Topic #5: Options Markets I: fundamentals. R. E. Bailey. Department of
Economics. University of Essex. Outline. Contents. 1 Call options and put options.
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EC372 Bond and Derivatives Markets Topic #5: Options Markets I: fundamentals R. E. Bailey Department of Economics University of Essex Outline
Contents 1
Call options and put options
1
2
Payoffs on options
3
3
Option-like assets
6
4
Upper and lower bounds for option prices
6
4.1
7
5
Lower bounds for European option premiums . . . . . . . . . . . . . . . . . . . . .
The put-call parity relationship
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Reading: Economics of Financial Markets, chapter 18
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Call options and put options
Call options and put options • Options provide a right but not an obligation to take an action • Call option: the right but not the obligation to buy, on or before the expiry date, an underlying asset for the exercise price • Put option: the right but not the obligation to sell, on or before the expiry date, an underlying asset for the exercise price • Holders (buyers) and Writers (issuers) – ‘Holder’ owns the option: has ‘long’ position (an asset) – ‘Writer’ has issued the option: has a ‘short’ position (a liability) • Two styles: – American style: exercise at any time prior to expiry date – European style: exercise at expiry date only • Options can die, unexercised – options are exercised only if it is profitable to do so
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Holder Writer
Holder Writer
: :
Call Option May buy asset for Exercise price from Writer. Must sell asset for Exercise price, at Holder’s discretion.
: :
Put Option May sell asset for Exercise price to Writer. Must buy asset for Exercise price, at Holder’s discretion.
Option contracts and markets • The option contract specifies: – Underlying asset – Expiry date, T – Exercise (or, strike) price, X – Whether the option is call or put – Whether the option is American or European – Exotic options: more complicated specification • The market determines: – Underlying asset price, S – Option premium ≡ option price: c = European call; C = American call; p = European put; P = American put. – Interest factor: R(t, T ) ($1 invested at t becomes R(t, T ) at T ) • Time to expiry: τ = T − t, where t = ‘today’ Trading in options • Exchange traded options: standardized, anonymous, guaranteed by exchange authorities. • Over The Counter (OTC) options: private agreements between named counter-parties. • Margin deposits – Option buyer pays the premium, no further obligation – Option writer makes margin deposit – collateral for contingent liability • Termination of an option contract: 1. Option dies, unexercised at T 2. Holder exercises the option 3. Offsetting trade (holder sells or writer buys)
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2
Payoffs on options
Payoffs on options • Assumption: frictionless markets • European call option payoff at T : cT = max[ST − X, 0] – cT = ST − X, if ST ≥ X – cT = 0, if ST < X (option dies) • European put option payoff at T : pT = max[X − ST , 0] – pT = X − ST , if ST ≤ X – pT = 0, if ST > X (option dies) • American options: can be exercised at any time, hence premiums cannot be less than payoff from immediate exercise: – American call C(t) ≥ max[S(t) − X, 0] at every t ≤ T – American put P (t) ≥ max[X − S(t), 0] at every t ≤ T
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Call option
Put option
Payoff
Payoff (a)
6
6
(b)
X @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
-
X
S
X
Net gain 6
-
S
Net gain (c)
6
(d)
X −p
X
@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ X @ @ @ @ @ @ @ @ @ −p
S -
−c
S -
Figure 1: Payoff at exercise for call and put options: long positions. The exercise of a call option by its holder involves purchase of the underlying asset at price X. Exercise of a call option would occur only if the market price of the asset, S, is at least as great as X: S ≥ X. The payoff equals S − X (panel (a)). The exercise of a put option involves the sale of the underlying asset, by the holder to the option-writer, at price X. Exercise of a put option would occur only if the exercise price, X, is at least as great as the market price of the asset, S: X ≥ S. The payoff then equals X − S (panel (b)). Panels (c) and (d) show the net gains for call and put options respectively, found by subtracting the premium paid from the payoff.
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Call option
Put option
(a)
Payoff
Payoff
S @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
X
(b) X
S -
−X
?
?
Net gain
Net gain
(c)
c @ @ @ @ @ @ @ @ @ @ @ X @ S @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
(d)
p -
X
S
p−X
?
?
Figure 2: Payoff at exercise for call and put options: short positions. The exercise of a call option requires the option writer to sell the underlying asset to the option holder at price X. Exercise of a call option would occur only if the market price of the asset, S, is at least as great as X: S ≥ X. The payoff then equals X − S (panel (a)). The exercise of a put option requires the option writer to purchase the underlying asset from the option holder at price X. Exercise of a put option would occur only if the exercise price, X, is at least as great as the market price of the asset, S,: X ≥ S. The payoff then equals S − X (panel (b)). Panels (c) and (d) show the net gain for call and put options respectively, found by adding the premium received to the payoff.
