# Outline

• The Assumptions of the Black-Scholes-Merton Model
• The Concepts Underlying Black-Scholes
• An Example
• Implied Volatility
• Dividends
• American Call

# The Assumptions of the Black-Scholes-Merton Model

• A stock price S
• In a short period of time of length, Δt
• The return on the stock ( Δ S / S ) is assumed
• To be normally distributed with mean μ Δt
• Standard deviation σ(Δt)½
• ( Δ S / S ) = ln St – ln St – 1
• A longer time means the mean and volatility will be larger
• The expected return is μ
• Volatility is σ
• The interest rate, r, will replace μ
• The Lognormal Distribution
• Returns usually are distributed log-normal distribution
• Total returns comes from multiplying daily returns
• Compounding
• These assumptions imply ln( ST ) is normally distributed
• mean = ln( S0 ) + ( μ – σ2 / 2 )T
• standard deviation = σ (T)½
• Because the logarithm of ST is normal while ST is log-normally distributed
• Notation
• ln ST ~ φ [ ln( S0 ) + ( μ – σ2 / 2 )T, σ2T ]
• ~ means distributed
• φ means normally distributed
• φ[ mean, variance ] is a normal distribution
• I can substract a number from ST that lowers the mean by the same amount
• ln ST – ln S0 ~ φ [ ln( S0 ) – ln S0 + ( μ – σ2 / 2 )T, σ2T ]
• ln( ST / S0 ) ~ φ [ ( μ – σ2 / 2 )T, σ2T ]
• Stock prices must always be positive
• Cannot take natural log of a negative number
• Lognormal distribution is below

• The Expected Return
• The expected value of the stock price at time T is S0 e μ T
• The return in a short period Δt is μ Δt
• The expected return on the stock with continuous compounding is μ – σ2 / 2
• This reflects the difference between arithmetic and geometric means
• Stock Return Example
• Suppose a stock has the following returns: 15%, 20%, 30%, -20% and 25%
• The arithmetic mean of the returns is 14%
• The returned that would actually be earned over the five years
• The geometric mean = 12.4%
• Geometric reflect compounding
• (1 + i1) (1 + i2) (1 + i3) …
• The Volatility
• The volatility is the standard deviation of the continuously compounded rate of return in 1 year
• The standard deviation of the return in time Δ t is σ ( Δ t )½
• If a stock price is \$80 and its volatility is 10% per year what is the standard deviation of the price change in one week?
• standard deviation (%) = 10% (1 / 52)½ = 1.3868%
• standard deviation (\$) = 0.013868 x \$80 = \$1.11
• Nature of Volatility
• Volatility is usually much greater when the market is open (i.e. the asset is trading) than when it is closed
• For this reason time is usually measured in "trading days" not calendar days when options are valued
• Estimating Volatility from historical data
• Take observations S0, S1, . . . , Sn at intervals of τ years
• For example, weekly data τ = 1 / 52
• Calculate the continuously compounded return in each interval as:

• Calculate the standard deviation, s, from the ui
• Calculate the historical volatility as:

# The Concepts Underlying Black-Scholes

• The option price and the stock price depend on the same underlying source of uncertainty
• We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty
• The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate
• Black-Scholes Equations are below:

• The N(x) Function
• N(x) is the probability that lies between 0 and 1
• A standard normal distribution with a mean of zero and a standard deviation of 1,or N(0, 1)
• See tables at the end of the book

