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PAK Condensed Summary Quantitative Finance and Investment Advanced (QFIA) Exam Fall 2015 Edition

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PAK Condensed Summary Quantitative Finance and Investment Advanced (QFIA) Exam Fall 2015 Edition PAK Condensed Summary Quantitative Finance and Investment Advanced (QFIA) Exam Fall 2015 Edition CDS Options Hedging Derivatives Embedded Options Interest Rate Models Performance Measurement Volatility Modeling Structured Finance Credit Risk Models Behavioral Finance Liquidity Risk Attribution PCA MBS QFIA Exam 2015 PAK Condensed Summary Ribonato-7 Ribonato-7 Empirical Facts about Smiles Fundamental and Derived Analysis The two potential strands of empirical analysis: 1. Fundamental analysis and 2. Derived Analysis In the fundamental analysis, it is recognized that the value of options and other derivatives contracts are derived from the dynamics of the underlying assets. The fundamental analysis is relevant to the plain vanilla options trader. The derived approach is of great interests to the complex derivatives trader. Not only the underlying assets but also other plain-vanilla options constitute the set of hedging instruments. In the derived approach, implied volatilities (or options prices) and the dynamics of the underlying assets are relevant. But mis-specification of the dynamics of the underlying asset is often considered to be a ‘second-order effect’. The complex trader usually engages in substantial vega and/or gamma hedging of the complex product using plain vanilla options. The delta exposure of the complex trade and of the hedging derivatives usually cancels out. Market Information About Smiles Direct Static Information The smile surface at time t is the function: : 퐼푚푝푙 2 휎푡 푅 → 푅 ( , ) ( , ) 퐼푚푝푙 푡 In principle, a better set of coordinate than the strike퐾 푇 (K)→ 휎can be퐾 a measure푇 of in-the-moneyness. For a fixed maturity, a common choice is to select points with the same of Black out-of-the-moneyness: 퐾 푙푛 � � When comparing points of the surface with the similar degree푆 of out-of-the-moneyness across maturities, it can be useful to use the quantity h such that: = 퐾 푙푛 � � ℎ 푆 휎√푇 1 QFIA Exam 2015 PAK Condensed Summary Ribonato-7 The risk-reversal statistic for a given maturity is defined as the difference 25-delta implied volatility for calls and puts of the same maturity: ( , ) = , , ( , , ) Where: 퐼푚푝푙 25∆푝 퐼푚푝푙 25∆푐 푅푅= 푡 푇 휎 �푡 퐾 푇 � 25− 휎 푡 퐾 푇 퐾25∆푝 푇ℎ푒 푠푡푟푖푘푒 푡ℎ푎푡 푔푖푣푒푠 푎 푑푒푙푡푎 푡표 푡ℎ푒 푝푢푡 푎푛푑 = 25 25∆푐 The straddle is calculated as: 퐾 푇ℎ푒 푠푡푟푖푘푒 푡ℎ푎푡 푔푖푣푒푠 푎 푑푒푙푡푎 푡표 푡ℎ푒 푐푎푙푙 ( , ) = , , + ( , , ) 2 ( , , ) Where: 푆푇 푡 푇 휎퐼푚푝푙�푡 퐾25∆푝 푇� 휎퐼푚푝푙 푡 퐾25∆푐 푇 − 휎퐼푚푝푙 푡 퐾퐴푇푀 푇 ( , , ) = 퐼푚푝푙 퐴푇푀 The risk reversal gives an indication휎 푡 퐾 about푇 the 푇asymmetryℎ푒 퐴푡 푡ℎ푒 푀표푛푒푦 of the 푖푚푝푙푖푒푑smile for푣표푙푎푡푖푙푖푡푦 a given maturity. The straddle gives information about its average curvature around the ATM level. Indirect Information: From the volatility surface to the pdf of the stock price Once a smooth volatility surface has been obtained, one can obtain the risk neutral probability density function of the stock price: ( , , , ) = [ × ( )] × 2 휕 퐶 Where: ∅ 푡0 푆0 푆푡 푡 푒푥푝 푟 푡 − 푡0 2� 휕퐾 퐾=푆푡 ( , , , ) = , 0 0 푡 ∅ 푡 푆 푆 푡 0 푡 푡ℎ푒 푡푖푚푒 푡 푝푟표푏푎푏푖푙푖푡푦 푑푒푛푠푖푡푦 푡ℎ푎푡 푡ℎ푒 푠푡표푐푘 푝푟푖푐푒 푤푖푙푙 푟푒푎푐ℎ 푣푎푙푢푒 푆 푎푡 푡푖푚푒 푡 푔푖푣푒푛 0 0 This pdf can be compared with the lognormal.