The Infinite Actuary s. Detailed Study Manual for the. QFI Core Exam. Zak Fischer, FSA CERA

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1 The Infinite Actuary s Detailed Study Manual for the QFI Core Exam Zak Fischer, FSA CERA Spring 2018 & Fall 2018

2 QFI Core Sample Detailed Study Manual You have downloaded a sample of our QFI Core detailed study manual. The full version covers the entire syllabus and is included with the online seminar. Each portion of the detailed study manual is available in PDF with a clickable table of contents. Each reading (and sub-chapters if applicable) are also bookmarked in the PDF for ease of navigation in your favorite desktop, tablet, or smartphone PDF viewer. If you have additional questions about the detailed study manual or any aspect of the exam, please me. Zak Fischer, FSA, CERA, MAAA zak@theinfiniteactuary.com

3 QFI Core Detailed Study Guide - Sample Problems and Solutions in Mathematical Finance: Ch Set Theory Background Introduction Definition Definition of a Field F Definition Definition of a σ-algebra F Comparing Fields and σ-algebras Coin Flipping Example FSA Exam Example Measureable and Probability Spaces Definition Probability Measure P Definition Filtration F Definition Real-Valued Random Variable X Definition Adapted Stochastic Process X t Sub-σ Algebras G of F Definition Conditional Expectation E(X G) Properties of Conditional Expectation Book Practice Problems Low Yield Curves and Absolute/Normal Volatilities Overview Quantiative Easing Negative Rates Pitfalls of the Black Formula Implied Volatility Measure Characteristics Displaced Lognormal Model Implications for Insurers Conclusion

4 Problems and Solutions in Mathematical Finance: Ch. 1 Eric Chin, Dian Nel, and Sverrir Olafsson This book is quite techical. It goes a bit deeper into measure theory and stochastic calculus than other books on the syllabus. The great news is that this book has lots of questions and solutions to give you practice. So does this detailed study guide. Set Theory Background This is not a section in the reading. I wanted to include a quick review of set theory because it is assumed throughout the book. This will hopefully be straightforward! A set is a collection of objects The empty set is denoted as = {} The notation A B determines if A is an element of B Examples: 1 {1} 1 {1, 2} 4 / {1, 2} { } {} {{}} / The notation A B determines if A is a subset of B Examples: {1} {1} {1} {1, 2} 1 {1, 2} Make sure to understand that 1 {1, 2} but 1 {1, 2}. There is a big difference between 1 and {1}! Two sets are disjoint if their intersection is empty. That is, A and B are disjoint if and only if A B =

5 Introduction Ω is the sample space, which is the set of all possible outcomes The sample space Ω is the set of all possible outcomes of an experiment or random trial. Any subset A of the sample space is known as an event, where an event is a set consisting of possible outcomes of an experiment A Ω The complement of A is denoted as A c = Ω A Both of the pieces of notation above are probably familiar already. The next definition might be new. If this new definition is confusing, take a look at the coin flipping example in the DSG to see an example. F is any collection of subsets of Ω Definition Definition of a Field F A field F is a collection (or family) F of subsets of Ω satisfying the following conditions: (a) F (b) A F A c F (c) F is closed under finite unions Definition Definition of a σ-algebra F A σ-algebra F is a collection (or family) F of subsets of Ω satisfying the following conditions: (a) F (b) A F A c F (c) F is closed under countable unions Comparing Fields and σ-algebras Informally, remember the size of sets from smallest to largest is: Finite < Countably Infinite < Uncountably Infinite Therefore, all σ-algebra s are fields, but not all fields are σ-algebra s When dealing with a finite sample space, there is no difference between a field and σ- algebra. This will often happen in simple examples (rolling a die once, flipping a coin once, etc.) c 2018 The Infinite Actuary, LLC Page 3 of 16

6 Coin Flipping Example Ω = {H, T} One possible F where F is a σ-algebra would be F = {, {H}, {T}, {H, T}} Note that the only sets in F that have measure are {H} and {T} since P({H}) = P({T}) =.5 This does imply P(Ω) = 1 if P is a probability measure Note that there are four possible subsets of Ω 1. Ω 2. {H} Ω 3. {T} Ω 4. {H, T} Ω There are 2 4 = 16 choices for F, but not all choices will give a field or σ-algebra. The reason there are 2 4 choices is that each of the four items above can be chosen to be included or not in F An example of an event is A = {H} Understanding σ-algebras Q: Consider the example of flipping a fair coin once. Which of the following are possible σ-algebras? (i) F = {, {H}, {T}, {H, T}} (ii) F = (iii) F = {{H}, {T}} (iv) F = {, {H, T}} (v) F = { } Solution: Only (i) and (iv). For (ii), this is not a σ-algebra since / (if this is confusing you need to brush up on set theory basics!) and so property (a) from Definition 1.2 is not satisifed. Similarly, (iii) fails property (a). Lastly (v) fails because c = {H, T} / F, so property (b) is not satisifed. Note that (iv) is a sub-σ algebra of (i).

