section3 - Derivative Securities Fall 2007 Section 3 Notes...

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Derivative Securities – Fall 2007– Section 3 Notes by Robert V. Kohn, extended and improved by Steve Allen. Courant Institute of Mathematical Sciences. Multiperiod Binomial Trees. We turn to the valuation of derivative securities in a time-dependent setting. We focus for now on multi-period binomial models, i.e. binomial trees. This setting is simple enough to let us do everything explicitly, yet rich enough to approximate many realistic problems. The material covered in this section is very standard (and very important). The treatment here tracks closely with Baxter and Rennie (Chapter 2). Hull addresses this topic in Chapter 11 (6th edition). Another good treatment is that of Jarrow and Turnbull (Chapter 5, 2nd edition), which includes many examples. In the Section 4 notes we’ll discuss how the parameters should be chosen to mimic the conventional (Black-Scholes) hypothesis of lognormal stock prices, and we’ll pass to the continuous-time limit. Binomial trees are widely used in practice, in part because they are easy to implement numerically. (Also because the scheme can easily be adjusted to price American options.) For a nice discussion of alternative numerical implementations, see the article “Nine ways to implement the binomial method for option valuation in Matlab,” by D.J. Higham, SIAM Review 44 (2002) 661-677. As we saw in Section 2, an option can be replicated by trading the underlying, or alterna- tively by trading forwards or futures on the underlying. We’ll focus first on replication by trading the underlying (since this is the first thing you’ll see in most books). Then we’ll make a second pass, replicating with futures (the viewpoint emphasized by Steve Allen’s version of these notes). The practical question is not how to replicate an option but rather how to hedge it. But replication and hedging are intimately connected. Simple example: when you sell a call with strike K and maturity T you receive cash for it now, but you owe ( s T - K ) + to the holder when the option matures. We’ll explain (in the binomial setting) how the apparent uncertainty in your final-time obligation can be entirely eliminated (hedged) by pursuing a suitable trading strategy. It is, of course, the strategy that replicates your final-time obligation ( s T - K ) + . ******************************* First pass: trading the underlying. A multi-period binomial model generalizes the single-period binomial model we considered in Section 2. It has just two securities: a risky asset (a “stock,” paying no dividend) and a riskless asset (“bond”); a series of times 0 ,δt, 2 δt,. ..,Nδt = T at which trades can occur; interest rate r i during time interval i for the bond; a binomial tree of possible states for the stock prices. 1
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1 3 2 4 7 5 6 10 11 12 14 13 9 8 15 Figure 1: States of a non-recombining binomial tree.
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section3 - Derivative Securities Fall 2007 Section 3 Notes...

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