MFE Study Guide

MFE Study GuideFall 2007 2nd EditionMFE Study Guide (Fall 2007)Notes from McDonalds Derivative MarketsWritten by Colby Schaeffer IntroductionThe material in this quick study guide has been done to the best of my knowledge. Some topics are only covered briefly (Delta-Hedging and Caps/Floors) while other topics have been omitted (Equity Linked Annuities and Compound Options w/One Discrete Dividend). All exam tips are marked in red! This the 1st Edition of my MFE study guide, and it may be updated in the future with better content. Comments and questions may be directed via PM to colby2152 on the Actuarial Outpost.2nd Edition Notes: Jraven and fractl helped me tweak some little errors in the study guide. An entirely revamped Spring 08 Edition will be out by March 2008 with more on Brownian Motion, a better understanding but less detailed look at the Greeks, an easier view of convexity, and perpetual options will be removed.Acknowledgements: Day Yi, Abraham Weishus, Bill Cross and AO Member “Jraven”Chapter 9 - Parity and Other Option RelationshipsOption Exercise Style American: any time European: end of maturityValue of otherwise identical options: European St K When exercising calls just prior to a dividend, early exercise is not optimal at any time where: K - PVt,T(K) PVt,T(DIV)Arbitrage Inequalities (for both American & European)THERE IS NO FREE LUNCH!K1 K2 K30 C(K1) C(K2) K2 K10 P(K2) P(K1) K2 K1*if options are European, then the difference in option premiums must be less than the present value of the difference in strikesPremiums decline at a decreasing rate as we consider calls with progressively higher strike prices. Premiums also decline for puts but when the strike price monotonically decreases.Convexity of option price w.r.t. strike priceC(K3) - C(K2)/(K3 - K2) StCall goes downPut goes upKr 0Vega: tests if volatility is sufficient, always 0, theta: option price change w.r.t. time to maturity change, usually 0, rho: sensitivity to risk free rate, psi: sensitivity to the dividend rateThe Greek measure of a portfolio is the sum of the Greeks of the individual portfolio componentsElasticity tells us the risk of the option relative to the stock in %termsFor a call 1, while for a put 0Risk Premium: r = ( r) Sharpe Ratio: Perpetual OptionsEach x value is the present value of 1 when a stock of value S rises or falls to price H, where the value is (S/H)xh1 = lower value of x, and h2 = higher value of xCallsValue: Maximum H: Puts just change (H K) to (K H) and change h1 to h2Chapter 13 Delta HedgingMarket makers want stable portfolios, so they use delta hedging as a method of controlling risk.Overnight ProfitDelta-Gamma-Theta Approximation*Delta and Delta-Gamma approximations are contained within the formulaWhere: Black-Scholes equationThis is different than the Black-Scholes FORMULA that was used for pricing options. Rather, this equation is a function of the greeks, stock price, volatility, and risk-free rate.Chapter 14 Exotic OptionsAsian Options based on the arithmetic/geometric average of underlying asset/strike price *useful for hedging currency exchange, variable annuities, and reducing volatilityGeometric(S) t1Arithmetic Brownian MotionZ(0) = 0, Z(t + s) Z(t) N(0, s), Z(t) is continuous: volatility or variance factor: drift factorOrnstein-Uhlembeck ProcessVariation of Arithmetic Brownian motionGeometric Brownian MotionItos LemmaMultiplication TabledtdZdt00dZ0dtSharpe Ratio= “expected return per unit risk”If AND THENChapter 24 Interest Rate ModelsA stochastic interest-rate model that assumes a flat yield curve cannot be arbitrage-free.Arithmetic: (similar to Arithmetic Brownian Motion)Problems: r St KAt the prices Sh = Su, Sd, a replicating portfolio will satisfy:(D Su edh ) + (B erh) = Cu(D Sd edh ) + (B erh) = CdBlack-Scholes PricingOption Greeks = e-t N(d1)ElasticityRisk Premium: r = ( r) Sharpe Ratio: Overnight ProfitDelta-Gamma-Theta ApproximationWhere: Itos LemmaOrnstein-Uhlembeck ProcessVariation of Arithmetic Brownian motionRisk Neutral Probability (of increase in stock)u e(r )h dCox Ross-RubinsteinLognormalSchroders MethodF = S PV(Div)Path-dependent optionsAsian based on average priceBarrierKnock-In + Knock-Out = Standard OptionOther Exotic OptionsCompoundCallOnOption PutOnOption = Gap: use trigger in d1, strike in PC parityExchange: Volatility depends on both assets: Perpetual OptionsValue: Maximum H: Vasicek: Cox-Ingersoll-Rand (CIR) Model: Black-Derman-Toy treeWritten by Colby Schaeffer17/17