푡 Thisℎ푎푡 푖푡pdf푖푠 is푆 a risk푎푡 푡푖푚푒-adjusted푡 one. Equities The most salient empirical features of equity smiles are the following: Fact 1: Smiles have greatly increased in magnitude after the 1987 equity crash Fact 2: The magnitude of the smile as a function of a fixed money strike tends to decrease for increasing option expiries: Short maturities display pronounced smiles, and distant maturities give rise to shallow smiles. Fact 3: The magnitude of the smile as a function of the degree of the out-of-the-moneyness is much more constant across different option expiries. Fact 4: the smile is much more pronounced going from the ATM level towards out-of-the-money puts than in the opposite direction. Going towards OTM calls the smile either becomes less steep, or is monotonically decreasing, or sometimes, it is even absent. Fact 5: the asymmetry in the smile tends to increase during periods of market turbulence (Associated with the negative skewness in the RN density). 2 QFIA Exam 2015 PAK Condensed Summary Ribonato-7 Subtler Effects 1. Negative Correlation between Changes in the Implied Volatility and the Level of the Underlying 2. Movements of call prices and underlying in the opposite directions 3. Level, Skew and Convexity as a function of maturity For each maturity, three levels of strike: , − 0 + Such that 0.25; 0.50 and 0.75 of the cumulative distribution퐾 퐾 푎푛푑 lies퐾 to their left. Their associated implied volatilities: , − 0 + The following three quantities are introduced: 휎 휎 푎푛푑 휎 ( ) = 푉표푙푎푡푖푙푖푡푦 푙푒푣푒푙 휎0 ( ) = = 휎+ − 휎− And 푉표푙푎푡푖푙푖푡푦 푆푘푒푤 휒 +휎0 2 ( ) = = 휎+ 휎− − 휎0 푉표푙푎푡푖푙푖푡푦 퐶표푛푣푒푥푖푡푦 휔 0 4. Persistence of the Smirk as a Function of At-the-Moneyness 휎 Let h be a plausible measure of moneyness: = 퐹 푙푛 � � Where: ℎ 퐾 = 휎√푇 퐹 = 퐹표푟푤푎푟푑 푣푎푙푢푒 퐾= 푇ℎ푒 푠푡푟푖푘푒 푝푟푖푐푒 = 푇 푇푖푚푒 푡표 푀푎푡푢푟푖푡푦 This definition of the moneyness 휎allows푇ℎ 푒a 푎푣푒푟푎푔푒natural link푣표푙푎푡푖푙푖푡푦 between표푓 the푡ℎ푒 slope푖푛푑푒푥 and curvature of a smile and the skewness and kurtosis of the associated risk-neutral distribution. When the volatility smile is plotted against this moneyness (h), it fails to flatten out, at least for maturities as long as two years. 5. Asymptotic Behavior of prices of Out-of-the-money Options As the strike of an option progressively moves towards the out-of-the-money region, all models for the underlying naturally predict that the option price shall go to zero. 3 QFIA Exam 2015 PAK Condensed Summary Ribonato-7 The speed of convergence to zero as a function of the option expiry can give information about the continuous or otherwise nature of the underlying process. 6. Floating vs Sticky Behavior of the Smile Surface The floating or sticky behavior of the smile surface depends on how the smile changes when the underlying changes. “Sticky smiles (volatility surface) are smiles that do not move when the underlying asset moves. Floating smiles move with the underlying asset. A trader who believes in a sticky smile world would label the x-axis of the volatility surface (volatility as a function of moneyness) with strikes and use the same graph whatever the spot value of the underlying is. On the contrary, a trader who believes in a floating smile world would label the x-axis with the ratio of K over S (K/S) and use the same graph whenever the spot value changes.” Interest Rates Fact 1: Before 1994, interest rate smiles used to be fundamentally flat across strikes. Fact 2: The earliest and most pronounced smiles began in JPY around 1994, when interest rates became substantially lower than in all other major currencies. Fact 3: Between 1994 and 1998 the smile surfaces of interest rate options in other currencies took the lead from JPY and became somewhat sloped. Fact 4: In this initial period the shape of the smile used to be monotonically decreasing from low to high strikes. Fact 5: The magnitude of the smile does not systematically decrease with increasing option maturity. Fact 6: Interest rate traders subscribe to a normal view of the distributional properties of forward rates. 4 .
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