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 1,250
  • Suppose 1 contract is exercised when the index level is 1,200
  • What is the payoff?
  • Solution
    • Going long on put
    • 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 is 1,500
  • 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 ) = 1,500 x 0.95 = 1,425
    • optimal contracts = β V _p / 100 S ₀ = 1x1,000,000 / (100 x 1,500) = 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 equals $1,000,000 in value and stock index stands at 1,200
  • 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) / ( 1,200 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

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
  • We can value European options by reducing the stock price to S ₀ e ^–qT and pretend the stock pays no dividend
    • S _new = S ₀ e ^–qT
  • 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 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 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 – r _f ) T}
• Solve for S ₀
• S ₀ = F ₀ e ^–rT e ^{–r _fT}
• 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

• Refer to tree below

• 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
    • a = e ^{( r - q )Δt}
  • Currency option
    • Set q = r _f , so a = e ^{( r - q )Δt} = e ^{( r - r _f )Δt}
  • Futures option
    • Set q = r, so a = e ^{( r - q )Δt} = e ^{( r - r )Δt} = e ⁰ = 1