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FINM2002/7041 (Applied) Derivatives Lecture 7 Options on Stock Indices and Currencies; Options on Futures Contracts Hull et al: Chapter 15 Review of Previous Lecture Last week we examined the evolution of stock prices, and found the expected price of a stock based on this evolution. We also used the Black-Scholes Model to price European put and call options on both dividend and non-dividend paying stocks. 2 Lecture Overview In today’s lecture we will discuss options on stock indices and currencies. We will focus on how to treat dividend yields when examining options. More specifically, with reference to stock index options and FX options, we will address necessary alterations to: – Lower bounds for options; – Put-call parity; – Binomial trees; – Black-Scholes Model for dividend yielding stocks; and, – Portfolio insurance. 3 1. Options and Dividend Yields How do we treat a stock which pays a known dividend yield – Take two identical stocks. Stock 1 pays no dividends, and stock 2 pays a dividend yield at rate q p.a. – Both stocks will give an identical overall return. The return from stock 1 will be in the form of capital gains, and the return from stock 2 will be in the form of capital gains and dividends. 4 1. Options and Dividend Yields The payment of a dividend will cause stock 2’s price to drop by an amount equal to the value of the dividend. So payment of a dividend yield at rate q causes the growth rate to be less than that it would otherwise be, by an amount of q. Thus if the price of a stock, with a dividend yield of q, grows from S0 today to ST at time T: – In the absence of dividends it would grow from S0 today to STeqT at time T; or, – In the absence of dividends it would grow from S0e-qT today to ST at time T. 5 1. Options and Dividend Yields Therefore, we get the same probability distribution for the stock price at time T in each of the following cases: – The stock price starts at price S0 and pays a dividend yield at rate q; and, – The stock price starts at price S0e-qT and pays no dividend yield. 6 1. Options and Dividend Yields The rule which stems from the above example is: – We can value European options lasting for time T on a stock paying a known dividend yield at rate q by reducing the stock price to S0e-qT , and then valuing the option as though the stock pays no dividend. 7 1. Options and Dividend Yields The above rule can be applied to any option on an underlying asset which pays a known dividend yield. In this lecture we focus on two particular options: – Options on Stock Indices; and, – Options on Currencies. 8 1. Options and Dividend Yields Stock indices paying a known dividend yield: – Exchange traded stock indices are available in most exchanges around the world; – In Australia, options on the ASX200 are traded on the Australian Securities Exchange; and, – In this lecture, we assume that dividends on stock indices are paid in terms of a dividend yield. They can also be paid in terms of a dollar value. 9 1. Options and Dividend Yields Options on currencies: – A foreign currency is analogous to a stock paying a known dividend yield. – The owner of the foreign currency option receives a yield equal to the risk-free rate of interest in the foreign currency. – The options are traded on both OTC markets and on an exchange, although they are mainly traded in OTC markets. – The Philadelphia Stock Exchange is the major exchange for FX options. – FX options can be either European or American. 10 2. Lower Bounds for Option Prices The payment of a dividend by the underlying asset, lowers the price of that asset. The reduction in price is good for the holder of a put option as it increases their payoff, while it is bad news for the holder of a call option as it reduces their payoff. Thus, the lower bounds for both call and put options are affected by the payment of dividends. 