EC372 Bond and Derivatives Markets Topic #5: Options Markets I: fundamentals

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1 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 Lower bounds for European option premiums The put-call parity relationship 8 Reading: Economics of Financial Markets, chapter 18 1 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 1

2 Call Option Holder : May buy asset for Exercise price from Writer. Writer : Must sell asset for Exercise price, at Holder s discretion. Put Option Holder : May sell asset for Exercise price to Writer. 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, 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, 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) 2

3 2 Payoffs on options Payoffs on options Assumption: frictionless markets European call option payoff at T : c T = max[ T, 0] c T = T, if T c T = 0, if T < (option dies) European put option payoff at T : p T = max[ T, 0] p T = T, if T p T = 0, if T > (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[(t), 0] at every t T American put P (t) max[ (t), 0] at every t T 3

4 Payoff Call option (a) Payoff Put option (b) Net gain (c) Net gain p (d) c p 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. Exercise of a call option would occur only if the market price of the asset,, is at least as great as :. The payoff equals (panel (a)). The exercise of a put option involves the sale of the underlying asset, by the holder to the option-writer, at price. Exercise of a put option would occur only if the exercise price,, is at least as great as the market price of the asset, :. The payoff then equals (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. 4

5 Call option Put option Payoff (a) Payoff (b) Net gain c (c) Net gain p (d) p 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. Exercise of a call option would occur only if the market price of the asset,, is at least as great as :. The payoff then equals (panel (a)). The exercise of a put option requires the option writer to purchase the underlying asset from the option holder at price. Exercise of a put option would occur only if the exercise price,, is at least as great as the market price of the asset,,:. The payoff then equals (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. 5

6 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 4 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 2. American options worth at least as much as European options: C c, P p. 3. A call option is never worth more than its underlying asset: c C. 4. A put option is never worth more than its exercise price: p P Also: p /R (European put never worth more than present value of its exercise price) 4.1 Lower bounds for European option premiums Lower bound for a European call option: c max Lower bound for a European put option: p max [ 0, ] R(t, T ) [ 0, ] R(t, T ) (1) (2) 6

7 c Call option AoAO Region /R p Put option AoAO Region /R /R Figure 3: Absence of Arbitrage Opportunities regions for European options The regions marked AoAO denote pairs of option and asset prices c and for calls, p and 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. If R > 1, /R < c /R Hence, C = c (no advantage for early exercise of American call) Generally, P > p If P = p then low values of an arbitrage opportunity Lower bound for European call option: numerical example uppose that: = $110, = $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 = /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 $ The stock price at the expiry date, T may be greater or less than the exercise price, $110. uppose, for the sake of definiteness, that T is either 120 or 100, and consider the following table. Initial Outcome: Outcome: outlay T = 120 T = 100 Buy one call option: hort sell stock: Make a loan: Net total: uppose that T = $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. uppose, instead, that T = $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 T if T < ). uch an outcome is incompatible with the arbitrage principle. 7

8 5 The put-call parity relationship The put-call parity relationship Put-Call Parity for European options c + ometimes 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. R(t, T ) = p + (3) The Modigliani-Miller theorem can be interpreted as an application of the put-call parity relationship. Put-call parity: numerical example CAE (A): c + /R < p +. uppose that: = $110, = $110, c = $15, p = $10, R(t, T ) = 1.1. This configuration violates the put-call parity relationship: 115 = /1.1 < = 120 c + R < p + 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 T = 120 or T = 100 and consider the following table: Initial Outcome: Outcome: outlay T = 120 T = 100 Write one put option: 10 0 ( ) Buy one call option: 15 ( ) 0 hort sell stock: Make a loan: Net total: uppose that T = $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 $ Thus, the net payoff on the strategy equals $5.50 (= ). uppose, instead, that T = $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 $ Once again, the net payoff on the strategy equals $5.50 (= ). 8

9 Notice that the magnitude of T (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 T = 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. uch an outcome is inconsistent with market equilibrium for a frictionless market in the absence of arbitrage opportunities. CAE (B): c + /R > p +. uppose now that: = $110, = $110, c = $20, p = $5, R = 1.1. This configuration clearly violates the put-call parity: $ /1.1 = 120 > 115 = 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 T = 120 or T = 100 and consider the following table: Initial Outcome: Outcome: outlay T = 120 T = 100 Buy one put option: 5 0 ( ) Write one call option: 20 ( ) 0 Buy stock: Borrow: Net total: uppose that T = $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 $ Thus, the net payoff on the strategy equals $5.50 (= ). uppose, instead, that T = $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 $ Once again, the net payoff on the strategy equals $5.50 (= ). Again, a portfolio has been constructed with zero initial outlay and which yields a positive payoff in every eventuality. uch 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. ummary ummary 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 9

10 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 and T written on the same underlying asset: c + /R = p +. 10

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