# Outline

• Stock Index Options
• European Stock Index Options
• Forward/Futures Prices on a Stock Index
• Implied Dividend Yields
• Currency Options
• The Binomial Model for American Options

# Stock Index Options

• The most popular U.S. indices underlying options include
• The S&P 100 Index (OEX and XEO)
• The S&P 500 Index (SPX)
• The Dow Jones Index times 0.01 (DJX)
• The Nasdaq 100 Index (NDX)
• Characteristics
• Contracts are usually 100 times the index
• They are settled in cash
• OEX is American
• The XEO and all other options are European.
• Index Option Example
• Consider a put option on an index with a strike price of 1250
• Suppose 1 contract is exercised when the index level is 1200
• What is the payoff?
• Solution
• Going long on pput
• Buy at spot and sell at exercise price
• payoff = ( 1250 – 1200 ) x 100 = 5,000
• Using Stock Index Options as Portfolio Insurance
• The index value equals S0 and the strike price is K
• The number of contracts required for portfolio insurance
• number contracts (puts) = β Vportfolio / ( 100 S0 )
• Choose a K that kicks in when the stock index falls below a certain level
• Note:
• Call option – protect or insure portfolio if stock prices rise above K.
• Put option – protect or insure portfolio if stock prices drop below K
• Example 1
• Portfolio has a β = 1.0
• Portfolio has a current worth of 1,000,000
• The index currently stands at 1500
• Calculate the trade that provides insurance against the portfolio value dropping below \$950,000?
• Solution
• %Δ = 100 x ( 950,000 – 1,000,000) / 1,000,000 = –5%
• Exercise price, k = S0 x ( 1 - %Δ ) = 1500 x 0.95 = 1425
• optimal contracts = β Vp / 100 S0 = 1x1,000,000 / (100 x 1500) = 6.67 or 7 contracts
• Example 2
• Beta does not equal one and both the stock index and portfolio earn dividends
• Portfolio has a β = 1..50
• Portfolio current worth equals \$1,000,000 and stock index stands at 1200
• The risk-free rate is 10% per annum
• The dividend yield equals 5% and portfolio return equals 4%
• How many put option contracts should be purchased for portfolio insurance?
• contracts = 1.5 (1,000,000) / ( 1200 x 100) = 12.5 or 13 contracts
• Calculate the expected portfolio value in 6 months
• If index rises to 1300,
• Index return = %Δ in index + dividends = 0.083+ 0.5 / 2 = 0.1083 in six months
• Use CAPM to calculate portfolio returns
• Rp = rf + β ( Rm – rf ) = 0.05 + 1.5 ( 0.1083 – 0.05 ) = 0.1375 in six months
• Deduct the portfolio dividends, so portfolio returns = 0.1375 – 0.02 = 0.1175
• Portfolio value=\$1,000,000 x (1 + 0.1175) = 1,117,500
• An option with a strike price of 1100 will provide protection against a 13.3% fall in the portfolio value
• Excel calculated table values so you may experience rounding error
 Index Value in 3 months Index Return in 3 months (%) CAPM minus dividend yield (%) Expected Porfolio Value in 3 months (\$) 1400 19.2% 24.3% 1,242,500 1300 10.8% 11.8% 1,117,500 1200 2.5% -0.8% 992,500 1100 -5.8% -13.3% 867,500 1000 -14.2% -25.8% 742,500 900 -22.5% -38.3% 617,500

