Transcription
Options PricingSpencer LinSebastian Ruf5/1/2012
Outline BackgroundModel of Stock EvolutionBlack-Scholes EquationBinomial Method
Options Contract/Agreement between two partiesDefining parameters: What underlying security (what stock)Quantity of the underlying (number of shares)Option Type (Put or Call)Strike PriceOption expiration dateStyle (European, American, etc.)Other legal terms (not important to us)
Option Options Put: Call: Buyer pays premium (to seller) for right to purchase theunderlying at strike priceSeller is obligated to sell the underlying to BuyerAmerican: Seller pays premium (to buyer) for the right to sell theunderlying at strike priceBuyer is obligated to purchase the underlying from SellerOption may be exercised at on any trading day beforeexpiration dateEuropean: Option may only be exercised on day of expiration
Black Scholes Model Assumptions No transaction costs or taxesTrading happens in a continuous mannerNo dividends or splitsRisk-free interest rate is constantOptions are European styleThe underlying stock behavior follows a geometricbrownian motion, with constant drift and volatility
Wiener Process Generalized Wiener Process , , : random, normally distributed value on [0,1]Figure from Hull J.C., “Options Futures and Other Derivatives”, p221
Stock Behavior : Stock value: expected rate of return : volatility
Ito’s Lemma If x follows a Generalized Wiener ProcessA function G(x,t) follows 1 2
Black Scholes Model 1 2
Solution to Black Scholes Process: Variable substitution Heat equation Self-similar solution
Solution to Black Scholes cont.
Simulations – raw dataGoogle Stock Price in 2011660640opening stock price [ ]620600580560540520500480460050100150200trading day of 2011250300
Simulations – changing strike priceStrike Price vs Option ValueGoogle 2011 DataStock Price 642.0, Volatility 29.2600PutCall500Option Value [ ]4003002001000300400500600700800900Strike Price [ ]1000110012001300
Simulations – Volatility and Expiration (Put)Changing Volatility and Expiration TimePut Option80option value [ ]60402000.80.610.80.40.60.40.2volatilityNominal Strike: 64200.20stock value: 642expiration time [yrs]risk free rate: 18% per annum
Simulations – Volatility and Expiration (Call)Changing Volatility and Expiration TimeCall Option200option value [ ]1501005000.80.610.80.40.60.40.2volatilityNominal Strike: 64200.20stock value: 642expiration time [yrs]risk free rate: 18% per annum
Simulations – Volatility and Strike (Put)Changing Volatility and Strike PricePut Option100option value [ tion: 0.5 years0620600stock value: 642strike price [ ]risk free rate: 18% per annum
Simulations – Volatility and Strike (Call)Changing Volatility and Strike PriceCall Option140120option value [ ation: 0.5 yearsstock value: 642600strike price [ ]risk free rate: 18% per annum
Simulations – Expiration and Strike (Put)Changing Expiration Time and Strike PricePut Option60option value [ ]5040302010017006800.5660640expiration time [yrs]Volatility: 29% per annum0620600stock value: 642strike price [ ]risk free rate: 18% per annum
Simulations – Expiration and Strike (Call)Changing Expiration Time and Strike PriceCall Option200option value [ ]15010050017006800.5660640expiration time [yrs]Volatility: 29% per annum0620600stock value: 642strike price [ ]risk free rate: 18% per annum
Binomial Method" # ! , " ! " , , , , , , , , , , , , 1 , !Figure modified from Hull J.C., “Options Futures and Other Derivatives”, p207 , ,
Binomial Method Simulation (Euro)Value of euro style put options, binomial method vs Black Scholes3634Value of option now [ ]3230Black Scholes2826242220010203040Number of time steps5060
Binomial Method Simulation (American)Value of amer style put options, binomial method vs Black Scholes3635Value of option now [ ]3433323130292827Black Scholes010203040Number of time steps5060
Further Work Exploration into Modification of Black-Scholes: Other Option StylesDividendsVarying Volatility
Options Pricing Spencer Lin Sebastian Ruf 5/1/2012. Outline Background Model of Stock Evolution Black-Scholes Equation Binomial Method. Options . volatility option value [ ] Simulations -Volatility and Strike (Put) Expiration: 0.5 years stock value: 642 risk free rate: 18% per annum Put Option 600 620 640 660 680 700 0 0.2 0.4 0.6 0.8 0 20 .