7 Understanding Fields Q: Consider the example of flipping a fair coin once. Which of the following are possible fields? (i) F = {, {H}, {T}, {H, T}} (ii) F = (iii) F = {{H}, {T}} (iv) F = {, {H, T}} (v) F = { } Solution: Only (i) and (iv). Since the sample space is finite, there is no distinction between a field and a σ-algebra. More σ-algebras! I Q: Let Ω = {a, b, c}. Construct the smallest σ-algebra F. Is it also a field? Solution: F = {, {a, b, c}}. Since n = 3 is finite, this is both a field and σ-algebra. More σ-algebras! II Q: Let Ω = {a, b, c}. Construct two more σ-algebras F beyond the one shown in the prior example. Solution: There are many possible examples: F = {, {a}, {b, c}, {a, b, c}} F = {, {b}, {a, c}, {a, b, c}} F = {, {c}, {a, b}, {a, b, c}} F = {, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}} c 2018 The Infinite Actuary, LLC Page 5 of 16

8 Closed Under Unions Implies Closed Under Intersections Q: For a σ-algebra F, prove that if A 1 F and A 2 F, then A 1 A 2 F Solution: Step Reasoning A 1 F, A 2 F Given A1 c F, Ac 2 F Property (b) of σ-algebras A1 c Ac 2 F Property (c) of σ-algebras (A 1 A 2 ) c F De Morgan s Law A 1 A 2 F Property (b) of σ-algebras Note: In this question we use n = 2, but this argument easily can be extended to any countable collection of objects. Therefore, this implies that a field is closed under finite intersections and a σ-algebra is closed under countable intersections. For additional practice, see Practice Problem #2 on page 4 in the book. FSA Exam Example Suppose that an ambitious (crazy) student decides to sit for QFI Core and QFI Advanced in the same sitting Let A 1 = {Core} represent the event that QFI Core is passed Let A 2 = {Adv} represent the event that QFI Advanced is passed Ω = {Core, Adv} F = {, {Core}, {Adv}, {Core, Adv}} This is actually very similar to the fair coin example, but the main difference is that all elements of the set above could have measure. For example, you can theoretically pass or fail both exams, but a coin flip always has exactly one result. Measureable and Probability Spaces The pair (Ω, F) is called a measurable space. The triple (Ω, F, P) is called a probability space. If you study advanced probability theory, this is a very important definition. A complete probability space is a special type of probability space that satifies the additional property that F contains all subsets of Ω with P-outer measure zero. Mathematically: P (B) = inf{p(a) : A F, B A} = 0

9 Inf means infimum In other words, for all B F with P(B) = 0 and all A B, A F. Definition Probability Measure P A probability measure P on a measurable space (Ω, F) is a function P [0, 1] such that: (a) P( ) = 0 (b) P(Ω) = 1 (c) For any countable collection of distinct events A i, P( i=1 A i) = Constructing a Probability Measure: Coin Flips i=1 P(A i ) Q: Suppose a biased coin is flipped once with probability of heads equal to p. (a) Define Ω. (b) Construct a probability measure P [0, 1]. (c) Verify that property (c) is met for distinct events A 1 = {H} and A 2 = {T}. Solution: (a) Ω = {H, T} (b) P( ) = 0 P(Ω) = P({H, T}) = 1 P({H}) = p P({T}) = 1 p (c) Note that {H} {T} = Ω and P({H}) + P({T}) = p + 1 p = 1 = P(Ω) Definition Filtration F Let Ω be a non-empty sample space and let T be a fixed positive number, and assume for each t [0, T] there is a σ-algebra F t. In addition, we assume that if s t, then every set in F s is also in F t. We call the collection of σ-algebras F t, 0 t T a filtration. c 2018 The Infinite Actuary, LLC Page 7 of 16