# European Stock Index Options

• Stock prices have the same probability distribution at time T for the following cases:
• The stock starts at price S0 and earns a dividend yield = q
• The stock starts at price S0 e–qT and grows into S0 without dividends
• We can value European options by reducing the stock price to S0 e–qT and pretend the stock pays no dividend
• Snew = S0 e–qT
• Plain vanilla ice cream options
• Lower Bound for European calls: c ≥ max{ S0 – K e–rT, 0 }
• Lower Bound for European puts: p ≥ max{ K e–rt – S0, 0 }
• Put Call Parity: c + K e–rT = p + S0
• Stock index equations
• Just substitute Snew = S0 e–qT into the relations
• Lower Bound for European calls: c ≥ max{ S0 e–qT – K e–rT, 0 }
• Lower Bound for European puts: p ≥ max{ K e–rt – S0 e–qT, 0 }
• Put Call Parity: c + K e–rT = p + S0 e–qT
• If q = 0, then you get the original plain vanilla conditions
• Black Sholes Equations for stock index
• Substitute Snew = S0 e–qT into plain vanilla Black Sholes Equations
• Set q = 0 for plain vanilla ice cream Black Scholes

• Do the same substitution for d1
• Set q = 0 for plain vanilla Black Sholes

• Derive p and d2 similarly
• Complete equations for Black Scholes Equations for stock index is below
• Set q = 0 for the original plain vanilla Black Scholes equations

# Forward/Futures Prices on a Stock Index

• We have a futures/forward on a stock index with dividends q
• F0 = S0 e( r – q ) T
• Solve for S0
• S0 = F0 e–rT eqT
• Use the special equations of Black Scholes with q% dividends
• Substitutes S0 into call option equation

• Substitute S0 into the d1 equation
• Notice the r and q drop out

• All options on forward / futures have two properties
• The terms for c and p are discounted
• The r and q disappears from d1 and d2
• The Black Scholes Equations for stock index futures are below

# Implied Dividend Yields

• Using European calls and puts with the same strike price and time to maturity
• Use Put-Call Parity to solve for q
• These formulas allow term structures of dividend yields OTC
• European options are typically valued using the forward prices
• Estimates of q are not then required)
• American options require the dividend yield term structure

# Currency Options

• Currency Options
• Investors trade currency options on the NASDAQ OMX
• Investors can actively trade over-the-counter (OTC) market
• Corporations use currency options as insurance when they have an foreign exchange rate (FOREX) exposure
• Range Forward Contracts
• Ensure the exchange rate paid or received lies within a certain range
• When holder receives a payment in a foreign currency, he or she sells a put with strike K1 and buys a call with strike K2
• Similar to a long forward contract with a flat range in the payoff
• It would equal a forward if K1 = K2

• When holder receives a currency, it involves buying a put with strike K1 and selling a call with strike K2
• Normally the price of the put equals the price of the cal

• Currency options
• We denote the foreign interest rate by rf
• If the exchange rate is USD / AUD
• The option buys or sells AUD by using USD
• Thus r is the U.S. interest rate while rf is the Australian interest rate
• The return measured in the domestic currency from investing in the foreign currency is rf times the value of the investment
• This shows that the foreign currency provides a yield at rate rf
• Valuing European Currency Options
• We can use the formula for an option on a stock paying a continuous dividend yield
• Set S0 = current exchange rate
• Set q = rƒ

• Black Scholes Equations for currency options
• Using the equation to price a currency forward
• F0 = S0 e( r – rf ) T
• Solve for S0
• S0 = F0 e–rT e–rfT
• Substitutes into Black Scholes for currency options
• Similar to stock index futures options
• For all options on futures / forwards
• Both terms in c and p are discounted
• r and rf disappears from d1 and d2
• Similar to q

# The Binomial Model for American Options

• Refer to tree below

• The option price at each node
• f = [ p fu + ( 1 – p ) fd ] e–rΔt
• The probability, up increment, and down increment are below

• All trees are calculated in the same manner
• They only differ in the the way probability is calculated
• Plain vanilla options
• Set q = 0, so a = e( r - q )Δt = e( r - 0 )Δt = er Δt
• Stock index options
• a = e( r - q )Δt
• Currency option
• Set q = rf, so a = e( r - q )Δt = e( r - rf )Δt
• Futures option
• Set q = r, so a = e( r - q )Δt = e( r - r )Δt = e0 = 1