11 2. Lower Bounds for Option Prices The effect on lower bounds for European calls: – In lecture 4 we saw that the lower bound for a European call on a non-dividend paying stock was: – Substituting S0e-qT for S0, the lower bound for a European call option on a stock which pays a dividend yield is: 0 rTc S Xe ≥ 0 qT rTc S e Xe ≥ 12 2. Lower Bounds for Option Prices We can prove the above result directly by considering the following two portfolios: – Portfolio A: one European call option plus an amount of cash equal to Xe-rT – Portfolio B: e-qT shares with dividends being reinvested in additional shares. – In portfolio A, if the cash is invested at the risk-free interest rate, it will grow to X at time T. If ST > X, the call option is exercised at time T, and portfolio A is worth ST. 13 2. Lower Bounds for Option Prices – If ST < X, the call option expires worthless, and the portfolio is worth X. – Hence, at time T portfolio A is worth: Max(ST,X) – Due to the reinvestment of dividends, portfolio B becomes one share at time T. Thus, it is worth ST at this time. – Therefore, portfolio A must be worth at least as much as portfolio B at time T. – Hence: 0 0 rT qT qT rT c Xe S e or c S e Xe + ≥ ≥ 14 2. Lower Bounds for Option Prices The effect on lower bounds for European puts: – In lecture 4 we saw that the lower bound for a European put on a non-dividend paying stock was: – Substituting S0e-qt for S0, the lower bound for a European put option on a stock which pays a dividend yield is: 0 rTp Xe S ≥ 0 rT qTp Xe S e ≥ 15 2. Lower Bounds for Option Prices As with the call option, this can be proved directly by forming the following two portfolios: – Portfolio C: one European put option plus e-qt shares with dividends on the shares being reinvested in additional shares – Portfolio D: an amount of cash equal to Xe-rT 16 3. Put-Call Parity In order to incorporate the effect of dividend yields on put-call parity: – In lecture 4 we saw that put-call parity for a European call and put option on a non-dividend paying stock was: – Replacing S0 with S0e-qT yields the following put-call parity relationship on a stock paying a dividend yield at rate q: 0 rTc Xe p S + = + 0 rT qTc Xe p S e + = + 17 3. Put-Call Parity We can prove the above result directly by considering the following two portfolios: – Portfolio A: one European call option plus an amount of cash equal to Xe-rT – Portfolio C: one European put option plus e-qT shares with dividends on the shares being reinvested in additional shares. – Both portfolios must be worth max(ST,X) at time T. They must therefore be worth the same today, and the put-call parity result above holds. 18 3. Put-Call Parity The above put-call parity relationship will hold for all options on underlying assets which pay a dividend yield. A slight alteration to the formula is necessary when considering foreign currency options. q is replaced by rf, where rf is the foreign interest rate, and r is the domestic interest rate. Thus, the relationship becomes: 0 fr TrTc Xe p S e + = + 19 4. Binomial Trees To examine the effect of a dividend yield equal to q on the binomial model discussed in Lecture 5, consider the following tree diagram: S0 f S0u fu S0d fd 20 4. Binomial Trees The stock price in this diagram starts at S0 and either moves up to S0u or down to S0d. We define p as the probability of an up movement in a risk-neutral world. The total return provided by the stock in a risk- neutral world must be the risk-free interest rate r. Our stock pays dividends equal to a return of q. Therefore, the return from capital gains must be equal to r-q. 21 4. Binomial Trees p must satisfy: ( ) ( )0 0 0 ( ) 1 r q T r q T pS u p S d S e or e dp u d + = = 22 4. Binomial Trees As we saw in Lecture 5, the value of the derivative at time zero is the expected payoff in a risk-neutral world discounted at the risk free rate: [ ](1 )rT u df e pf p f = + 23 4. Binomial Trees Example: – Consider a six-month put option on a stock with a strike price of $40. Suppose that the initial stock price is $50 and the stock price will either move up to $60 or down to $35 during a six-month period. The risk-free interest rate is 6% p.a. and the stock is expected to provide a dividend yield of 2% p.a. 24 4. Binomial Trees – Solution: u = 1.2 and d = 0.7 If the stock price moves up the payoff is zero, if it moves down the payoff is $5. The value of the option is therefore: (0.06 0.02) 6 /12 0.7 0.6404 1.2 0.7 ep × = = [ ]0.06 6/12 0.6404 0 0.3596 5 1.74e × × + × = 25 5. Black-Scholes for Dividend Yielding Stocks Substituting S0e-qT for S0 in the Black-Scholes formulas we covered in lecture 6 yields the following formula’s to price both European calls and puts on dividend yielding stocks: - 0 1 2 - 2 0 1 - 0 0 0 1 0 2 1 ( ) ( ) ( ) ( ) As: ln =ln 2ln( / ) ( / 2) 2ln( / ) ( / 2) qT rT rT qT qT c S e N d Xe N d p Xe N d S e N d S e S qT X X S X r q Td T S X r q Td d T T σ σ σ σ σ = = + + = + = = 26 5. Black-Scholes for Dividend Yielding Stocks A slight alteration is necessary to the previous Black-Scholes Model when considering FX options. q is replaced by rf, where rf is the foreign interest rate, and r is the domestic interest rate. Black-S

