Stock Index and Currency Options
Read Chapter 15 in Hull
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 S_{0} and the strike price is K
 The number of contracts required for portfolio insurance
 number contracts (puts) = β V_{portfolio} / ( 100 S_{0} )
 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 = S_{0} x ( 1  %Δ ) = 1500 x 0.95 = 1425
 optimal contracts = β V_{p} / 100 S_{0} = 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 riskfree 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
 R_{p} = r_{f} + β ( R_{m} – r_{f} ) = 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 S_{0} and earns a dividend yield = q
 The stock starts at price S_{0} e^{–qT} and grows into S_{0} without dividends
 We can value European options by reducing the stock price to S_{0} e^{–qT} and pretend the stock pays no dividend
 Plain vanilla ice cream options
 Lower Bound for European calls: c ≥ max{ S_{0} – K e^{–rT}, 0 }
 Lower Bound for European puts: p ≥ max{ K e^{–rt} – S_{0}, 0 }
 Put Call Parity: c + K e^{–rT} = p + S_{0}
 Stock index equations
 Just substitute S_{new} = S_{0} e^{–qT} into the relations
 Lower Bound for European calls: c ≥ max{ S_{0} e^{–qT} – K e^{–rT}, 0 }
 Lower Bound for European puts: p ≥ max{ K e^{–rt} – S_{0} e^{–qT}, 0 }
 Put Call Parity: c + K e^{–rT} = p + S_{0} e^{–qT}
 If q = 0, then you get the original plain vanilla conditions
 Black Sholes Equations for stock index
 Substitute S_{new} = S_{0} e^{–qT} into plain vanilla Black Sholes Equations
 Set q = 0 for plain vanilla ice cream Black Scholes
 Do the same substitution for d_{1}
 Set q = 0 for plain vanilla Black Sholes
 Derive p and d_{2} 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
 F_{0} = S_{0} e^{( r – q ) T}
 Solve for S_{0}
 S_{0} = F_{0} e^{–rT} e^{qT}
 Use the special equations of Black Scholes with q% dividends
 Substitutes S_{0} into call option equation
 Substitute S_{0} into the d_{1} 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 d_{1} and d_{2}
 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 PutCall 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 overthecounter (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 K_{1} and buys a call with strike K_{2}
 Similar to a long forward contract with a flat range in the payoff
 It would equal a forward if K_{1} = K_{2}
 When holder receives a currency, it involves buying a put with strike K_{1} and selling a call with strike K_{2}
 Normally the price of the put equals the price of the cal
 Currency options
 We denote the foreign interest rate by r_{f}
 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 r_{f} is the Australian interest rate
 The return measured in the domestic currency from investing in the foreign currency is r_{f} times the value of the investment
 This shows that the foreign currency provides a yield at rate r_{f}
 Valuing European Currency Options
 We can use the formula for an option on a stock paying a continuous dividend yield
 Set S_{0} = current exchange rate
 Set q = r_{ƒ}
 Black Scholes Equations for currency options
 Using the equation to price a currency forward
 F_{0} = S_{0} e^{( r – rf ) T}
 Solve for S_{0}
 S_{0} = F_{0} 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 r_{f} disappears from d_{1} and d_{2}
 Similar to q
The Binomial Model for American Options
 The option price at each node
 f = [ p f_{u} + ( 1 – p ) f_{d} ] 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} = e^{r Δt}
 Stock index options
 Currency option
 Set q = r_{f}, 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} = e^{0} = 1
