Valuing European Options

Read Chapter 13 in Hull

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 S t – ln S t – 1
  • A longer time means the mean and volatility will be larger
  • The expected return is μ
  • The interest rate, r, will replace μ
  • Volatility is σ
  • The Lognormal Distribution
    • Returns usually are distributed log-normal 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 log-normally distributed
    • Notation
      • ln S T ~ φ [ ln( S 0 ) + ( μ – σ 2 / 2 )T, σ 2T ]
      • ~ means distributed
      • φ means normally distributed
      • φ[ mean, variance ] is a normal distribution
      • I can subtract 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 2T ]
        • ln( S T / S 0 ) ~ φ [ ( μ – σ 2 / 2 )T, σ 2T ]
    • Stock prices must always be positive
      • Cannot take natural log of a negative number
      • Lognormal distribution is below

lognormal distribution

mean variance for lognormal distribution

  • 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
    • Although the equation looks peculiar by subtracting σ, this is a particular form of Brownian motion
  • 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
    • Then take the n-th root to calculate the geometric average
      • (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 return

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

calculate volatility

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
  • We call the portfolio delta neutral
  • Black-Scholes Equations are below:

Black-Scholes Equations

  • 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

lognormal distribution

    • 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 Black-Scholes 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 σ becomes very large?
    • What happens as T becomes very large?
  • Risk-Neutral Valuation
    • The interest rate replaces the variable μ 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 = S T – K
    • Expected payoff in a risk-neutral 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 option price for Black Scholes

  • Calculate the values for a normal distribution
    • Use interpolation from the normal distribution
    • Interpolation allows you to use four digits from a normal distribution with 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 two 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 Put-Call Parity
    • c + K e –rT = p + S 0

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

The volatility index (VIX) 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
    • S new = S old – 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 an option maturing just before the final ex-dividend date
        • Investor would exercise call before the stock price drops