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Black Scholes Option Pricing Model

Black Scholes Option Pricing Model. Finance (Derivative Securities) 312 Tuesday, 10 October 2006 Readings: Chapter 12. Random Walk Assumption. Consider a stock worth S

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Black Scholes Option Pricing Model

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  1. Black Scholes Option Pricing Model Finance (Derivative Securities) 312 Tuesday, 10 October 2006 Readings: Chapter 12

  2. Random Walk Assumption • Consider a stock worth S • In a short period of time, Dt, change in stock price is assumed to be normal with mean mSDt and standard deviation wherem is expected return, s is volatility

  3. Lognormal Property • Implies that ln ST is normally distributed with mean: and standard deviation: • Since logarithm of STis normal, ST is lognormally distributed

  4. Lognormal Property where Φ[m,s] is a normal distribution with mean m and standard deviation s

  5. Lognormal Distribution

  6. Expected Return • Expected value of stock price is S0emT • Expected return on stock with continuous compounding is m– s2/2 • Arithmetic mean of returns over short periods of length Dt is m • Geometric mean of returns is m–s2/2

  7. Volatility • Volatility is standard deviation of the continuously compounded rate of return in one year • Standard deviation of return in time Dt: • If stock price is $50 and volatility is 30% per year what is the standard deviation of the price change in one week? • 30 x √(1/52) = 4.16% => 50 x 0.0416 = $2.08

  8. Estimating Volatility • Using 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 values • Historical volatility estimate is:

  9. Concepts Underlying Black Scholes • Option price and stock price depend on the same underlying source of uncertainty • Can form a portfolio consisting of the stock and option which eliminates this source of uncertainty • Portfolio is instantaneously riskless and must instantaneously earn the risk-free rate

  10. Black Scholes Formulae

  11. N(x) Tables • 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 • See tables at the end of the book

  12. Black Scholes Properties • As S0 becomes very large ctends to S –Ke–rTand p tends to zero • As S0 becomes very small c tends to zero and p tends to Ke–rT– S

  13. Black Scholes Example • Suppose that: • Stock price in six months from expiration of option is $42 • Exercise price of option is $40 • Risk-free rate is 10%, volatility is 20% p.a. • What is the price of the option?

  14. Black Scholes Example • Using Black Scholes: • d1 = 0.7693, d2 = 0.6278 • Ke–rT = 40e–0.1(0.5) = 38.049 • If European call, c = 4.76 • If European put, p = 0.81 • Thus, stock price must: • rise by $2.76 for call holder to breakeven • fall by $2.81 for put holder to breakeven

  15. Risk-Neutral Valuation • m does not appear in the Black-Scholes equation • equation is independent of all variables affected by risk preference, consistent with the risk-neutral valuation principle • Assume the expected return from an asset is the risk-free rate • Calculate expected payoff from the derivative • Discount at the risk-free rate

  16. Application to Forwards • 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

  17. Implied Volatility • Implied volatility of an option is the volatility for which the Black-Scholes price equals the market price • One-to-one correspondence between prices and implied volatilities • Traders and brokers often quote implied volatilities rather than dollar prices

  18. Effect of Dividends • European options on dividend-paying stocks valued by substituting stock price less PV of dividends into Black-Scholes formula • Only dividends with ex-dividend dates during life of option should be included • “Dividend” should be expected reduction in the stock price expected

  19. Effect of Dividends • Suppose that: • Share price is $40, ex-div dates in 2 and 5 months, with dividend value of $0.50 • Exercise price of a European call is $40, maturing in 6 months • Risk-free rate is 9%, volatility is 30% p.a. • What is the price of the call?

  20. Effect of Dividends • PV of dividends • 0.5e–0.09(2/12) + 0.5e–0.09(5/12) = 0.9741 • S0 is 39.0259 • d1 = 0.2017, N(d1) = 0.5800 • d2 = –0.0104, N(d2) = 0.4959 • c = 39.0259 x 0.5800 – 40e–0.09(0.5) x 0.4959 = $3.67

  21. American Calls • American call on non-dividend-paying stock should never be exercised early • American call on dividend-paying stock should only ever be exercised immediately prior to an ex-dividend date • Set American price equal to maximum of two European prices: • The 1st European price is for an option maturing at the same time as the American option • The 2nd European price is for an option maturing just before the final ex-dividend date

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