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3
Option-like assets
Option-like assets • Exotic options, allow for many varieties: – Underlying assets, e.g. futures contracts, currencies – Exercise dates, e.g. several specified dates – Payoff rules, amount paid if option is exercised • Option-like assets include: – Warrants, option to buy new equity from a company – Callable bonds: bond issuer’s option to redeem bond before maturity – Convertible bonds, option holder may exercise it to convert into another asset – Rights: form of dividend that confers option to buy new equity
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Upper and lower bounds for option prices
Bounds on option prices • Assumptions: – Frictionless markets, giving force to the Arbitrage Principle – No dividend paid on underlying asset during option’s life • Four simple bounds: 1. Option prices are non-negative: c = 0, p = 0, C = 0, P = 0 Why? Because the option holder is under no obligation to exercise the option, which may always be allowed to die, unexercised. 2. American options worth at least as much as European options: C = c, P = p. Why? Because the holder of an American option has more flexibility (the right to exercise the option before its expiry) than the holder of the equivalent European option. Such flexibility will never impose an extra cost. But note: under the assumptions stated, an American option is always worth at least as much alive as dead, i.e., its market premium is at least as much as the payoff from exercise, C = S − X and P = X − S. There is no extra gain to be had from exercising the option before expiry, T , rather than selling it in the market. Why? Because if C < S − X or P < X − S, in a frictionless market a gain could always be made by buying the option and exercising it immediately (i.e., an arbitrage opportunity). Implication: if exercise before expiry of an American option is observed, infer the existence of market frictions (or, in the case of a call option, the payment of a dividend on the underlying asset before the option’s expiry date). 3. A call option is never worth more than its underlying asset: c 5 C 5 S. Why? If not, there is an arbitrage opportunity: write the option and buy the underlying asset; then, even if the option is exercised a gain is guaranteed; if it is not exercised, the gain is even bigger. 6
4. A put option is never worth more than its exercise price: p 5 P 5 X Also: p 5 X/R (European put never worth more than present value of its exercise price) Why? If not, there is an arbitrage opportunity: write the option and invest the proceeds at the risk-free rate; then even if he option is exercised a gain is guaranteed; if it is not exercised, the gain is even bigger. For the European put: p 5 X/R. Suppose not, p > X/R. As for the American put option, write the option and invest the proceeds, p. The option may be exercised only at expiry in which case the value of the risk-free investment is Rp, which by assumption exceeds X. Hence, a net gain will be made even if the option is exercised, and an even bigger gain, Rp, if it is not.
4.1
Lower bounds for European option premiums
• Lower bound for a European call option:
X c ≥ max 0, S − R(t, T )
(1)
Remember: C = S − X (American call option) • Lower bound for a European put option: p ≥ max 0,
X −S R(t, T )
(2)
Remember: P = X − S (American put option) • With frictionless markets, C = c specialises to C = c. Here’s a proof that C = c. Suppose to the contrary that C > c. Then write one American option and buy one European option, investing the difference C − c at the risk-free rate. At expiry, either (a) both options are exercised, or (b) neither option is exercised (remember that, under the stated assumptions, the American call will not be exercised early, i.e., before expiry). If both options are exercised, the net gain is R(C − c), which is also the gain if neither option is exercised. Thus C > c provides an arbitrage opportunity. See footnote 7 on p. 453 of Economics of Financial Markets. What if C > c is observed? This implies the presence of market frictions or payment of a dividend on the underlying asset prior to expiry of the option. • Generally, P > p If P = p then low values of S ⇒ an arbitrage opportunity Lower bound for European call option: numerical example Suppose that: S = $110, X = $110, c = $5, R(t, T ) = 1.1. (This would be the case, for instance, if the interest rate is 10% per period, T − t = 1, and interest accrues, without compounding, only at date T .) This configuration violates the lower bound, for $5 < 10 = 110 − 110/1.1. Construct the following portfolio: (i) short sell one unit of stock for $110; (ii) buy one call option for $5; and (iii) lend $105 at the risk-free interest rate. At the expiry date, T , the loan is worth $115.50. The stock price at the expiry date, ST may be greater or less than the exercise price, $110. Suppose, for the sake of definiteness, that ST is either 120 or 100, and consider the following table. 7
c 6
Call option
Put option
p 6
X/R
AoAO Region
@ @ @ @ @ @ @ AoAO @ @ @ @ @ @ Region @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
-
X/R
X/R
S
-
S
Figure 3: Absence of Arbitrage Opportunities regions for European options The regions marked AoAO denote pairs of option and asset prices — c and S for calls, p and S for puts — such that it is impossible for investors to make positive arbitrage profits. If option and asset and prices occur outside these regions, it is possible to design investment strategies (i.e. trading in options, the underlying asset and risk-free borrowing or lending) that guarantee arbitrage profits.
Buy one call option: Short sell stock: Make a loan: Net total:
Initial outlay −5 110 −105 0
Outcome: ST = 120 120 − 110 −120 115.5 5.5
Outcome: ST = 100 0 −100 115.5 15.5
Suppose that ST = $120. In this event the option is exercised, the asset acquired being delivered to redeem the short sale. The cash deposited on loan more than covers the exercise price of $110 and the strategy yields a net payoff of $5.5. Suppose, instead, that ST = $100. In this event the option dies unexercised but the unit of stock is purchased (to redeem the short sale) for $100, an amount which, again, is more than covered by the deposit. A net payoff of $15.5 is obtained. Whatever the stock price at the expiry date, the chosen portfolio (which requires a zero initial outlay) yields a positive net payoff in every state (though the size of the payoff depends on the ST if ST < X). Such an outcome is incompatible with the arbitrage principle.
