Index, Currency and Futures Options Finance (Derivative Securities) 312 Tuesday, 24 October 2006 Readings: Chapters 13 & 14

Known Dividend Yield  Same probability distribution for stock price at time T if: Stock starts at price S 0 and provides a dividend yield = q Stock starts at price S 0 e –qT and provides no income  Reduce current stock price by dividend yield, then value option as though stock pays no dividends

Option Pricing  Lower Bound for Calls c  S 0 e –qT –Ke –rT  Lower Bound for Puts p  Ke –rT – S 0 e –qT  Put-call Parity p + S 0 e –qT = c + Ke –rT

Black Scholes

Binomial Model  In a risk-neutral world the stock price grows at r – q rather than at r when there is a dividend yield q  The probability, p, of an up movement must therefore satisfy pS 0 u + (1 – p)S 0 d = S 0 e (r-q)T so that:

Index Options  Suppose that: Current value of index is 930, dividend yields of 0.2% and 0.3% expected in first and second months European call option with exercise price of 900 expires in two months Risk-free rate is 8%, volatility is 20% p.a.  What is the price of the option?

Index Options  Using Black Scholes: d 1 = , d 2 = N(d 1 ) = , N(d 2 ) = c = 930 x e –0.03(2/12) – 900 x e –0.082/12 = $51.83

Portfolio Insurance  Suppose the value of the index is S 0 and the strike price is K If a portfolio has a  of 1.0, the portfolio insurance is obtained by buying 1 put option contract on the index for each 100 S 0 dollars held If  is not 1.0, the portfolio manager buys  put options for each 100 S 0 dollars held  K is chosen to give the appropriate insurance level

Portfolio Insurance  Suppose that: Portfolio has a beta of 1.0, worth $500,000 Index currently stands at 1000 Risk-free rate is 12%, dividend yield is 4%, volatility is 22% p.a. Option contract is 100 times the index  What trade is necessary to provide insurance against the portfolio value falling below $450,000 in the next three months?

Portfolio Insurance  Using Black Scholes, p = $6.48 Cost of insurance = 5 x 100 x 6.48 = $3,240  If index drops to 880: Portfolio drops to $440,000 Option payoff = 5 x (900–880) x 100 = $10,000

Portfolio Insurance  What if beta was 2.0? Choose K = 960 (Table 13.2) p = $19.21 Since beta is 2.0, two put contracts required for each $100,000 Cost of insurance = 10 x 100 x = $19,210  If index drops to 880: Portfolio drops to $370,000 Option payoff = 10 x (960–880) x 100 = $80,000 Cost of hedging is higher (more put options, higher K )

Currency Options  Denote foreign interest rate by r f  When a U.S. company buys one unit of the foreign currency it has an investment of S 0 dollars  Return from investing at the foreign rate is r f S 0 dollars  Foreign currency provides a “dividend yield” at rate r f

Currency Option Pricing  Lower Bound for Calls c  S 0 e –r f T –Ke –rT  Lower Bound for Puts p  Ke –rT – S 0 e –r f T  Put-Call Parity p + S 0 e –r f T = c + Ke –rT

Black Scholes

Futures Options  Call futures option allows holder to acquire: Long position in futures Cash amount equal to excess of futures price over strike price at previous settlement  Put futures option enables holder to acquire: Short position in futures Cash amount equal to excess of strike price over futures price at previous settlement

Payoffs  If futures position is closed out immediately: Payoff from call = F 0 – K Payoff from put = K – F 0 where F 0 is futures price at time of exercise

Advantages over Spot Options  Futures contract may be easier to trade than underlying asset  Exercise of the option does not lead to delivery of the underlying asset  Futures options and futures usually trade in adjacent pits at exchange  Futures options may entail lower transaction costs

Put-Call Parity  Strategy I: buy a European call on a futures contract and invest Ke -rT of cash F T ≤ K F T > K Buy Call 0F T – K Invest Ke –rT K K Total K F T

Put-Call Parity  Strategy II: buy a European put futures option, enter a long futures contract, and invest F 0 e -rT F T ≤ K F T > K Long Futures F T – F 0 F T – F 0 Buy Put K – F T 0 Invest F 0 e -rT F 0 F 0 Total K F T

Put-Call Parity  If two portfolios provide the same return, they must cost the same to set up, otherwise an opportunity for arbitrage exists c + Ke -rT = p + F 0 e -rT

Binomial Pricing  Suppose that: 1-month call option on futures has a strike price of 29 In one month the futures price will be either $33 or $28 Futures Price = $33 Option Price = $4 Futures Price = $28 Option Price = $0 Futures price = $30 Option Price = ?

Binomial Pricing  Consider a portfolio: Long  futures, short 1 call futures option  Portfolio is riskless when 3  – 4 = – 2    =  – 4 –2–2

Binomial Pricing  Riskless portfolio: Long 0.8 futures, short 1 call futures option  Value of the portfolio in one month: 3 x  0.8 – 4 = –1.6  Value of portfolio today ( r = 6%): –1.6 e –0.06  1/12) = –1.592  Value of futures is zero, so value of option must be $1.592

Generalisation  A derivative lasts for time T and is dependent on a futures contract F 0 u ƒ u F 0 d ƒ d F0ƒF0ƒ

Generalisation  Consider the portfolio that is long  futures and short 1 derivative  The portfolio is riskless when: F 0 u  F 0  – ƒ u F 0 d  F 0  – ƒ d

Generalisation  Value of portfolio at time T: F 0 u  – F 0  – ƒ u  Value of portfolio today: (F 0 u  – F 0  – ƒ u )e –rT  Cost of portfolio today: – f  Hence ƒ = – [F 0 u  – F 0  – ƒ u ]e –rT

Generalisation  Substituting for  we obtain ƒ = [ pƒ u + (1 – p)ƒ d ]e –rT where

Dividend Yield  Valuing futures is similar to valuing an option on a stock paying a continuous dividend yield Set S 0 = current futures price ( F 0 ) Set q = domestic risk-free rate ( r )  Setting q = r ensures that the expected growth of F in a risk-neutral world is zero

Dividend Yield  Futures contracts require no initial investment  In a risk-neutral world the expected return should be zero  Expected growth rate of futures price is therefore  zero  Futures price can therefore be treated like a stock paying a dividend yield of r

Black’s Model

Futures Price v Spot Price  European Options If a European call (put) futures option matures before futures contract, and futures prices exceed spot prices, it is worth more (less) than the corresponding spot option When futures prices are lower than spot prices (inverted market) the reverse is true

Futures Price v Spot Price  American Options If futures prices are higher than spot prices, an American call (put) on futures is worth more (less) than a similar American call (put) on spot When futures prices are lower than spot prices (inverted market) the reverse is true