valuing stock options the black scholes model n.
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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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the black scholes random walk assumption
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)

the lognormal property
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)

the lognormal property continued
The Lognormal Property(continued)

wherem,s] is a normal distribution with mean m and standard deviation s

고려대 경제학과 대학원 (05-02)

the lognormal distribution
The Lognormal Distribution

고려대 경제학과 대학원 (05-02)

the expected return
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)

the volatility
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)

estimating volatility from historical data
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)

nature of volatility
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)

the concepts underlying black scholes
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)

the black scholes formulas
The Black-Scholes Formulas

고려대 경제학과 대학원 (05-02)

the n x function
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)

risk neutral valuation
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)

applying risk neutral valuation
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)

valuing a forward contract with risk neutral valuation
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)

dividends
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)

dividends example
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)

american calls
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)

slide19
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)