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The put-call parity relationship
The put-call parity relationship • Put-Call Parity for European options c+
X =p+S R(t, T )
• Sometimes called the option conversion relationship • Applies only to European, not American options – Why not? – Because P > p, it is not permissible to substitute P for p. 8
(3)
• The Modigliani-Miller theorem can be interpreted as an application of the put-call parity relationship. Put-call parity: numerical example C ASE (A): c + X/R < p + S. Suppose that: S = $110, X = $110, c = $15, p = $10, R(t, T ) = 1.1. This configuration violates the put-call parity relationship: 115 = 15 + 110/1.1 < 10 + 110 = 120 X c+ < p+S R Construct a portfolio as follows: (i) write one put option; (ii) buy one call option; (iii) short sell one unit of stock; (iv) lend the balance at interest. Assume that the stock price at expiry is either ST = 120 or ST = 100 and consider the following table:
Write one put option: Buy one call option: Short sell stock: Make a loan: Net total:
Initial outlay 10 −15 110 −105 0
Outcome: ST = 120 0 (120 − 110) −120 115.50 5.50
Outcome: ST = 100 (100 − 110) 0 −100 115.50 5.50
Suppose that ST = $120. In this event: (i) the put option dies, costing the investor nothing; (ii) the call option is exercised, with payoff $10; (iii) a unit of stock is purchased for $120 and returned to its lender in fulfilment of the short-sale; (iv) the payoff on the loan, with interest, equals $115.50. Thus, the net payoff on the strategy equals $5.50 (= 10 − 120 + 115.50). Suppose, instead, that ST = $100. In this event: (i) the put option is exercised, with a loss of $10; (ii) the call option dies unexercised; (iii) a unit of stock is purchased for $100 and returned to its lender in fulfilment of the short-sale; (iv) the payoff on the loan, with interest, equals $115.50. Once again, the net payoff on the strategy equals $5.50 (= −10 − 100 + 115.50). Notice that the magnitude of ST (either $120 or $100 in the example) is irrelevant. In each case its value cancels out from the option that is exercised and the purchase of the asset at date T . (The singular case ST = X is trivial, both put and call options having exactly zero value at expiry.) In summary, a portfolio has been constructed with zero initial outlay and which yields a positive payoff in every eventuality. Such an outcome is inconsistent with market equilibrium for a frictionless market in the absence of arbitrage opportunities. C ASE (B): c + X/R > p + S. Suppose now that: S = $110, X = $110, c = $20, p = $5, R = 1.1. This configuration clearly violates the put-call parity: $20 + 110/1.1 = 120 > 115 = 5 + 110. Construct a portfolio as follows: (i) buy one put option; (ii) write one call option; (iii) buy one unit of stock; (iv) borrow the funds needed for a zero initial outlay. Assume once again that the stock price at expiry is either ST = 120 or ST = 100 and consider the following table: 9
Buy one put option: Write one call option: Buy stock: Borrow: Net total:
Initial outlay −5 20 −110 95 0
Outcome: ST = 120 0 (110 − 120) 120 −104.5 5.5
Outcome: ST = 100 (110 − 100) 0 100 −104.5 5.5
Suppose that ST = $120. In this event: (i) the put option dies, unexercised; (ii) the call option is exercised, at a cost to the investor of $10; (iii) the unit of stock is sold for $120; (iv) the loan is repaid, with interest, at cost of $104.50. Thus, the net payoff on the strategy equals $5.50 (= −10 + 120 − 104.50). Suppose, instead, that ST = $100. In this event: (i) the put option is exercised, with a gain of $10; (ii) the call option dies unexercised; (iii) the unit of stock is sold for $100; (iv) the loan is repaid, with interest, at cost of $104.50. Once again, the net payoff on the strategy equals $5.50 (= 10 + 100 − 104.50). Again, a portfolio has been constructed with zero initial outlay and which yields a positive payoff in every eventuality. Such an outcome is inconsistent with market equilibrium for a frictionless market in the absence of arbitrage opportunities. This justifies the put-call parity relationship for European style options.
Summary Summary 1. An option provides its holder with the right but not the obligation to take a specified action at a date, or range of dates, in the future. 2. Call options give their holders the right to buy. Put options give their holders the right to sell 3. European style options can be exercised only at expiry. American style options can be exercised before and at expiry 4. An option holder can allow the option to die, unexercised. The writer is obliged to buy (put) or sell (call) until expiry 5. The arbitrage principle places upper and lower bounds on the prices of options traded in a frictionless market 6. The put-call parity relationship for European style options links option prices with the same X and T written on the same underlying asset: c + X/R = p + S.
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