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73323357-17-Basic-Numerics

73323357-17-Basic-Numerics - B asic utnerical Procedures...

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Basic utnerical Procedures This chapter discusses three numerical procedures for valuing derivatives when exact formulas are not available. The first involves representing the asset price movements in the form of a tree and was introduced in Chapter 11. The second involves Monte Carlo simulatiop., which we encountered briefly in Chapter 12 when explaining stochastic processes. The third involves finite difference methods. . Monte Carlo simulation is usually used for derivatives where the payoff is dependent on the history of the underlying variable or where there are several underlying variables. Trees and finite difference methods are usually used for American options and other derivatives where the holder has early exercise decisions to make prior to maturity. In addition to valuing a derivative, all the procedures can be used to calculate Greek letters such as delta, gamma, and vega. The basic procedures we discuss in this chapter can be used to handle most of the derivatives valuations problems that are encountered in practice. However, sometimes they have to be adapted to cope with particular situations. We discuss this in Chapter 24. 17.1 BINOMIAL TREES We introduced binomial trees in Chapter 11. They can be used to value either European or American options. The Black-Scholes formulas and their extensions that we pre- sented in Chapters 13 and 14 provide analytic valuations for European options.! There are no analytic valuations for American options. Binomial trees are therefore most useful for valuing these types of options. 2 As explained in Chapter 11, the binomial tree valuation approach involves dividing the life of the option into a large number of small time intervals of length ~t. It assumes that in each time interval the price of the underlying asset moves from its initial value of 1 The Black-Scholes formulas are based on the same set of assumptions as binomial trees. As one might expect, in the limit as the number of time steps is increased, the price given for a European option by the binomial method converges to the Black-Scholes price. 2 Some analytic approximations for valuing American options have been suggested. The most well-known one is the quadratic approximation approach. See Technical Note 8 on the author's website for a description of this approach. 391
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392 CHAPTER 17 Figure 17.1 Asset price movements in time !:it under the binomial model. S Sit Sd S to one of two new values, Su and Sd. The approach is illustrated in Figure 17.1. In general, u > 1 and d < 1. The movement from S to Su, therefore, is an "up" movement and the movement from S to Sd is a "down" movement. The probability of an up movement will be denoted by p. The probability of a down movement is 1 - p. Risk-Neutral Valuation The risk-neutral valuation principle, explained in Chapters 11 and 13,states that an option (or other derivative) can be valued on the assumption that the world is risk neutral. This means that for valuation purposes we can use the following procedure: 1.
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