Tài chính doanh nghiệp - Topic 13A: Black - Scholes - merton option pricing model, implied vols, and volatility estimation

Financial ModelingTopic #13a: Black-Scholes-Merton Option Pricing Model, Implied Vols, and Volatility EstimationL. Gattis1Learning ObjectivesValue options using historical vol, moving average vol (MAV), exponentially weighted moving average (EWMA), and generalized autoregressive conditional heteroskedasticity (GARCH).Calculate option model implied volatility surfaces -- time skew (a.k.a. terms structure of volatility), and strike skew (Smiles and Smirks)Understand what volatility surfaces reveal about option prices, volatility, and the models.2Black-Scholes-Merton FunctionsFunction BSCall(s, k, v, r, t, d) d_1 = (Application.Ln(s / k) + (r - d + (v ^ 2) / 2) * t) / (v * t ^ 0.5) nd1 = Application.NormSDist(d_1) d_2 = d_1 - v * t ^ 0.5 nd2 = Application.NormSDist(d_2)BSCall = s * Exp(-d * t) * nd1 - k * Exp(-r * t) * nd2End FunctionFunction BSPut(s, k, v, r, t, d) d_1 = (Application.Ln(s / k) + (r - d + (v ^ 2) / 2) * t) / (v * t ^ 0.5) minus_nd1 = Application.NormSDist(-d_1) d_2 = d_1 - v * t ^ 0.5 minus_nd2 = Application.NormSDist(-d_2)BSPut = -s * Exp(-d * t) * minus_nd1 + k * Exp(-r * t) * minus_nd2End Function3Google Options as of 2/28/2012GOOG Spot = $614.27 Bloomberg:GOOG Equity OMON5Calculating Implied VolatilityThe March, $615 Strike, call and put are selling for $10.05 and $11.05 (Average of Bid & Ask) respectively?Implied volatility is the standard deviation input that equates the BSM model option price to the market price.Start with volatility guess (say 10%), then run solver to equate market model prices6Calculating Implied VolatilityThe March, $615 Strike, call and put are selling for $10.05 and $11.05 (Average of Bid & Ask) respectively?Implied volatility is the standard deviation input that equates the BSM model option price to the market price.Start with volatility guess (say 10%), then run solver to equate market model prices7Implied Volatility VBA FunctionsFunction putvol(s, k, r, t, d, mkt)hivol = 1 lovol = 0 Do While (hivol - lovol) > 0.0001 If BSPut(s, k, (hivol + lovol) / 2, r, t, d) > mkt Then 'if bsput > Mkt price hivol = (hivol + lovol) / 2 'lower vol Else: lovol = (hivol + lovol) / 2 'raise vol End If Loopputvol = (hivol + lovol) / 2End FunctionFunction callvol(s, k, r, t, d, mkt)hivol = 1 lovol = 0 Do While (hivol - lovol) > 0.0001 If BScall(s, k, (hivol + lovol) / 2, r, t, d) > mkt Then 'if bscall value is too high hivol = (hivol + lovol) / 2 'lower vol Else: lovol = (hivol + lovol) / 2 'raise vol End If Loopcallvol = (hivol + lovol) / 2End Function8Uses of Implied VolatilityPrice quotes – Option sellers will often quote the implied vol to communicate the price of an option so that buyers can recalculate the option price when the spot price of the underlying asset changes.Volatility estimate for portfolio optimizationValue-at-Risk (VaR) parameterOne can use the implied volatility from an option with an observable price to calculate the price of another option on the same underlying assetTest of Black-Scholes ModelAccording to BSM, underlying asset return volatility is constant (homoskedasticity), therefore the implied vol for a particular stock should be the same for all maturities, all calls and puts, and the same for all strike prices.9Strike Skew, Smiles, and SmirksIn practice, implied volatilities of ITM, ATM, and OTM strike prices are different resulting in the volatility or strike skew. Implied vols of OTM and ITM option are often higher than ATM options, creating a vol “smile” or “smirk”. The smile may reflect traders beliefs that the distribution of stock returns have fatter tails than the normal curve implies (evidence against BS assumption)For example, Suppose a OTM put strike (say 5 standard deviations away from the mean) is so low that it is unlikely to ever be exercised and the model prices is extremely low. However, traders observe that events (like -5σ moves) happen more often than the normal curve implies, so they bid up the prices of the put.The higher price causes a higher implied volatility, and thus a vol smirkTwo Views of BSVol smirks indicate a fatter left tail than normalPutVol smiles indicate fatter tails than normal10Vol Term Structure (Maturity)The volatility term structure (or time skew) shows the implied vols for options of differing