Tài chính doanh nghiệp - Topic 13: Option contracts and hedging, monte carlo valuation, and black - Scholes

Financial ModelingTopic #13: Option Contracts and Hedging, Monte Carlo Valuation, and Black-ScholesL. Gattis1Learning ObjectivesCompute the payoffs and profits of plain vanilla option contractsValue options using Monte Carlo Simulation and Black-Scholes modelsComputed hedged and unhedged cashflows using options and forwardsValue arithmetic Asian optionsUse @Risk to value options and compute position risk2Option ContractA call option is a contract to buy an asset in the future in which the asset, price, quantity, delivery place and location, are specified in the contract. The owner of the call has the choice to exercise the option, pay the exercise (a.k.a., strike) price, and take delivery of the asset. The seller of the call is obligated to deliver the asset and receive the strike price if the buyer exercises.A put option is a contract to sell an asset in the future in which the asset, price, quantity, delivery place and location, are specified in the contract. The owner of the put has the choice to exercise the option, receive the exercise (a.k.a., strike) price, and delivers the asset.The seller of the put is obligated to take delivery and pay the strike price if the buyer exercises.There is a cost to acquiring a call or put. The buyer of the call or put pays the seller a premium.3Option CharacteristicsUnderlying asset: Stock, Bond, Currency, Commodity, Futures contractOption Cost (a.k.a., Premium, Price)Option Type: Call or PutExercise TypeEuropean: Exercise only at maturityAmerican: Exercise anytime up to maturityOption MaturityOption StrikeATM: Strike = Current SpotITM: Strike Spot for putOTM: Strike > Spot for call, Strike X or allow option to expireProfit = Max(St-X, 0)Q-cQWhere c = call premiumLong (buyer) PutPayoff = Max(X-St,0)QExercise if X>St or allow option to expirePayoff = Max(X-St,0)Q-pQWhere p = put premiumSt is the spot price at contract maturity, f is the forward price, X is the option strike priceValuing OptionsThe valuation of option premiums is a difficult problem that requires the following informationOption Type: Call or PutOption Exercise TypeOption MaturityOption StrikeSpot Price of Underlying Asset (Stock Price)Annualized Volatility of Underlying Asset ReturnsAnnualized Dividend Yield of underlying assetTreasury Yield with the same maturity as the option6Google Options as of 2/28/2012GOOG Spot = $614.27 Bloomberg:GOOG Equity OMONGoogle Stock Return Volatility:40 Day Trailing Vol = 30%Bloomberg:GOOG Equity HVG40 Day TrailingGoogle Equity – (No Dividends)Bloomberg:GOOG Equity DESU.S. Gov Yield Curve: 1 Mo = .10%10Bloomberg:IYC1U.S.Summary11Copy into excel12Monte Carlo Simulation and Options ValuationContinuous Comp. Random Walk ModelFunction stocksim_cont(s, r, d, v, h, z)stocksim_cont = s * Exp((r - d - 0.5 * v ^ 2) * h + v * z * h ^ 0.5)End Function----------------------------------------------------------------------------------S: Current Stock Price; d: annualized dividend yield; z: random ~N(0,1); h: time (in years) between stock price changes; if you model stock prices change only once between now and maturity; T=h). You could assume stock prices change daily or monthly, but does not affect the valuation for a european style option.R: Annualized rate of return with same maturity as option. v: Expected annualized volatility of underlying asset returns over life of option13Note: ert is a continuous time future value factor similar to the discrete version (1+r)t E.g. FV=$100e.05*(1/12) = $100.417 and FV=$100*(1.05)^(1/12)= $100.407Random Walk Example: When S=100, r =5%, d= 2%, v=.20, h = 1/12 (1 month) and Z = 1 (and e =2.71828) Continuous Time RW ModelS(t+h)=St*Exp((r-d-0.5* v ^ 2)*h+v*z*h^0.5)=100*Exp((.05-.02-0.5*.2 ^ 2)*(1/12)+.2*1*(1/12)^0.5)=$106.03Discrete Time RW ModelS(t+h)=St*(1+(r-d)*h+Z*v*h^.5)=100*(1+(.05-.02)*(1/12)+1*.2*(1/12)^.5)=$106.02Monte Carlo Valuation ProcessSelect a model for stock prices (GBM-Continuous compound) -- 2.Simulate many possible stock prices -- Generate many Zs and stock prices3. Option Value= PV of the average payoffs where payoffs are Call= max(0,S-K) Put = max(0,K-S) Note: e-rt is a continuous time present value factor similar to the discrete version 1/(1+r)t E.g. PV=$100e-.05*.25 = $98.76 and PV=$100/(1.05)^.25= $98.78 @Risk Monte Carle Valuation10,000 iterations, one-step model15The simulated stock return assumes that the expected return is the risk free rate for a risky asset. This is a computational short-cut to value options. This approach is called “risk-neutral” pricingLaunch @RiskAlternative: Macro Simulation16Black-Scholes-Merton Option Pricing ModelThe Black–Scholes or Black–Scholes–Merton (BSM) model is a mathematical model of a financial market containing certain derivative investment instruments. From the model, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options.  