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# Section2 - Derivative Securities Fall 2007 Section 2 Notes by Robert V Kohn extended and improved by Steve Allen Courant Institute of Mathematical

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Derivative Securities – Fall 2007– Section 2 Notes by Robert V. Kohn, extended and improved by Steve Allen. Courant Institute of Mathematical Sciences. Binomial and trinomial one-period models. This section explores the implications of arbitrage for the pricing of contingent claims in a one-period setting. We do not claim, of course, that any market can realistically be modelled using a one- period binomial or trinomial framework. But we will argue soon that many markets can be modelled using multiperiod trees, in much the same way that diﬀusion can be modelled by random walk on a lattice. A good understanding of the single-period setting will lead, with just a little extra work, to an understanding of the multiperiod models. Many books discuss only the binomial case. I discuss also the trinomial setting, because it provides the simplest example of a market that’s not complete. Moreover it permits me to introduce (ever so brieﬂy) the connection between risk-neutral pricing and the duality theory of linear programming. However this is “enrichment” material: my discussion of the trinomial model, and the related discussion of linear programming duality, will not be tested by the HW’s or exams. We’ll do two passes. In the ﬁrst, we’ll focus (like most texts, including Hull and Baxter- Rennie) on hedging by a combination of a bond and the “underlying.” In the second, we’ll discuss how hedging can alternatively be achieved by buying or selling just a forward or futures contract. The second viewpoint (hedging by forwards or futures) is in some ways better than the ﬁrst, as we’ll explain when we begin that discussion. (For this reason, Steve Allen’s version of Section 2 focuses entirely on hedging by forwards or futures.) **************** The binomial model. We consider a one-period market which has just two securities: a stock (paying no dividend, initial unit price per share s 1 dollars) and a bond (interest rate r , one bond pays one dollar at maturity). just one maturity date T just two possible states for the stock price at time T : s 2 and s 3 ,w ith s 2 <s 3 (see the ﬁgure). We could suppose we know the probability p that the stock will be worth s 3 at time T . This would allow us to calculate the expected value of any contingent claim. However we will make no use of such knowledge. Pricing by arbitrage considerations makes no use of information about probabilities – it uses just the list of possible events. The reasonable values of s 1 ,s 2 3 are not arbitrary: the economy should permit no arbitrage. This requires that s 2 1 e rT 3 . 1

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S 1 S 3 S 2 Figure 1: Prices in the one-period binomial market model. It’s easy to see that if this condition is violated then an arbitrage is possible. The converse is extremely plausible; a simple proof will be easy to give a little later.
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## This note was uploaded on 01/02/2012 for the course FINANCE 347 taught by Professor Bayou during the Fall '11 term at NYU.

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Section2 - Derivative Securities Fall 2007 Section 2 Notes by Robert V Kohn extended and improved by Steve Allen Courant Institute of Mathematical

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