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

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

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

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

Intermediate Investments F303 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

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

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

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

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

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

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

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

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