Valuing Stock Options:The Black-Scholes Model

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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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### 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

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 STis normal, ST is lognormally distributed

고려대 경제학과 대학원 (05-02)

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

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

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?

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:

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

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

고려대 경제학과 대학원 (05-02)

The Black-Scholes Formulas

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

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

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

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

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

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

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

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?

고려대 경제학과 대학원 (05-02)