and the Society of Actuaries' exams - and there is no reason why you should get any of these questions wrong. <br/><br/>2. Learn as much as possible about the intermediate-level material. Approximately another 25% of the exam should be composed of material pertaining to delta-hedging and exotic options. All of this material is also manageable, although the variety of problems that could be asked is somewhat greater.<br/>Much of your success at hedging questions will be assured if you memorize the formula for the delta-gamma-theta approximation: Cnew = Cold + єΔ + (1/2)є2Γ + hθ. The delta-gamma approximation is just the delta-gamma-theta approximation without the hθ term. You should also be 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's gamma zero, you can always compensate for the leftover delta by adding or subtracting shares of stock (as each stock has a delta of 1). The mathematics behind these problems is simple arithmetic, and again, there is no reason to get any of them wrong. Sometimes, you might be given a delta in disguise; remember that Δ = e-δTN(d1). If you are given δ, T, and N(d1), you can find Δ using this formula. <br/><br/>Also remember the formulas for option elasticity (Ω = SΔ/C) and option volatility (σoption = σstock*│Ω│). These are easy to memorize and apply. <br/><br/>Less likely but quite possible are questions regarding the return and variance of the return to a delta-hedged market-maker. To be able to answer those questions, memorize the formulas <br/><br/>Rh,i = (1/2)S2σ2Γ(xi2-1)h for the return and Var(Rh,i) = (1/2)(S2σ2Γh)2 for the variance of the return. Also remember that you can model the stock price movement in any time period h as σSt√(h). You might be asked to determine said stock price movement, given h, St, and some function of σ - probably N(d1) - from which you will be able to figure out σ. <br/><br/>It also would not hurt to memorize the Black-Scholes partial differential equation: rC(St) = (1/2)σ2St2Γt + rSt∆t + θ.<br/>For exotic options, each type of option is not difficult to understand conceptually, but there are a lot of types to keep in mind. For Asian options, make sure you remember that they are path-dependent and that you know how to take a geometric average. Pay attention to the difference between geometric average price options and geometric average strike options. For barrier options, remember the parity relationship Knock-in option + Knock-out option = Ordinary option. 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 are priced just like ordinary options via the Black-Scholes formula, with the exception that the trigger price rather than the strike price is used in the formula for d1, while the strike price is still used in the formula for the overall option price. For pricing exchange options, the Black-Scholes formula holds as well, with two significant differences. Sigma in the formula for exchange options is σ = √[σS2 + σK2 - 2ρσSσK]; you will be given the individual volatilities of the underlying asset and the strike asset, as well as their correlation ρ. Also, S is your underlying asset price, K is your strike asset price, and you will be given the dividend yields δS and δK pertaining to these assets. Then your Black-Scholes exchange call price will be <br/><br/>C= Se-(δ_S)TN(d1) - Ke-(δ_K)TN(d2). <br/><br/>3. Memorize shortcut approaches to Brownian motion and interest rate models. The most difficult - and most open-ended - topics on the exam will be based on Chapters 20 and 24 of McDonald's Derivative Markets - dealing respectively with Brownian motion and interest rate models. The best way to approach these topics is not from the vantage point of theory or derivation from first principles. Rather, you will save yourself a lot of time and stress by memorizing the general form of results obtained by doing specific kinds of problems. <br/><br/>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)). <br/><br/>Ito's Lemma can be easily memorized in the form <br/><br/>dC(S, t) = CSdS + (1/2)CSS(dS)2 + Ctdt, which is applicable to geometric Brownian motion. It is doubtful that you will be asked to apply Ito's Lemma to non-geometric kinds of Brownian motion. 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 Brownian motions Z and Z', then dZ * dZ' = ρ. <br/><br/>If you memorize some results derived using Ito's Lemma, then you will not have to go through the derivation on the test. For example, if dX(t)/X(t) = αdt + σdZ(t) (i.e., X follows a geometric Brownian motion, then d[ln(X)] = (α - 0.5σ2)dt + σdZ(t) (i.e., ln(X) follows an arithmetic Brownian motion with the exact same volatility.) Just memorizing the latter formula can make several currently known prior exam questions extremely easy to solve. <br/><br/>Do not forget about the formula for the Sharpe ratio (φ = (α - r)/σ) and the fact that the Sharpe ratios of two perfectly correlated Brownian motions are equal.<br/>To figure out whether a particular Brownian motion has zero drift (or what kind of drift factor it has), you will need to apply Ito's Lemma to the given Brownian motion and see what the resultant dt term is. <br/><br/>Also recall that the quadratic variation of a Brownian process is expressible as<br/>n→∞limi=1nΣ(Z[ih] - Z[(i -1)h])2 = T from time 0 to time T. Here, Z is the Standard Brownian Motion. In X is some other Brownian motion with volatility factor σ, then <br/><br/>n→∞limi=1nΣ(Y[ih] - Y[(i -1)h])2 = σ2T. <br/><br/>Other useful facts to know are as follows. <br/><br/>Var[σdZ(t)│Z(t)] = σ2Var[dZ(t)│Z(t)] = σ2dt. <br/><br/>For any arithmetic Brownian motion X(t), the random variable [X(t + h) - X(t)] is normally distributed for all t ≥ 0, h > 0, and has a mean of X(t) + αh and a variance of σ2h. <br/><br/>The Black-Scholes option pricing framework is based on the assumption that the underlying asset follows a geometric Brownian motion:<br/>dS(t)/S(t) = αdt + σdZ(t). <br/><br/>When X(t) follows an arithmetic Brownian motion, the following equation holds: <br/>X(t) = X(a) + α(t - a) + σ√(t-a)ξ. <br/><br/>When X(t) follows a geometric Brownian motion, the following equation holds: <br/>X(t) = X(a)exp[(α - 0.5σ2)(t - a) + σ√(t-a)ξ]. <br/><br/>If given either of those two equations, you should be able to recognize arithmetic and geometric Brownian motion. <br/><br/>Be sure to know how to value claims on power derivatives (of the formS(T)a). This is a newly added topic and is thus likely to be tested on at least one question. Memorize the following formulas. <br/><br/>d(Sa)/Sa = (a(α - δ) + 0.5a(a-1)σ2)dt + aσdZ(t) <br/><br/>γ = a(α - r) + r, where r is the annual continuously compounded risk-free interest rate. <br/><br/>δ* = r - a(r - δ) - 0.5a(a-1)σ2<br/>F0,T[S(T)a] = S(0)aexp((a(r - δ) + 0.5a(a-1)σ2)T) <br/>FP0,T[S(T)a] = e-rTS(0)aexp((a(r - δ) + 0.5a(a-1)σ2)T)<br/>While many actuarial students might think that Brownian motion is the most difficult topic on the 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 regarding them. 