By Alonso Peña
This e-book will introduce you to the foremost mathematical types used to cost monetary derivatives, in addition to the implementation of major numerical types used to resolve them. specifically, fairness, foreign money, rates of interest, and credits derivatives are mentioned. within the first a part of the publication, the most mathematical types utilized in the realm of monetary derivatives are mentioned. subsequent, the numerical equipment used to unravel the mathematical types are offered. ultimately, either the mathematical versions and the numerical equipment are used to unravel a few concrete difficulties in fairness, currency, rate of interest, and credits derivatives.
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Term structure of spot rates of interest at t=0 are EURIBOR 3M = 1 percent pa, EURIBOR 6M = 2 percent pa, EURIBOR 9M = 3 percent pa, and EURIBOR 12M = 4 percent pa. First, we need to compute the initial forward rates by bootstrapping as observed from time t=0. We obtain this using the following equation: / / / / Then using the iterative equation 9, we compute the forward the rates into the future as follows: / / / 0 / / 0 / 0 / / 0 / 0 / 0 corresponds to We populate the initial forward rates in the left-most column and advance column-by-column to the right until we have all the values we need, as shown on the right.
The current risk-free rate is 5 percent pa. How do we proceed to solve this problem using BT? 25. The tables in the following screenshot illustrate the numerical values for each of the three steps applied to this problem: Example of Binomial Trees pricing. [ 42 ] Chapter 3 We first compute the up and down factors as well as the up probability p. In numerical terms, these are calculated using the following equations: X H[SV 'W H[S G H[SV 'W H[S The following are the probabilities of going up and down respectively: S H[SU 'W G XG H[S u S With all these parameters, we can now proceed to construct our tree in three phases, as follows: 1.
Algorithm of FDM The application of FDM to the preceding PDE requires the first derivative with respect to time and the second derivative with respect to x, which leads to the following equations: wX 'X | wW 'W w X ' § 'X · | ¨ ¸ w[ '[ © '[ ¹ XL M XL M 'W XL M XL M XL M '[ The preceding approximations can be derived from a Taylor series expansion. See (Wilmott et al. 1995) as we did in the preceding section. [ 46 ] Chapter 3 In order to do this, we need to discretize the domain of the function to a discrete set of nodes.
Advanced Quantitative Finance with C++ by Alonso Peña