10 Filtration Example One of the most natural ways to define a filtration F t is to incorporate all past and current information up to time t. F t incorporates all information up to time t It is then clear that for s t, F s F t F t F s is the additional information between times s and time t Definition Real-Valued Random Variable X Let Ω be a non-empty sample space and let F be a σ-algebra of subsets of Ω. A real-valued random variable X is a function X : Ω R such that {ω Ω : X(ω) x} F for each x R and we say X is F-measurable. The word measurable is used often - so become familiar with it! A random variable X can be measurable with respect to a family of subsets F. Note: The choice of σ-algebra F is important. See the following examples to hammer this point down. Constant Real-Valued Random Variable Example Q: Consider the simple case where X is equal to a constant c. (a) State the smallest generic σ-algebra F. (b) Is X a real-valued random variable with respect to σ-algebra F? (a) F = {, Ω}. By property (a) of σ-algebra s, F. By property (b) of σ-algebra s, c = Ω F. Therefore, these two elements must always be in F. Moreover, the set containing these two elements is always a σ-algebra. (b) Yes. There are two cases to validate: (i) If x < c, then {ω Ω : X(ω) = c x} = F (ii) If x c, then {ω Ω : X(ω) = c x} = Ω F

11 Constant Real-Valued Random Variable Example Q: Consider the simple case where X is equal to a constant c. Prove that if F is a σ-algebra, then X is F-measurable. You may assume that X(ω) = c for all ω Ω. Solution: This is a simple extension from the prior example. We have already proved that F and Ω F for any σ-algebra F. Therefore, the two cases below (also analyzed in the prior example) are met: 1. If x < c, then {ω Ω : X(ω) = c x} = F 2. If x c, then {ω Ω : X(ω) = c x} = Ω F Note that the only difference from the prior example is that F may contain more elements, but the main point is that X is still F-measurable. Measurable Example Let F contain all subsets of Ω. Consider the real-valued random variable X : Ω R. Answer the following: (a) Is F a σ-algebra? (b) Is X F-measurable? Solution: Yes to both (a) and (b). This is the largest σ-algebra and is always measurable, since it contains all possible subsets of Ω. c 2018 The Infinite Actuary, LLC Page 9 of 16

12 Definition Adapted Stochastic Process X t Let Ω be a non-empty sample space with filtration F t, t [0, T] and let X t be a collection of random variables indexed by t. The collection of random variables X t is an adapted stochastic process if for each t, the random variable X t is F t measurable. Adapted Stochastic Process Example Let X t be the stock price at time t F t contains the stock prices between times 0 and t (inclusive) Then X t is an adapted stochastic process Sub-σ Algebras G of F G is a sub-σ-algebra of F if sets in G are also in F. Sub σ-algebra Example Example: G = {, {H, T}} is a sub-σ algebra of F = {, {H}, {T}, {H, T}} Definition Conditional Expectation E(X G) Let (Ω, F, P) be a probability space and let G be a sub-σ-algebra of F. Let X be an integrable and non-negative random variable. Then the conditional expectation of X given G, denoted by E(X G) is any random variable that satisfies: (a) E(X G) is G-measurable (b) For every set A G, we have the partial averaging property: A E(X G)dP = A XdP Properties of Conditional Expectation Given the definition above, it s possible to prove that conditional expectation has some nice properties (most of which you probably already know from the prelims!).

13 The book goes through and proves most of these through practice problems, so if you want some extra practice I recommend going through those. The key idea for these is to start off by applying the partial averaging property. I ll go through the conditional expectation example below. Throughout this section, (Ω, F, P) is a probability space and G is a sub-σ-algebra of F. X and Y are integrable random variables. The word measurable is used often - so become familiar with it! A random variable X can be measurable with respect to a set G. We already saw this in Definitions 1.5 and 1.6. X will represent a quantity to be estimated and G contains limited information about X. An example of G could be past stock prices and the current stock price where X is the stock price. Another example of G would be the history of a series of coin flips and X contains the number of heads minus the number of tails observed from a series of flips of a fair coin. In both cases, E(X G) = X. That is, the future expected value is equal to its current value. Properties: Conditional probability: E(1 A G) = P(A G) Linearity: E( n i=1 c i X i G) = n c i E(X i G) i=1 Positivity: X 0 almost surely E(X G) 0 almost surely Monotonicity: If X Y almost surely then E(X G) E(Y G) Computing expectations by conditioning: E[E(X G)] = E(X) (see practice problem below!) Measurability: If X is G-measurable then E(X G) = X Taking out what is known: If X is G-measurable, then E(XY G) = X E(Y G) Tower property: E[E(X G) H] = E(X H) if H is a sub-σ-algebra of G Independence: If X is independent of G then E(X G) = E(X) Note that independence is normally something you would think of between two random variables, but this is independence between a random variable and a set This means that given all the information in G, you have no additional information about X As an example, G could contain information about past coin flips and X could be the value of the next flip. If it is truly a fair coin, there would be independence. c 2018 The Infinite Actuary, LLC Page 11 of 16

