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section1 - Derivative Securities Fall 2007 Section 1 Notes...

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Derivative Securities – Fall 2007 – Section 1 Notes by Robert V. Kohn, extended and improved by Steve Allen. Courant Institute of Mathematical Sciences. Forwards, puts, calls, and other contingent claims. This section discusses the most basic examples of contingent claims, and explains how considerations of arbitrage determine or restrict their prices. This material is in Chapters 1 and 3, Sections 8.1 and 8.2, and Chapter 10 of Hull (6th edition). For a more concise, less distributed treatment see also Chapters 2 and 3 of Jarrow and Turnbull. We concentrate for simplicity on European options rather than American ones, on forwards rather than futures, and on deterministic rather than stochastic interest rates. We close with a detailed discussion about the pricing of forwards, corresponding to Chapter 5 of Hull. *********************** The most basic instruments: Forward contract with maturity T and delivery price K. buy a forward ho ldalongforward holder is obliged to buy the underlying asset at price K on date T . European call option with maturity T and strike price K. buy a call ldalongca l l holder is entitled to buy the underlying asset at price K on date T . European put option with maturity T and strike price K. buy a put hold a long put holder is entitled to sell the underlying asset at price K on date T . These are contingent claims , i.e. their value at maturity is not known in advance. Payoﬀ formulas and diagrams (value at maturity, as a function of S T =value of the underlying) are shownintheF igure . Any long position has a corresponding (opposite) short position: Buyer of a claim has a long position seller has a short position. Payoﬀ diagram of short position = negative of payoﬀ diagram of long position. 1

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Forward S - K (S -K) (K-S ) Put Call S KK S K S T T T+ + T T T Figure 1: Payoﬀs of forward, call, and put options. An American option diﬀers from its European sibling by allowing early exercise. For exam- ple: the holder of an American call with strike K and maturity T has the right to purchase the underlying for price K at any time 0 t T . A discussion of American options must deal with two more-or-less independent issues: the unknown future value of the underlying, and the optimal choice of the exercise time. By focusing initially on European options we’ll develop an understanding of the ﬁrst issue before addressing the second. ************************ Why do people buy and sell contingent claims? Brieﬂy, to hedge or to speculate . Examples of hedging: A US airline has a contract to buy a French airplane for a price ﬁxed in Euros, payable one year from now. By going long on a forward contract for Euros (payable in dollars) it can eliminate its foreign currency risk. The holder of a forward contract has unlimited downside risk. Holding a call limits the downside risk (but buying a call with strike K costs more than buying the forward with delivery price K). Holding one long call and one short call costs less, but gives up some of the upside beneﬁt: ( S T - K 1 ) + - ( S T - K 2 ) + K 1 <K 2 This is known as a “bull spread”. (See the ﬁgure.)
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section1 - Derivative Securities Fall 2007 Section 1 Notes...

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