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
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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)
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Black-Scholes Option Valuation
• Inputs needed to use B-S method are:
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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
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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
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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
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Black-Scholes Example
• Given the following information, use BlackScholes to price the option:
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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?
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Black-Scholes Example
• Given the following information, use BlackScholes to price the option:
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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?
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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)]
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Using Black Scholes to Value a Put - Example
• Assume the following:
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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
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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)
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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)
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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
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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
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