maturitiesVolatility term structures tend to be upward sloping when short-dated volatilities are at historically low levels (and expected to rise) --- mean reversion of volVolatility term structures tend to be downward sloping when short-dated volatilities are at historically high levels (and expected to fall) --- mean reversion of volAnother strike against BS assumption – constant volOption MaturityI.V.Downward sloping Term structure indicates volatility will return to lower levels11GOOG Time and Strike Skew12GOOG Time and Strike SkewActual GOOG Strike Skew13 Skew and BSMThe time and strike skew appear to violate the BSM assumption of homoskedasticity (constant volatility)However, BS is still widely used to value and quote option pricesThe Two Views of Black, Scholes, and MertonDemigodsFalse Gods1415Estimating volatility is the central problem in option tradingUsing historical data, you must decide on the frequency of data collection and the sample sizeA common solution is to use 1-2 years of daily or weekly returns for longer term options or less for short-term optionsWeightingEqually weighting (normal σ formula)Unequal weightingThe exponentially weighted moving average (EWMA) volatility is one particular weighting scheme which weights recent data more heavily. GARCH – Average of long-term avg and recent.GOOG Historical Vol of Continuously Comp Return(2-years of Daily Prices, finance.yahoo.com)16Copy data into excel and insert graph of sample and MAV volsEWMA Volatility EstimatesEmpirical studies have found autocorrelation in volatility which means high vol is likely to be followed by high vol (“Volatility Clustering”).Sample and population volatility measures equally weight historical data when computing volatilityExponential Weighted Moving Average Volatility (EWMA) is a volatility estimate that weighs recent data more heavily than older data when computing historical volatility1718EWMAHistorical standard deviation is the square root of variance -- variance is the simple average of squared deviations a. Historical “Equal Weight” Vol (σ) where each t is weighted 1/n b. EWMA: Exponentially Weighted Moving Average Volatility weights more recent deviations more heavilyWhere λ is a constant usually between .90 and .99. - Mean is often assumed to be zeroλ is called the decay factor ; 1- λ is the weight applied to most recent dataGOOG EWMA Vol19EWMA and GARCHEWMA AdvantagesUses all the data. Analysts do not have to decide how much data to captureMatches empirical observation of volatility clustering (autocorrelation) – large changes tend to be followed by large changes, and small changes by small changes.Often used to forecast short-term volatility (days/weeks)EWMA DisadvantagesDoes not incorporate empirical finding of mean reversion that shows that volatility tends to move back to the long term average. This explains the term structure of volatility (time skew)Does not have a time dimension EWMA forecast is for next day and next year.GARCH (Generalized Autoregressive Heteroskedasticity) is a volatility model similar to EWMA that incorporates mean reversionGARCH is considered the standard model for forecasting medium and long term volatility.Addresses the disadvantages of EWMA20EWMA vs. GARCH21EWMA estimate regardless of forecast termGARCH estimates forecast short-term vol trend will continue (Lile EWMA), but will revert to long-term mean over time22GARCH(1,1)Autoregressive Conditional Heteroskedasticity GARCH(1,1) estimate for variance for historical day t σ2t = γV + αr2t-1 + βσt-1 α,β,γ : coefficients that sum to 1 (γ=1-α-β), estimated by finding values that best predict historical volatility r2t-1: lagged squared return (zero mean deviation) σt-1: lagged GARCH variance estimateGARCH(1,1) estimate for future period i σ2t+i = V + (α+β)i(σ2t – V)Long term varianceYesterday’s deviationYesterday’s GARCH ForecastAdjustment for difference between long-term and short-term estimateLong term variance23GOOG GARCH(1,1) Volatility ForecastGARCH Term estimates for n days: Forecasts Upward sloping vol term structure24Buy, Sell, or Hold 615 March CallVolsVolatility ForecastsHistorical (equal weight)Implied volatilityEWMAGARCHImplied volatilities are preferred if options prices are available and reliable.GARCH is the preferred/standard historical model due to its inclusion of mean reversion, term, and autocorrelation. 25