• Example
• c = S0 N(d1) – K e–rT N(d2)
• If N(d1) = N(d2) = 1
• Then c = S0 – K e–rT
• Which is the amount the call option is in the money
• Example 2
• p = K e–rT N(–d2) – S0 N(–d1)
• If N(d1) = N(d2) = 1
• Then p = K e–rT – S0
• Which is the amount the put option is in the money
• Properties of Black-Scholes Formula
• As S0 becomes very large, c becomes very large S0 – Ke–rT while p approaches zero
• As S0 becomes very small c tends to zero and p tends to Ke–rT – S0
• What happens as s becomes very large?
• What happens as T becomes very large?
• Risk-Neutral Valuation
• The variable μ does not appear in the Black-Scholes equation
• The equation is independent of all variables affected by risk preference
• This is consistent with the risk-neutral valuation principle
• Applying Risk-Neutral Valuation
• Assume that the expected return from an asset is the risk-free rate
• Calculate the expected payoff from the derivative
• Discount at the risk-free rate
• Valuing a Forward Contract with Risk-Neutral Valuation
• Forward contract, payoff = ST – K
• Expected payoff in a risk-neutral world, payoff = S0 erT – K
• Present value of expected payoff is
• E[payoff] = e–rT[ S0 erT – K ] = S0 – Ke–rT

# An Example

• The European call has the following parameters
• Stock price, S0 = 100
• Exercise price, K = 98
• Volatility, σ = 15%
• Interest rate, r = 10%
• Maturity, T = 6 months
• Calculate the European option prices for a call, c, and a put, p
• First, calculate the d1 and d2

• Calculate the values for a normal distribution
• Use interpolation from the normal distribution
• Interpolation allows you to use four digits from a normal distribution wih two decimal places
• For call option
• N(d1) = N(0.71) + 0.49 [ N(0.72) – N(0.71) ] = 0.7611 + 0.49 [ 0.7642 – 0.7611 ] = 0.7626
• The N(0.71) comes from the normal distribution table
• Then look up the next highest value, which is N(0.72)
• The 0.49 is the next digits of d1
• N(d2) = N(0.60) + 0.88 [ N(0.61) – N(0.60] = 0.7257 + 0.88 [ 0.7291 – 0.7257 ] = 0.7287
• For the put option
• Use the property, N(–d1) = 1 – N(d1)
• N(–d1) = N(–0.7149) = 1 – 0.7626 = 0.2374
• N(–d2) = N(–0.6088) = 1 – 0.7287 = 0.2713
• Calculate the price of the call
• c = S0 N(d1) – K N(d2) e-rT = 100 (0.7626) – 98 (0.7287) e–0.1x0.5 = 8.3302
• Calculate the price of the put
• p = K N(–d2) e-rT – S0 N(–d1) = 98 (0.2713) e–0.1x0.5 – 100 (0.2374) = 1.5507
• Note: You can also use the Put-Call Parity
• c + K e–rT = p +S0

# Implied Volatility

• The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price
• Use a method to calculate the volatility of an option
• Put parameters into Black Scholes
• Market price = Black-Scholes price
• This is a one-to-one correspondence between prices and implied volatilities
• Traders and brokers often quote implied volatilities rather than dollar prices
• Volatility Index (VIX)
• The Chicago Board of Options Exchange (CBOE)
• Represents a measure of risk and volatility, otherwise known as the investor's fear gauge.
• If the VIX equals 20, the investors expect the S&P 500-stock index to swing by 20% over the next 12 months.
• If the VIX increases, then investors become more pessimistic and the financial markets become more volatile.
• Some economists and analysts use the VIX as a recession indicator
• During the 2008 Financial Crisis, the VIX peaked at 60, and the stock markets lost roughly half their market value during 2009
• The VIX Index of S&P 500 from Yahoo Finance

# Dividends

• European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into the Black-Scholes-Merton formula
• Only dividends with ex-dividend dates during life of option should be included
• The "dividend" should be the expected reduction in the stock price on the ex-dividend date
• Snew = Sold – PV(dividends)

# American Call

• An American call on a non-dividend-paying stock should never be exercised early
• Based on a famous proof
• Only for plain vanilla options
• Thus, American and European calls are equivalent, C=c
• An American call on a dividend-paying stock should only ever be exercised immediately before an ex-dividend date
• Spot price for stock falls after corporation pays dividend
• Black's Approximation for American Calls with dividends
• Calculate the American call price equal to the maximum of two European prices
• The 1st European price is the option maturing at the same time as the American option
• Investors exercises call on the same date as the European
• The 2nd European price is ther an option maturing just before the final ex-dividend date
• Investor exercises call before the stock price drops