choles becomes: – 0 1 2 – 2 0 1 – 0 0 0 1 0 2 1 ( ) ( ) ( ) ( ) As: ln =ln 2ln( / ) ( / 2) 2ln( / ) ( / 2) f f f r T rT r TrT r T f f f c S e N d Xe N d p Xe N d S e N d S e S r T X X S X r r T d T S X r r T d d T T σ σ σ σ σ = = + + = + = = 27 5. Black-Scholes for Dividend Yielding Stocks As the forward exchange rate can be written as: [ ] [ ] ( ) 0 0 – 0 1 2 2 0 1 0 1 0 2 1 BSOPM becomes: ( ) ( ) ( ) ( ) Where: 2ln( / ) / 2 2ln( / ) / 2 fr r T rT rT F S e c e F N d XN d p e XN d F N d F X Td T F X Td d T T σ σ σ σ σ = = = + = = = 28 6. Portfolio Insurance Portfolio managers can use index options as insurance against the value of their portfolios falling. Consider a manager of a well diversified portfolio which mimics the ASX 200 Index, and as such has similar returns and dividend yield. – The percentage changes in the value of the portfolio can be expected to be approximately the same as the percentage changes in the value of the index. – We know that an index option on the ASX 200 is on 25 times the index, and as such, a long put option on the ASX 200 would allow a manager to hedge against the risk of the value of their portfolio falling. Suppose that the manager’s portfolio is worth $100 million, and the value of the index is currently 5,000. The volatility of the index is 25% p.a., the risk-free interest rate is 9% p.a., the manager wants to hedge the portfolio for the next 3 months, the dividend yield on the index is 5% p.a., and put options can be bought with an exercise price of 5,000. – How many options would the manager need to purchase – How much would this insurance cost – Show that the strategy works 29 6. Portfolio Insurance 1. The manager would need to buy put options on the ASX 200. This would give them a payoff on the option should the value of the index fall. 2. The manager would need to purchase the following number of put options: Value to be hedged 100,000,000 800 Value one put option will hedge 5000 25x = = 30 6. Portfolio Insurance The total cost of the insurance to protect the $100 million portfolio is calculated as follows: 0.09 0.25 -0.05 0.25 2 1 1 2 1 5000 ( ) 5000 ( ) 211.36 800 25 $4,227,200 2ln(5000 / 5000) (0.09 0.05 0.25 / 2)0.25 0.14 0.25 0.25 0.25 0.25 0.02 x xp e N d e N d x x d d d = = = + + = = = = 31 6. Portfolio Insurance To illustrate that the insurance works, consider the situation where the value of the index drops to 4,000 in three months. – The portfolio will be worth: 4,000/5,000 x $100 million = $80 million – The payoff from the options will be: 800 x (5000-4000) x 25 = $20 million – This brings the total value of the portfolio up to the insured value of $100 million (or $95.77 million when the cost of the options is accounted for). 32 Options on Futures Contracts Hull et al: Chapter 16 Lecture Overview In today’s lecture we will discuss options on futures. We will focus on the following topics: – What are futures options ; – Why are futures options popular ; – Put-call parity with futures options; – Bounds for futures options; – Using binomial trees with futures options (Lecture 8); and, – Blacks model for valuing futures options (Lecture 8). 34 1. What are Futures Options A futures option is the right, but not the obligation, to enter into a futures contract at a certain futures price by a certain date. The underlying security of a futures option is a futures contract, not a physical commodity. Exercising these options gives the holder a position in a futures contract. 35 1. What are Futures Options A call futures option: – is the right to enter into a long futures contract at a certain price; and, – when the holder exercises the option they receive a cash amount equal to the excess of the futures price over the strike price. A put futures option: – is the right to enter into a short futures contract at a certain price; and, – when the holder exercises the option they receive a cash amount equal to the excess of the strike price over the futures price. 36 1. What are Futures Options Example: You have bought a December call futures option on greasy wool with a strike price of $10 per bale. – The asset underlying one contract is 100 bales of wool. – As a result of purchasing this call futures option, you have the right to enter into a futures contract at the strike price, which in this case is $10 per bale. We know that most futures options are American, which means you can exercise them before their expiry. You decide to exercise your call futures option on greasy wool and the most recent settlement price for the December futures contract is $14. This means that the futures price when you exercise the call futures option is $14. Hence, F = $14. 