lt is widely used, although often with adjustments and corrections, by options market participants.  Many empirical tests have shown that the Black–Scholes price is "fairly close" to the observed prices, although there are well-known discrepancies.The Black–Scholes was first published by Fischer Black and Myron Scholes in their 1973 paper, "The Pricing of Options and Corporate Liabilities", published in the Journal of Political Economy. They derived a stochastic partial differential equation, now called the Black–Scholes equation, which estimates the price of the option over time. The key idea behind the model is to hedge the option by buying and selling the underlying asset in just the right way and, as a consequence, to eliminate risk. This type of hedge is called delta hedging and is the basis of more complicated hedging strategies such as those engaged in by investment banks and hedge funds.Robert C. Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model, and coined the term "Black–Scholes options pricing model". Merton and Scholes received the 1997 Nobel Prize in Economics for their work. Though ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish Academy.The Black–Scholes formula has only one parameter that cannot be observed in the market: the average future volatility of the underlying asset. Since the formula is increasing in this parameter, it can be inverted to produce an implied “volatility surface" that is then used to calibrate other models.Scholes and Merton went founded the infamous Long Capital Management hedge fund17Source: wikipediaBlack-Scholes FormulaBS Yields a similar value for options as the Monte Carlo approach In Fact, BS is a RW model where prices are continuously changing. This assumption allows BS to use Ito calculus to compute value using a formula.BS is the standard model for valuing plain vanilla, European style, calls and puts even though the assumptions of normality and constant and know volatility are widely criticized.BS is also used to compute implied volatilities of asset returns.18Black-Scholes 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 Function1920Monte Carlo vs. BS ModelValuing Currencies using BSMEnter the spot price of the currency ($/Fx)Substitute the foreign country’s risk-free sovereign rate for the dividend yield21For indirect quotes, such as the Yen/$, convert spot and strike to direct quotes by dividing by 1, and again entering the U.S. sovereign yield for r, and the Japanese sovereign yield for dHedging Euro Sale with Option vs. ForwardSuppose Apple plans to earn €800M euros in 1 year from the launch of the new iPhone in Europe. Apple is considering hedging the position with a options or short forward to sell euros.Furthermore, Apple is considering hedging less than the full amountEstimate the expected USD revenue and hedged values using ATM puts and/or calls222324Asian (Exotic) Option ValuationConsider a call option in which at the end of three months makes a payment based on the arithmetic average of the stock price at the end of months 1, 2, and 3. (This is an arithmetic average price Asian option)We compute the stock prices based on three random numbers Z(1), Z(2), and Z(3). (will need 3 randoms)S1=S0e(r-d-.5*σ^2)*h+ σ*sqrt(h)*Z(1))S2= S1e(r-d-.5*σ^2)*h+ σ*sqrt(h)*Z(2))S3= S2e(r-d-.5*σ^2)*h+ σ*sqrt(h)*Z(3))The payoff of the callPayoffasian=max[(S1+S2+S3)/3-K,0]Repeat 1,000 times and the value of the call isCasian=e-rtAverage{(max[(S1+S2+S3)/3-K,0])}25Example:3-Month, Arithmetic Asian CallLearning ObjectivesCompute the payoffs and profits of plain vanilla option contractsValue options using Monte Carlo Simulation and Black-Scholes modelsComputed hedged and unhedged cashflows using options and forwardsValue arithmetic Asian optionsUse @Risk to value options and compute position risk26