Bài giảng Financial Markets - Lecture 21: Options Markets

Lecture 21: Options MarketsOptionsWith options, one pays money to have a choice in the futureEssence of options is not that I buy the ability to vacillate, or to exercise free will. The choice one makes actually depends only on the underlying asset priceOptions are truncated claims on assetsOptions ExchangesOptions are as old as civilization. Option to buy a piece of land in the cityChicago Board Options Exchange, a spinoff from the Chicago Board of Trade 1973, traded first standardized optionsAmerican Stock Exchange 1974, NYSE 1982Terms of Options ContractExercise dateExercise priceDefinition of underlying and number of sharesTwo Basic Kinds of OptionsCalls, a right to buyPuts, a right to sellTwo Basic Kinds of OptionsAmerican options – can be exercised any time until exercise dateEuropean options – can be exercised only on exercise dateBuyers and WritersFor every option there is both a buyer and a writerThe buyer pays the writer for the ability to choose when to exercise, the writer must abide by buyer’s choice Buyer puts up no margin, naked writer must post marginIn and Out of the MoneyIn-the-money options would be worth something if exercised nowOut-of-the-money options would be worthless if exercised nowPut-Call Parity RelationPut option price – call option price = present value of strike price + present value of dividends – price of stockFor European options, this formula must hold (up to small deviations due to transactions costs), otherwise there would be arbitrage profit opportunitiesLimits on Option PricesCall should be worth more than intrinsic value when out of the moneyCall should be worth more than intrinsic value when in the moneyCall should never be worth more than the stock priceBinomial Option PricingSimple up-down case illustrates fundamental issues in option pricingTwo periods, two possible outcomes onlyShows how option price can be derived from no-arbitrage-profits conditionBinomial Option Pricing, Cont.S = current stock priceu = 1+fraction of change in stock price if price goes upd = 1+fraction of change in stock price if price goes downr = risk-free interest rate Binomial Option Pricing, Cont.C = current price of call optionCu= value of call next period if price is upCd= value of call next period if price is downE = strike price of optionH = hedge ratio, number of shares purchased per call soldHedging by writing callsInvestor writes one call and buys H shares of underlying stockIf price goes up, will be worth uHS-CuIf price goes down, worth dHS-CdFor what H are these two the same?Binomial Option Pricing FormulaOne invested HS-C to achieve riskless return, hence the return must equal (1+r)(HS-C)(1+r)(HS-C)=uHS-Cu=dHS-CdSubst for H, then solve for CFormula does not use probabilityOption pricing formula was derived without regard to the probability that the option is ever in the money!In effect, the price S of the stock already incorporates this probabilityFor illiquid assets, such as housing, this formula may be subject to large errorsBlack-Scholes Option PricingFischer Black and Myron Scholes derived continuous time analogue of binomial formula, continuous trading, for European options onlyBlack-Scholes continuous arbitrage is not really possible, transactions costs, a theoretical exerciseCall T the time to exercise, σ2 the variance of one-period price change (as fraction) and N(x) the standard cumulative normal distribution function (sigmoid curve, integral of normal bell-shaped curve) =normdist(x,0,1,1) Excel (x, mean,standard_dev, 0 for density, 1 for cum.)Black-Scholes FormulaImplied VolatilityTurning around the Black-Scholes formula, one can find out what σ would generate current stock price. σ depends on strike price, “options smile”Since 1987 crash, σ tends to be higher for puts or calls with low strike price, “options leer” or “options smirk”VIX Implied VolatilityWeekly, 1992-2004Implied and Actual Volatility Monthly Jan 1992-Jan 2004Actual S&P500 Volatility Monthly1871-2004Using Options to HedgeTo put a floor on one’s holding of stock, one can buy a put on same number of sharesAlternatively, one can just decide to sell whenever the price reaches the floorDoing the former means I must pay the option price. Doing the latter costs nothingWhy, then, should anyone use options to hedge?Behavioral Aspects of Options DemandThaler’s mental categories theoryWriting an out-of-the-money call on a stock one holds, appears to be a win-win situation (Shefrin)Buying an option is a way of attaining a more leveraged, risky positionLottery principle in psychology, people inordinately attracted to small probabilities of winning bigMargin requirements are circumvented by optionsOption DeltaOption delta is derivative of option price with respect to stock priceFor calls, if stock price is way below exercise price, delta is nearly zeroFor calls, if option is at the money, delta is roughly a half, but price of option may be way below half the price of the stock.For calls, if stock price is way above the exercise price, delta is nearly one and one pays approximately stock price minus pdv of exercise price, like buying stock with credit pdv(E)