14 Conditional Jensen s Inequality: If ϕ : R R is a convex function, then E[ϕ(X) G] ϕ[e(x G)] Remember that (unconditional) Jensen s Inequality is also covered in FAQ Q23 Computing Expectations by Conditioning Q: Prove that E[E(X G)] = E(X) Solution: As always with these kinds of questions, use the partial averaging property! For every set A G, we have the partial averaging property: E(X G)dP = XdP A A We know that Ω G, since Ω is in every σ-algebra. Therefore: E(X G)dP = XdP E[E(X G)] = E(X) Ω Ω Book Practice Problems Pages Questions 4-5 Q3 - Q Q Q4 - Q5

15 QFI Core Low Yield Curves Low Yield Curves and Absolute/Normal Volatilities Moody s Analytics This is a fairly short article advocating the use of a new implied volatility measure over the Black-Scholes formula. The Black-Scholes formula has substantially more noise in a low interest-rate environment. Overview Because Black s formula assumes that rates are lognormally distributed, the formula becomes infinitely sensitive to price changes as rates tend to zero. The formula cannot be solved at all for negative strikes or forward rates. Starting around 2009, using Black s formula became difficult, and the market solution for this was to focus on Normal/Absolute volatilities The Lognormal/Black SDE is unstable because F T t is very close to zero in many economies The Displaced Lognormal SDE is not objective because it involves an additional, arbitrary displacement parameter from using F instead of F Choice of Implied Volatility Measure Implied Volatility Model Stochastic Diff Eq Stable? Objective? Author s Choice? Lognormal/Black Displaced Lognormal dft T Ft T d F t T F t T = σdw t = σdw t Absolute/Normal df T t = σdw t Quantiative Easing QE was to force longer term yields lower by purchasing government bonds and other longer dated financial assets, in combination with clear future guidance that policy rates would stay low for a prolonged period of time. Used in US/UK in 2008, Japan in 2011 Negative Rates Nowadays, negative real rates are quite common In some markets (e.g. Switzerland), nominal rates are even negative. This tends to be more of a concern for shorter terms on the yield curve c 2018 The Infinite Actuary, LLC Page 13 of 16

16 QFI Core Low Yield Curves Swap Rates and Swaption Implied Vol Over Time Pitfalls of the Black Formula As rates dropped in 2009, the implied volatility from the Black model spiked drastically to unreasonable values The market consensus was that this value was far too large, and the value implied by normal/absolute volatility modeling was significantly more reasonable When there is a skew of swaption volatilities for different strikes away-from-the-money, Black volatilities have shown a marked dependence on the strike level whereas by contrast the absolute/normal volatilities are relatively constant Remember that the implied volatility should be roughly the same across strikes if the Black-Scholes model is correct No smile/skew is expected under Black-Scholes; there ideally would be a flat/constant volatility Therefore, this is a weakness for the Black model and an advantage for the Absolute/Normal model

17 QFI Core Low Yield Curves Normal/Absolute Volatilities Since 1996 Implied Volatility Measure Characteristics Ideally, an implied volatility measure should vary in an intuitive way when: 1. Strike and forward rates change. 2. Maturities and tenors vary. 3. Different types of instruments are analyzed (for example puts and calls, payers and receivers) via put-call parity. Displaced Lognormal Model Additional displacement parameter Relatively easy to use, only takes minor tweaks on the Black formula c 2018 The Infinite Actuary, LLC Page 15 of 16

18 QFI Core Low Yield Curves Implications for Insurers Whether the value of guarantees will increase or decrease from the move to the normal/absolute methodology depends on whether the base yield curve is above or below the swap curve Base Yield Curve < Swap Curve Value(Lognormal) < Value(Normal) Example: Government Curve Base Yield Curve > Swap Curve Value(Lognormal) > Value(Normal) Example: EIOPA, Solvency II, AA Evaluating fit is easier under the normal/absolute methodology Correlations between yield curve and implied volatility: Black Model: strongly negative Absolute/Normal Model: weakly positive Many banks have already started using normal/absolute methodology Conclusion Author advocates use of absolute/normal volatility over the traditional Black formula because it is more robust under a variety of different interest rate environments