37 1. What are Futures Options As a result of exercising this call futures option, you enter into a long futures position at $14 (the most recent settlement price). – You also receive a cash amount equal to the excess of the most recent settlement price for the December futures contract over the strike price: ($14-$10)x100 bales=$400. This follows the formula (F-X) Now, when you decide to close out your December futures contract, the December futures price has risen to $15. The mechanics of your profit from closing out this futures position are identical to the profits from holding an ordinary futures contract. – The gain on the futures contract is the excess of the December futures price at time of exercise over the price of the long futures contract you currently hold: ($15-$14)x100 bales=$100. Thus you make a total gain of $500. $400 on the exercise of the option and the $100 on closing out the position. 38 2. Put-Call Parity In lecture 4 we saw that put-call parity for a European call and put option on a non-dividend paying stock was: To derive a put-call parity relationship for European futures options we consider the following two portfolios: – Portfolio A: a European call futures option plus an amount of cash equal to Xe-rT – Portfolio B: a European put futures option plus a long futures contract plus an amount of cash equal to F0e-rT 0 rTc Xe p S + = + 39 2. Put-Call Parity The cash in portfolio A is invested at the risk-free rate, and grows to X at time T. Let FT be the futures price at maturity of the option. – If FT > X, the call option in portfolio A is exercised and portfolio A is worth FT. – If FT < X, the call is not exercised and portfolio A is worth X. Therefore, the value of portfolio A at time T is therefore: max(FT,X) 40 2. Put-Call Parity The cash in portfolio B is invested at the risk-free rate, and grows to F0 at time T. The put option provides a payoff of max(X - FT,0). The futures contract provides a payoff of FT - F0. The value of portfolio B at time T is therefore: F0 + (FT - F0) + max(X - FT,0) = max(FT,X) 41 2. Put-Call Parity Both portfolios have the same value at time T, namely: max(FT,X). As they are European options and cannot be exercised early, they must have the same value today. The value of portfolio A today is: In portfolio B, the futures contract is worth zero today due to marking to market, so portfolio B is worth: Hence, put-call parity for a futures options is: 0 rT rTc Xe p F e + = + rTc Xe + rT op F e + 42 2. Put-Call Parity For American futures options, the put-call parity relation is: 0 0 rT rTF e X C P F Xe < < 43 2. Put-Call Parity Example: – Imagine that the price of a European call option on greasy wool futures for delivery in six-months is $1 a bale, when the exercise price is $10. – Assume that the greasy wool futures price for delivery in six months is currently $9, and the risk-free rate for an investment that matures in six months is 5% p.a. – What should be the price of European put option on greasy wool futures with the same maturity and exercise date as the call option 44 2. Put-Call Parity Using: We have: 1.00 + 10.00e-0.05x6/12 - 9.00e-0.05x6/12 = $1.98 Therefore, the price of a European put futures option on greasy wool should be $1.98. 0 rT rTc Xe p F e + = + 45 3. Lower Bounds for Futures Options The put-call parity relationship we highlighted above, provides the lower bounds for European call and put options. As the price of a put option cannot be negative, the lower bound for a European call must be: 0 0( ) rT rT rT c Xe F e or c F X e + ≥ ≥ 46 3. Lower Bounds for Futures Options Also, as the price of a call option cannot be negative, the lower bound for a put option is: As American options can be exercised at any time, the lower bounds for American options are: 0 0( ) rT rT rT Xe F e p or p X F e ≤ + ≥ 0 0 C F X and P X F ≥ ≥ 47 Conclusion In today’s lecture we have considered options on underlying assets which pay a dividend yield. With particular reference to stock index options and FX options, we have addressed necessary alterations to: – Lower bounds for option prices; – Put-call parity; – BSOPM pricing formulas; – Binomial trees; and, – Portfolio insurance. We have also discussed options on futures contracts. – Put-call parity with futures options; – Bounds for futures options. 48