Lecture_11

Lecture_11 - 11 Options on Bonds This chapter studies the...

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1 11 Options on Bonds This chapter studies the arbitrage-free pricing of various types of options on bonds. Two types of structures are considered: distribution independent and distribution dependent. Distribution independent methods are those that do not depend on a particular evolution for the term structure of interest rates, while distribution dependent structures do. Distribution independent models generate very general results, often not specific enough to generate exact pricing and hedging descriptions. Usually, only pricing inequalities and bounds on synthetic constructions can be obtained. Hence, the need for distribution dependent models.
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2 A Distribution Free Option Theory This section studies option pricing theory without invoking any assumptions relating to the evolution for the term structure of interest rates. For this reason, the insights obtained are general, not specific, and pertaining to the relations between the various types of options: calls, and puts both European and American.
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3 For pedagogical reasons, we concentrate on options written on zero-coupon bonds. The analysis is more general and the results apply to all the other options studied in this book. Let the underlying zero-coupon bond have maturity date 2 T . Its time t price is denoted ). 2 , ( T t P
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4 Consider a European call option with strike price K and maturity date 2 1 T T written on this zero- coupon bond. Its time t price is denoted ) 2 : , 1 , ( T K T t C . The otherwise identical American call option’s price is denoted ) 2 : , 1 , ( T K T t A . A European put option with strike K and maturity date 2 1 T T written on this same zero-coupon bond’s time t price is denoted ) 2 : , 1 , ( T K T t P . The otherwise identical American put’s price is denoted ) 2 : , 1 , ( T K T t A .
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5 1 Call Options To derive these relations, we show how to synthetically construct one type of option or portfolio using another type of option, sometimes throwing away value. Then, the price of the traded option or portfolio must be no greater than the synthetic option. First, we consider two European call options differing in their strike prices, . 2 1 K K The call option with the smaller strike price ) 1 ( K is atleast as valuable as the call option with the larger strike price ) 2 ( K . The reason is that the holder of the call option with smaller strike ) 1 ( K can always throw away value and create the larger strike call ) 2 ( K by paying more upon exercise.
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6 Hence, it must be the case that 2 1 ) 2 : 1 , 1 , ( ) 2 : 2 , 1 , ( K K if T K T t C T K T t C . (11.1) By an identical argument, we have that this relation also holds for American calls, 2 1 ) 2 : 1 , 1 , ( ) 2 : 2 , 1 , ( K K if T K T t A T K T t A . (11.2)
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7 The next arbitrage-free pricing relation is that ) 2 : , 1 , ( ) 1 , ( ) 2 , ( T K T t C T t KP T t P . (11.3) This follows because the portfolio on the left side (holding the underlying bond and selling K zero-coupon bonds of maturity 1 T ) can be constructed from the call by (perhaps) throwing away value.
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This note was uploaded on 10/31/2010 for the course NBA 5550 taught by Professor Jarrow,robert during the Fall '08 term at Cornell.

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Lecture_11 - 11 Options on Bonds This chapter studies the...

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