1 Further Pricing Relationships on Options

Transcription

1 1 Further Pricing Relationships on Options 1.1 The Put Option with a Higher Strike must be more expensive Consider two put options on the same stock with the same expiration date. Put option #1 has a strike price of X 1 and its price is given by p 1 : Put option #2 has a strike price of X 2 and its price is given by p 2 r > 1: Suppose that X 1 > X We must have p 1 > p 2 whenever X 1 > X 2 What this means is that the price of the put option with a higher strike price must be always higher. To prove this suppose this is not true. In particular, suppose that we have two puts options on the same stock with the same expiration date, one with a higher strike price than the other, i.e., X 1 > X 2, but the put option with a lower strike price is more expensive, that is p 2 > p 1. In this case the following arbitrage position is possible. Buy the put option with strike price X 1 at p 1 Sell the put option with strike price X 2 at p 2 Invest p 2 p 1 > 0 at r% If S T < X 2 ; payo = (X 1 S T ) (X 2 S T ) + r(p 2 p 1 ) = (X 1 X 2 ) + r(p 2 p 1 ) > 0 If X 2 < S T < X 1 ; payo = (X 1 S T ) + r(p 2 p 1 ) > 0 If S T > X 1 > X 2 payo = + r(p 2 p 1 ) > 0 which establishes that we make an arbitrage pro t whenever p 2 > p 1 : But this cannot occur in equilibrium and we must have p 1 > p 2 whenever X 1 > X 2 that is the put option with a higher strike price must be more expensive. 1

2 1.1.2 An Example: Consider two put options on the same stock with the same expiration date. Put option #1 has a strike price of X 1 = 100 and its price is given by p 1 = 8: Put option #2 has a strike price of X 2 = 90 and its price is given by p 2 = 12 r > 1:1: Show that arbitrage is possible. Answer: The arbitrage position is Buy the put option with strike price X 1 = 100 at p 1 = 8: Sell the put option with strike price X 2 = 90 at p 2 = 12 Invest p 2 p 1 = 4 at r = 1:1 If S T < 90; payo = (100 S T ) (90 S T ) + (1:1) 4 = 14:4 > 0 If 90 < S T < 100; payo = (100 S T ) + (1:1) 4 > 0 If S T > 100 payo = (1:1) 4 > The Call Option with a Higher Strike must be cheaper Consider two call options on the same stock with the same expiration date. Call option #1 has a strike price of X 1 and its price is given by c 1 : Call option #2 has a strike price of X 2 and its price is given by c 2 r > 1: Suppose that X 1 > X 2. 2

3 1.2.1 We must have c 2 > c 1 whenever X 1 > X 2 What this means is that the price of the call option with a higher strike price must be always lower. To prove this suppose this is not true. In particular, suppose that we have two call options on the same stock with the same expiration date, one with a higher strike price than the other, i.e., X 1 > X 2, but the call option with a higher strike price is more expensive, that is c 1 > c 2. In this case the following arbitrage position is possible. Buy the call option with strike price X 2 at c 2 Sell the call option with strike price X 1 at c 1 Invest c 1 c 2 > 0 at r% If S T < X 2 ; payo = r(c 1 c 2 ) > 0 If X 2 < S T < X 1 ; payo = (S T X 2 ) + r(c 1 c 2 ) > 0 If S T > X 1 > X 2 payo = (S T X 2 ) (S T X 1 ) + r(c 1 c 2 ) = (X 1 X 2 ) + r(c 1 c 2 ) > 0 which establishes that we make an arbitrage pro t whenever c 1 > c 2 : But this cannot occur in equilibrium and we must have c 2 > c 1 whenever X 1 > X 2 An Example: Consider two call options on the same stock with the same expiration date. Call option #1 has a strike price of X 1 = 60 and its price is given by c 1 = 10: Call option #2 has a strike price of X 2 = 40 and its price is given by c 2 = 8 r > 1:1: Show that arbitrage is possible. 3

4 Answer: Buy the call option with strike price X 2 = 40 at c 2 = 8 Sell the call option with strike price X 1 = 60 at c 1 = 10 Invest c 1 c 2 = 2 at r = 1:1 If S T < 40; payo = (1:1 2) = 2:2 If 40 < S T < 60; payo = (S T 40) + (1:1 2) > 0 If S T > 60 payo = (S T 40) (S T 60) + (1:1 2) = 22:2 which establishes that we make an arbitrage pro t whenever c 1 > c 2 : But this cannot occur in equilibrium and we must have c 2 > c 1 whenever X 1 > X 2 that is the call option with a higher strike price must be cheaper. 1.3 Two Put Options with Di erent Strike Prices Consider two put options on the same stock with the same expiration date. Put option #1 has a strike price of X 1 and its price is given by p 1 : Put option #2 has a strike price of X 2 and its price is given by p 2 r > 1: Suppose that X 1 > X We must have p 1 p 2 X 1 X 2 What this means is that the price di erential of the two put options must be lower than the di erential in strike prices. 4

5 To prove this suppose this is not true. In particular, suppose that we have two puts options on the same stock with the same expiration date, one with a higher strike price than the other, i.e., X 1 > X 2, but we have p 1 p 2 > X 1 X 2 (1) In this case the following arbitrage position is possible. SELL the put option with strike price X 1 at p 1 BUY the put option with strike price X 2 at p 2 Invest p 1 p 2 > 0 at r% If S T < X 2 ; payo = -(X 1 S T ) + (X 2 S T ) + r(p 1 p 2 ) = (X 2 X 1 ) + r(p 1 p 2 ) > 0 (BECAUSE OF (1)) If X 2 < S T < X 1 ; payo = (X 1 S T ) + r(p 1 p 2 ) > 0 If S T > X 1 > X 2 payo = r(p 1 p 2 ) which establishes that we make an arbitrage pro t whenever p 1 p 2 > X 1 X 2 : But this cannot occur in equilibrium and we must have p 1 p 2 X 1 X 2 An Example: Consider two put options on the same stock with the same expiration date. Put option #1 has a strike price of X 1 = 100 and its price is given by p 1 = 15: Put option #2 has a strike price of X 2 = 96 and its price is given by p 2 = 8 r > 1:1: Show that arbitrage is possible. 5

6 SELL the put option with strike price X 1 = 100 at p 1 = 15: BUY the put option with strike price X 2 = 96 at p 2 = 8 Invest p 1 p 2 = 7 > 0 at r = 1:1 If S T < 96; payo = -(100 S T ) + (96 S T ) + 1:1(7) = 3:7 > 0 If 96 < S T < 100; payo = (100 S T ) + 1:1(7) > 3:7 > 0 If S T > 100 payo = 1:1(7) = 7:7 > 0 6