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 to memorize. The past exam questions I have seen have not asked for these formulas. Instead, they tended to involve various "auxiliary" formulas for these models. <br/><br/>Of course, it is essential to know the Brownian motions associated with the Vasicek and CIR models. 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 important implications, however, as discussed in Section 72 of my study guide. Make sure you read this section to find out about the essential similarities and differences between these models as well as why the time-zero yield curve for either of these models cannot be exogenously prescribed (a learning objective on the exam syllabus). <br/><br/>For both the Vasicek model, the following "auxiliary" equations are important to know, as questions involving them have appeared on prior exams. <br/><br/>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 the same whenever the difference between T and t is the same, even if the values of T and t are different. <br/><br/>Now say you have a Vasicek model where <br/><br/>dP[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(r1, t1, T1) and are asked to find α(r2, t2, T2). All you need to do is to solve the following equation: <br/>[α(r1, t1, T1) - r1]/(1 - exp[-a(T1 - t1)]) = [α(r2, t2, T2) - r2]/(1 - exp[-a(T2 - t2)])<br/><br/>The derivation of this formula is quite involved and is provided in Section 71. You would be well advised to simply memorize the end result (and read the derivation, if you are interested in the reasoning, but do not repeat the derivation each time unless you enjoy pain and suffering). <br/><br/>(It would not hurt to also memorize that q[r(t), t, T] = -σB(t, T) in the case above. A question using this formula is not likely to be asked, and if it is, I hope that σ and B(t, T) will be given.)<br/>Important: dZ * dZ' = ρ*dt<br/>The Black-Derman-Toy (BDT) interest rate model has easy and difficult applications. The easy applications involve pricing caplets and interest rate caps, provided that you have had practice with the BDT discounting procedure. Remember that the risk-neutral probability for the BDT model is always 0.5 and the interest rates at the nodes in any given time period differ by a constant multiple of exp[2σi√(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 will need to account for payments made each time period when the interest rate exceeds the cap strike rate. <br/><br/>The difficult parts of using the BDT model involve finding yield volatilities and actually constructing BDT binomial trees. I am aware of a scant few exam-style questions on the former and of none on the latter - which I hope will be indicative of the composition of future MFE exams. There is no way around memorizing the formula for yield volatility in the BDT model: <br/><br/>Yield volatility = 0.5ln(y[h, T, ru]/y[h, T, rd]) = 0.5ln([P[h, T, ru]-1/(T-h) - 1]/[P[h, T, rd]-1/(T-h) - 1]) <br/><br/>You will likely be given a multi-period BDT tree and asked to calculate the volatility in time period 1 of the bond used in constructing the tree. I am hard-pressed to see how the formula above might be used in finding the volatilities for time periods greater than 1, since the formula relies on the presence of two and only two nodes in the binomial tree. <br/><br/>For constructing binomial trees, I hope that, if a question is asked, it will involve constructing a tree for a one-period model. To do this, learn the following formulas: <br/><br/>If Ph is the price of an h-year bond, where h is one time period in the BDT model, then <br/>Ph = 1/(1 + r0) and thus r0 = 1/Ph - 1. If P2h is the price of an 2h-year bond, then Rh = rd and σh meet the following conditions: r0 = σhand P2h = 0.5Ph(1/(1 + Rhexp[2r0]) + 1/(1 + Rh)). Anything beyond this is intense algebraic busy work, and I hope that the SOA and CAS are kind enough not to embroil students in it. <br/><br/>Also remember to review the Black formula for pricing options on futures contracts (Section 37) and caplets (Section 75). <br/><br/>4. Do not forget the "miscellaneous" topics! I can think of two "miscellaneous" topics that appear seldom or not at all in McDonald's text but are highly likely to be tested. One of these is computing 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 the syllabus. Nonetheless, you are likely to be asked to use one or both of the following formulas to discover option payoffs in disguise:<br/>Formula 80.1: max(AB, C) = B*max(A, C/B) <br/>Formula 80.2: max(A, B) = A + max(0, B - A) <br/><br/>Section 80 gives some equity-linked insurance problems that you might find beneficial. <br/><br/>5. Take practice tests under exam conditions several days before the actual exam. Taking practice exams will give you an idea of your current knowledge level as well as areas to which you might need to devote additional attention. If you have studied well and you do well on the practice exams, this will give you a lot of confidence coming into the actual test. I expect the passing threshold for future sessions of Exam 3F/MFE to be around 13 or 14 out of 20, since the Fall 2007 exam - a particularly difficult test - had a passing threshold of 12. <br/><br/>

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