INSEAD Library

Includes bibliographical references and index

Digitized

Options Princing: Black-Scholes Made Easy Contents The true logic of this world is in the calculus of probabilities Getting Up and Running Volatility Stock prices are volatile Volatility means that a stock's future price path is uncertain The more volatile a stock, the more uncertain its future value An option can make you a ton of money or you can lose it all To understand European options, we use Black-Scholes. To understand American options, we add Black's approximation. 17 19 21 23 27 ix 1 15 Every Financial Forecast Is a Probability Distribution A forecast for a stock is a bell-shaped curve Different forecasts look different You can translate your estimate of possible future prices into a forecast You are 99.7% certain the outcome will be within the curve One chance in 10 that price will be in any given decile You can translate a forecast into potential price paths Monte Carlo simulations show relationship between paths and forecast From stock's historical returns, calculate historical standard deviation 31 33 35 37 39 41 43 45 29 Black-Scholes Assumptions [Part I) Animations drawn embody several assumptions Continuously compounded returns are normally distributed Stock-price changes are lognormally distributed Price paths are characterized by geometric brownien motion Volatility is constant over the investment horizon 49 51 53 55 57 47 Working with Geometric or Continuously Compounded Rates of Return May not be your customary way of thinking Lose all your money, rate of return is negative infinity Continuously compounded return on portfolio? Convert simple interest to continuously compounded Find the present value of a future dollar amount 61 63 65 69 71 59 Expected Return Expected return is average of all returns in probability distribution Stock's expected return is median plus half standard deviation squared Expected return varies with time Uncertainty varies with square root of time Why volatility varies not with time but with square root of time Is your portfolio manager talking holding-period returns? 75 77 79 81 83 85 73 How Dividends Affect Price Paths and Forecasts Lock random seed lets you create the same price path with variations Dividend payments reduce the price of a stock Dividends shift price probability distribution down A dividend yield shifts price probability distribution down 89 91 93 95 87 Option Outcomes, Probability Distributions. and Expected Returns A call gives you the right to buy a stock at a pre-set price Simulate potential outcomes of investing in a call Histogram approximates probability distribution for option Option's expected return is average of returns in probability distribution Color deciles link stock forecast to option forecast At extremes of probability distributions, divide into more intervals Put gives you right to sell a stock at a pre-set price Simulate potential outcomes of investing in put Calculate put's probability of profit and expected return If you're thinking and counting trading days, set days per year to 252 Does the dxpression probability density function make your brain hurt? 99 101 104 105 107 109 111 113 115 117 119 97 Option Pricing Making options valuation intuitively accessible An option's probability-weighted net present value It's like doing discounted cash-flow analysis in corporate finance Black-Scholes value is expected cost of setting up hedge Delta hedging keeps exposures in balance at no cost beyond setup Animation calculates cost of setting up delta hedge 123 125 127 131 133 143 121 Black-Scholes Assomptions (Part II) Delta hedging brings in additional assumptions Delta hedging in practice differs from in theory Black-Scholes assumptions envision a risk-neutral world Black-Scholes sets expected return equal to risk-free rate For strike prices at extremes of wide distributions, use more intervals Black-Scholes value of a put What to remember about hedging 147 149 151 153 155 157 163 145 Value of Early Exercise of American Options Black's Approximation for Valuing American Options Right of early exercise gives American options greater value than European If option has tinte value, don't exercise it early Out-of-the-money options have only tinte value As put goes deep into the money, may be advantageous to exercise Option's value may be its early-exercise value--Black's approximation When to exercise deep-in-money put if underlying pays lumpy dividends? May be optimal to exercise on last ex-dividend date Option value depends upon location of little squares relative to strike price What if put goes deep into money and underlying pays dividend yield? What if cal' goes deep into money and underlying pays lumpy dividends? Maybe exercise on last day before underlying goes ex-dividend for last tinte What if call goes deep into money and underlying pays dividend yield? Depends on yield, time value, volatility, expected return, risk-free rate If call on underlying that pays no dividends, never exercise early If no dividends, European and American calls have same value 167 169 171 173 175 177 179 181 183 185 165 187 189 191 193 193 Sensitivity of Option Values to Changes in Volatility, Spot Price of Underlying, Time to Expiration, and Risk-Free Rate How many little squares are above or below the strike price? How far? Deeper into money, less sensitive option value is to changes Farther out of money, more sensitive to changes , Vega--If volatility increases, value of call goes up A Delta--When spot price increases, value of call goes up 0, Theta--If underlying pays no dividends, call value goes down over time P, Rho--Increase in risk-free rate increases median return. Call value goes up. V, Vega--If Volatility increases, distribution spreads and drops. Put value goes up. A, Delta--When spot price increases, value of put goes clown 0, Theta--As time passes, put's value goes down. Usually! P, Rho--Increase in risk-free rate increases median return. Put value goes down. 197 199 201 201 203 205 207 209 211 213 215 195 Using Options to Leverage Your Expected Return From Black-Scholes Value, extract stock's implied volatility Implied volatility is market-equilibrium estimate of uncertainty Draw risk-neutralized, market-equilibrium forecast for stock If agree, then stock and option have same expected return If disagree, then use option to leverage expected return 219 219 221 223 225 217 Black-Scholes Assomptions (Part III) Does behavior of financial markets conform to assumptions? Bid and ask prices give different implied volatilities Do option prices imply constant volatility? Different strike prices give us volatility smile Different expiration dates give us term structure of volatility Be sensitive to ways in which markets may not conform Theoreticians keep building alternative models 229 231 233 235 237 239 241 227 Using the Animations to Assess Option Opportunities Use options to leverage your expected return Market-equilibrium forecasts Calculate your forecast without dividends for a call's underlying Simulate potential price paths of a call's underlying Enter dividend schedule for call's underlying Calculate call's probabilities of profit and expected returns Simulate call's potential investment outcomes If you think somebody's bubble is about to burst, buy puts Calculate your forecast without dividends for a put's underlying Enter dividend schedule for put's underlying Calculate puts' probabilities of profit and expected returns Simulate put's potential investment outcomes 245 247 251 255 257 259 261 263 267 269 271 273 243 What We Didn't Tell You You can sell what you've got.You can sell naked. Sell covered. Construct spreads, strips, straps, and strangles. 277 279 275 Using the Animation to 5e11 Options to Others Conversation with animations 283 281 Risk Management and Value at Risk Your value at risk 295 293 An Investment Strategy An investment strategy that aliows you to express your views and have your portfolio's value never go down Invest risk free an amount that interest will grow back to original portfolio value Translate your beliefs into a forecast Invest foregone interest in options Does this strategy make sense? 299 301 305 309 309 297 Navigating Through the Animations Index 311 332