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Valuing Stock Options:The Black-Scholes Model

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 D t the change in the stock price is assumed to be normal with mean m S D t and standard deviation

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Valuing Stock Options:The Black-Scholes Model

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  1. Valuing Stock Options:The Black-Scholes Model ECO760

  2. 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 고려대 경제학과 대학원 (05-02)

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

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

  5. The Lognormal Distribution 고려대 경제학과 대학원 (05-02)

  6. 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 고려대 경제학과 대학원 (05-02)

  7. 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? 고려대 경제학과 대학원 (05-02)

  8. 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: 고려대 경제학과 대학원 (05-02)

  9. 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 고려대 경제학과 대학원 (05-02)

  10. 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 고려대 경제학과 대학원 (05-02)

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

  12. 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 고려대 경제학과 대학원 (05-02)

  13. 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 고려대 경제학과 대학원 (05-02)

  14. 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 고려대 경제학과 대학원 (05-02)

  15. 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 고려대 경제학과 대학원 (05-02)

  16. 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 고려대 경제학과 대학원 (05-02)

  17. 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 고려대 경제학과 대학원 (05-02)

  18. 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 고려대 경제학과 대학원 (05-02)

  19. 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? 고려대 경제학과 대학원 (05-02)

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