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Black-Scholes Option Valuation

Black-Scholes Option Valuation. In order to continue on and use the Black-Scholes Option Valuation model we must assume that: The risk free interest rate is constant over the life of the option

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Black-Scholes Option Valuation

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  1. Black-Scholes Option Valuation • In order to continue on and use the Black-Scholes Option Valuation model we must assume that: • The risk free interest rate is constant over the life of the option • Stock price volatility as measured by the stock’s standard deviation is constant over the life of the option • Using Black-Scholes we will also discuss the Intrinsic Value of an option • Intrinsic value is the stock price minus the exercise price or the profit that could be attained by immediate exercise of an in-the-money call option • The actual value of an in-the-money call option will approach the intrinsic value of the option as the stock price increases Intermediate Investments F303

  2. Black-Scholes Option Valuation • The Black-Scholes formula in a world with no dividends is C0 = S0N(d1) – Xe-rTN(d2) Where: • N(d) is, loosely speaking, the probability thathte option will expire in the money (cumulative Normal distribution see pp. 552-3) • C0 is the current option value • e = 2.7128 • d1 = ln(S0/X) + (rf + SD2/2)T ) / (SD * SQRT of T) • d2 = d1 – (SD * SQRT T) Intermediate Investments F303

  3. Black-Scholes Option Valuation • Inputs needed to use B-S method are: • S0 = the current stock price • X = the exercise price • r = the risk free interest rate • T = Time to maturity • SD = stock’s Standard Deviation The first 4 variables can be known with certainty, while standard deviation can be estimated based on historical data. We have already used th e first 4 inputs in the Binomial Pricing Model Intermediate Investments F303

  4. Black-Scholes Option Valuation • To review, N(d) is the probability that a random draw from a normal distribution will be less than d in a cumulative normal distribution, or loosely speaking, the probability that the option will expire in the money • If both N(d) terms are close to 1, you can assume the option will expire in the money and the call will be exercised • In this case, C0 = S0 – Xe-rT • If S0 – X is the Intrinsic value, the above is the Adjusted Intrinsic Value • If both N(d) terms are close to 0, then the value of C0 will be 0 Intermediate Investments F303

  5. Implied Volatility • B-S can be used to find the value of options • If we assume that B-S is an accurate method of pricing options, we can also use B-S, given the market price of the option, to predict the unknown variable • Since Standard deviation can be estimated but not known with certainty, B-S can be used to show the underlying assumption regarding volatility that must be used in the market’s pricing of the option Intermediate Investments F303

  6. Black-Scholes Example • Given the following information, use Black-Scholes to price the option: • Stock Price = $100.00 • Exercise price = $95.00 • Risk free rate = 10% • Dividend Yield = 0% • Time to expiration = 3 months • Standard deviation of stock = 50% • What is the value of d1? d2? Intermediate Investments F303

  7. Black-Scholes Example • Given the following information, use Black-Scholes to price the option: • Stock Price = $14.00 • Exercise price = $10.00 • Risk free rate = 5% • Dividend Yield = 0% • Time to expiration = 6 months • Standard deviation of stock = 50% • What is the value of d1? d2? Intermediate Investments F303

  8. Using Black Scholes to Value a Put • In addition to Put-Call parity you can also use B-S to value a Put P = Xe-rT * [1-N(d2)] – S0 * [1-N(d1)] Intermediate Investments F303

  9. Using Black Scholes to Value a Put - Example • Assume the following: • Time to maturity = 6 months • Standard deviation = 50% per year • Exercise price = $50 • Stock price = $50 • Risk free rate = 10% • Value of a call option = $8.13 • Value the Put using Put-Call Parity • Value the Put using Black- Scholes Intermediate Investments F303

  10. Black-Scholes In-class Exercise • Consider the following: • On February 2, 1996 Microsoft stock closed at $93/share • The one year T-bill rate was 4.82% • Standard deviation on the stock was approximately 32% • Use Black-Scholes to price both a put and a call where: • Exercise price = $100 • Maturity is April 1996 (77 days) Intermediate Investments F303

  11. Black-Scholes In-class Exercise • Consider the following: • On December 20, 1996 Compaq stock closed at $76.75/share • The 6 month T-bill rate was 5.50% • Standard deviation on the stock was approximately 41% • Use Black-Scholes to price both a put and a call where: • Exercise price = $75 • Maturity is April 1997 (120 days) Intermediate Investments F303

  12. Review of Black-Scholes Assumptions and Approach • Black-Scholes Assumptions are: • Perfect Capital Markets, no taxes, transaction costs etc. • Stock does not pay a dividend over the course of the option (although the formula can be adjusted to include dividends) • The Risk free rate and the variance of the stock are: • Constant • Completely predictable • Stock prices are continuous Intermediate Investments F303

  13. Review of Black-Scholes Assumptions and Approach • The Black-Scholes approach is to: • Use a stock and bond to replicate eh value of the call • No arbitrage pricing • Formula is very well known and actually used Intermediate Investments F303

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