Valuing European Options
Read Chapter 13 in Hull
Outline
 The Assumptions of the BlackScholesMerton Model
 The Concepts Underlying BlackScholes
 An Example
 Implied Volatility
 Dividends
 American Call
The Assumptions of the BlackScholesMerton 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 S_{t} – ln S_{t – 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 lognormal distribution
 Total returns comes from multiplying daily returns
 Compounding
 These assumptions imply ln( S_{T} ) is normally distributed
 mean = ln( S_{0} ) + ( μ – σ^{2} / 2 )T
 standard deviation = σ (T)^{½}
 Because the logarithm of S_{T} is normal while S_{T} is lognormally distributed
 Notation
 ln S_{T} ~ φ [ ln( S_{0} ) + ( μ – σ^{2} / 2 )T, σ^{2}T ]
 ~ means distributed
 φ means normally distributed
 φ[ mean, variance ] is a normal distribution
 I can substract a number from S_{T} that lowers the mean by the same amount
 ln S_{T} – ln S_{0} ~ φ [ ln( S_{0} ) – ln S_{0} + ( μ – σ^{2} / 2 )T, σ^{2}T ]
 ln( S_{T} / S_{0} ) ~ φ [ ( μ – σ^{2} / 2 )T, σ^{2}T ]
 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 S_{0} e ^{μ T}
 The return in a short period Δt is μ Δt
 The expected return on the stock with continuous compounding is μ – σ^{2} / 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 + i_{1}) (1 + i_{2}) (1 + i_{3}) …
 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 S_{0}, S_{1}, . . . , S_{n} 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 u_{i}
 Calculate the historical volatility as:
The Concepts Underlying BlackScholes
 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 riskfree rate
 BlackScholes 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 = S_{0} N(d_{1}) – K e^{–rT} N(d_{2})
 If N(d_{1}) = N(d_{2}) = 1
 Then c = S_{0} – K e^{–rT}
 Which is the amount the call option is in the money
 Example 2
 p = K e^{–rT} N(–d_{2}) – S_{0} N(–d_{1})
 If N(d_{1}) = N(d_{2}) = 1
 Then p = K e^{–rT} – S_{0}
 Which is the amount the put option is in the money
 Properties of BlackScholes Formula
 As S_{0} becomes very large, c becomes very large S_{0} – Ke^{–rT} while p approaches zero
 As S_{0} becomes very small c tends to zero and p tends to Ke^{–rT} – S_{0}
 What happens as s becomes very large?
 What happens as T becomes very large?
 RiskNeutral Valuation
 The variable μ does not appear in the BlackScholes equation
 The equation is independent of all variables affected by risk preference
 This is consistent with the riskneutral valuation principle
 Applying RiskNeutral Valuation
 Assume that the expected return from an asset is the riskfree rate
 Calculate the expected payoff from the derivative
 Discount at the riskfree rate
 Valuing a Forward Contract with RiskNeutral Valuation
 Forward contract, payoff = S_{T} – K
 Expected payoff in a riskneutral world, payoff = S_{0} e^{rT} – K
 Present value of expected payoff is
 E[payoff] = e^{–rT}[ S_{0} e^{rT} – K ] = S_{0} – Ke^{–rT}
An Example
 The European call has the following parameters
 Stock price, S_{0} = 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 d_{1} and d_{2}
 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(d_{1}) = 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 d_{1}
 N(d_{2}) = 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(–d_{1}) = 1 – N(d_{1})
 N(–d_{1}) = N(–0.7149) = 1 – 0.7626 = 0.2374
 N(–d_{2}) = N(–0.6088) = 1 – 0.7287 = 0.2713
 Calculate the price of the call
 c = S_{0} N(d_{1}) – K N(d_{2}) e^{rT} = 100 (0.7626) – 98 (0.7287) e^{–0.1x0.5} = 8.3302
 Calculate the price of the put
 p = K N(–d_{2}) e^{rT} – S_{0} N(–d_{1}) = 98 (0.2713) e^{–0.1x0.5} – 100 (0.2374) = 1.5507
 Note: You can also use the PutCall Parity
 c + K e^{–rT} = p +S_{0}
Implied Volatility
 The implied volatility of an option is the volatility for which the BlackScholes price equals the market price
 Use a method to calculate the volatility of an option
 Put parameters into Black Scholes
 Market price = BlackScholes price
 This is a onetoone 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 500stock 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 dividendpaying stocks are valued by substituting the stock price less the present value of dividends into the BlackScholesMerton formula
 Only dividends with exdividend dates during life of option should be included
 The "dividend" should be the expected reduction in the stock price on the exdividend date
 S_{new} = S_{old} – PV(dividends)
American Call
 An American call on a nondividendpaying 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 dividendpaying stock should only ever be exercised immediately before an exdividend 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 exdividend date
 Investor exercises call before the stock price drops
