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 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

derivation for d1 for Black Scholes equations for stock index

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

derivation for d1 for Black Scholes equations for stock index

  • 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

derivation for d1 for Black Scholes equations for stock index

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

derivation of Black Scholes for stock index futures

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

derivation of Black Scholes for stock index futures

  • 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

Black Scholes equations for stock index futures

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

Derivation of implied dividend yield

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

long range forward

    • 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

short range forward

  • 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 option

  • 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

Black Scholes equations for currency futures option

The Binomial Model for American Options

  • Refer to tree below

Binomial tree

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

Binomial tree equations

  • 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
 

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