Valuing Stock Options:The Black-Scholes Model ECO760

The Black-Scholes Random Walk Assumption Consider a stock whose price is S In a short period of time of length Dt the change in the stock price is assumed to be normal with mean mSDt and standard deviation m is expected return and s is volatility 재무경제학특수연구 고려대 경제학과 대학원 (05-02)

The Lognormal Property These assumptions imply ln ST is normally distributed with mean: and standard deviation: Because the logarithm of ST is normal, ST is lognormally distributed 재무경제학특수연구 고려대 경제학과 대학원 (05-02)

The Lognormal Property (continued) where m,s] is a normal distribution with mean m and standard deviation s 재무경제학특수연구 고려대 경제학과 대학원 (05-02)

The Lognormal Distribution 재무경제학특수연구 고려대 경제학과 대학원 (05-02)

The Expected Return The expected value of the stock price is S0emT The expected return on the stock with continuous compounding is m – s2/2 The arithmetic mean of the returns over short periods of length Dt is m The geometric mean of these returns is m – s2/2 재무경제학특수연구 고려대 경제학과 대학원 (05-02)

The Volatility The volatility is the standard deviation of the continuously compounded rate of return in 1 year The standard deviation of the return in time Dt is If a stock price is $50 and its volatility is 30% per year what is the standard deviation of the price change in one week? 재무경제학특수연구 고려대 경제학과 대학원 (05-02)

Estimating Volatility from Historical Data Take observations S0, S1, . . . , Sn at intervals of t years Define the continuously compounded return as: Calculate the standard deviation, s , of the ui ´s The historical volatility estimate is: 재무경제학특수연구 고려대 경제학과 대학원 (05-02)

Nature of Volatility Volatility is usually much greater when the market is open (i.e. the asset is trading) than when it is closed For this reason time is usually measured in “trading days” not calendar days when options are valued 재무경제학특수연구 고려대 경제학과 대학원 (05-02)

The Concepts Underlying Black-Scholes The option price and the stock price depend on the same underlying source of uncertainty We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate 재무경제학특수연구 고려대 경제학과 대학원 (05-02)

The Black-Scholes Formulas 재무경제학특수연구 고려대 경제학과 대학원 (05-02)

The N(x) Function N(x) is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x Tables for N can be used with interpolation For example, N(0.6278) = N(0.62) + 0.78[N(0.63) - N(0.62)] = 0.7324+0.78*(0.7357-0.7324) = 0.7350 재무경제학특수연구 고려대 경제학과 대학원 (05-02)

Risk-Neutral Valuation The variable m does not appear in the Black-Scholes equation The equation is independent of all variables affected by risk preference This is consistent with the risk-neutral valuation principle 재무경제학특수연구 고려대 경제학과 대학원 (05-02)

Applying Risk-Neutral Valuation Assume that the expected return from an asset is the risk-free rate Calculate the expected payoff from the derivative Discount at the risk-free rate 재무경제학특수연구 고려대 경제학과 대학원 (05-02)

Valuing a Forward Contract with Risk-Neutral Valuation Payoff is ST – K Expected payoff in a risk-neutral world is SerT – K Present value of expected payoff is e-rT[SerT – K]=S – Ke-rT 재무경제학특수연구 고려대 경제학과 대학원 (05-02)

Dividends European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into the Black-Scholes formula Only dividends with ex-dividend dates during life of option should be included The “dividend” should be the expected reduction in the stock price expected 재무경제학특수연구 고려대 경제학과 대학원 (05-02)

Dividends –Example Consider a European call on a stock with ex-dividend dates in two months and five months. The div. on each ex-div date is expected to be $0.50. The current share price is $40, the exercise price is $40, the stock price volatility is 30% pa, the risk-free interest rate is 9% pa, and maturity is six months. Calculate the call price 재무경제학특수연구 고려대 경제학과 대학원 (05-02)

American Calls An American call on a non-dividend-paying stock should never be exercised early An American call on a dividend-paying stock should only ever be exercised immediately prior to an ex-dividend date 재무경제학특수연구 고려대 경제학과 대학원 (05-02)

Quiz Calculate the price of a three-month European put option on a non-dividend-paying stock with a strike price of $50 when the current stock price is $50, the risk-free interest rate is 10% pa, and the volatility is 30% pa What if a dividend of $1.50 is expected in two months? 재무경제학특수연구 고려대 경제학과 대학원 (05-02)