Black-Scholes Option Valuation

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 stocks
  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

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)

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 stocks 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

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

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 markets pricing of the
  option

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?

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?

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)

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

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)

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)

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

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