By Alonso Peña
This ebook will introduce you to the major mathematical versions used to cost monetary derivatives, in addition to the implementation of major numerical types used to unravel them. particularly, fairness, foreign money, rates of interest, and credits derivatives are mentioned. within the first a part of the booklet, the most mathematical versions utilized in the realm of monetary derivatives are mentioned. subsequent, the numerical equipment used to resolve the mathematical types are awarded. ultimately, either the mathematical types 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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Extra resources for Advanced Quantitative Finance with C++
In literature, this set of equations is called a stencil. 3. Next collocate the stencil to all the nodes of the domain. We now apply the stencil to all the nodes in the domain with the exception of the nodes that represent the initial and boundary conditions. For these nodes, we know the value is a priori, and, hence, it does not need to be computed. [ 47 ] Numerical Methods 4. Iterate the solution in time with the stencil until we cover the full domain. In explicit FDM, you simply advance and compute the values for the unknown function u.
We use N=500 and M=10,000. 34 seconds. The value of the premium and the execution time will vary from computer to computer. Note that this code can be easily modified to price other payoffs by simply changing STEP 4 in the algorithm. In terms of a C++ implementation, this concept can be incorporated using a class to define the payoff. Also, STEP 4 can be slightly modified to include an estimate of the accuracy in the Monte Carlo approximation. Please refer to the website for downloadable implementations containing these features.
Following BGM, we use Geometric Brownian Motion (BGM) following (Brace, Gatarek, and Musiela 1997) to describe each of these stochastic processes as follows: G/L PL /L GW V L /L G:L 4 L 1 Equation 8 We now further simplify the model and use a single factor driving all the forward rates. This simplification can be later relaxed into a multifactor LMM. It can be shown that by choosing the last rate as a terminal measure, the drift has the form as follows: N dLi = − ∑ k = i +1 α k ɍ k (Tn )Lk (Tn ) ɍi (t)Li (Tn )dT+ ɍi (Tn )dW 1+ α k Lk (Tn ) Note that the drift in the preceding GBM equation is a function of the forward rate, thus not constant but state-dependent.
Advanced Quantitative Finance with C++ by Alonso Peña