book

Quantitative Finance

Name: Quantitative Finance
Author: Erik Schlogl
ISBN: 9781439897959

by Erik Schlogl

November 2013

Intermediate to advanced

354 pages

8h 46m

English

CRC Press

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Preliminaries
Dedication
Preface
Disclaimer
Acknowledgements
Chapter 1 A brief review of the C++ programming language
1.1 Getting started1.2 Procedural programming in C++1.2.1 Built-in data types and the definition and declaration of variables1.2.2 Statements, expressions and operators1.2.3 Control flow1.2.4 Functions1.2.5 Dynamic memory allocation1.3 Object–oriented features of C++1.3.1 Classes1.3.2 Composition and inheritance1.3.3 Polymorphism1.4 Templates1.5 Exceptions1.6 NamespacesFigure 1.1Figure 1.2Table 1.1Table 1.2Table 1.3Table 1.4Listing 1.1Listing 1.2Listing 1.3Listing 1.4Listing 1.5Listing 1.6Listing 1.7Listing 1.8Listing 1.9Listing 1.10Listing 1.11
Chapter 2 Basic building blocks
2.1 The Standard Template Library (STL)2.1.1 Mathematical functions2.1.2 Complex numbers2.1.3 Strings2.1.4 Containers and iterators2.1.5 The <algorithm> header2.1.6 Streams2.2 The Boost Libraries2.2.1 Smart pointers2.2.2 Functions2.2.3 Probability distributions2.3 Numerical arrays2.3.1 Manipulating vectors, matrices and higher–dimensional arrays2.3.2 Linear algebra2.4 Numerical integration2.4.1 Gaussian quadrature2.4.2 Cubature2.4.3 The multivariate normal distribution2.5 Optimisation and root search2.5.1 A simple bisection template2.5.2 Newton/Raphson in multiple dimensions2.5.3 Brent’s line search and Powell’s minimisation algorithm2.6 The term structure of interest rates2.6.1 Fixed income terminology2.6.2 Term Structure Interpolation2.6.3 Implementing interest rate term structures2.6.4 Fitting the term structure to the marketFigure 2.1Figure 2.2Figure 2.3Figure 2.4Table 2.1Listing 2.1Listing 2.2Listing 2.3Listing 2.4Listing 2.5Listing 2.6Listing 2.7Listing 2.8Listing 2.9Listing 2.10Listing 2.11Listing 2.12Listing 2.13
Chapter 3 Lattice models for option pricing
3.1 Basic concepts of pricing by arbitrage3.1.1 A simple example3.1.2 The general single period case3.2 Hedging and arbitrage–free pricing in a binomial lattice model3.2.1 The one –period case3.2.2 Binomial lattices3.2.3 The binomial option pricing formula3.2.4 Dividend –paying assets in the binomial model3.3 Defining a general lattice model interface3.4 Implementing binomial lattice models3.5 Lattice models for the term structure of interest rates3.5.1 General construction3.5.2 Concrete modelsFigure 3.1Figure 3.2Figure 3.3Figure 3.4Figure 3.5Figure 3.6Figure 3.7Figure 3.8Figure 3.9Figure 3.10Figure 3.11Figure 3.12Figure 3.13Figure 3.14Figure 3.15Listing 3.1Listing 3.2Listing 3.3Listing 3.4
Chapter 4 The Black/Scholes world
4.1 Martingales4.2 Option pricing in continuous time4.2.1 The Black/Scholes partial differential equation4.2.2 The probabilistic approach4.2.3 Time–varying interest rates and dividend yields4.3 Exotic options with closed form solutions4.3.1 M–Binaries and the “Quintessential Option Pricing Formula”4.3.2 M–Binaries with exercise dimension one4.3.3 M–Binaries with higher exercise dimensions4.3.4 Options on maxima or minima4.3.5 Continuous monitoring and the method of images4.4 Implementation of closed form solutions4.4.1 Representing the deterministic volatility of assets with Black/Scholes–type dynamics4.4.2 Prices and hedge sensitivities of options with discrete event dates4.5 American options4.5.1 The Method of Barone-Adesi and Whaley4.5.2 Geske and Johnson’s approximation using compound optionsFigure 4.1Table 4.1Table 4.2Table 4.3Table 4.4Table 4.5Table 4.6Table 4.7Table 4.8Table 4.9Table 4.10Listing 4.1
Chapter 5 Finite difference methods for partial differential equations
5.1 The object-oriented interface5.2 The explicit finite difference method5.3 The implicit finite difference method5.4 The Crank/Nicolson schemeListing 5.1Listing 5.2Listing 5.3Listing 5.4Listing 5.5Listing 5.6Listing 5.7
Chapter 6 Implied volatility and volatility smiles
6.1 Calculating implied distributions6.2 Constructing an implied volatility surface6.3 Stochastic volatilityFigure 6.1Table 6.1Table 6.2Listing 6.1Listing 6.2Listing 6.3Listing 6.4

Chapter 7 Monte Carlo simulation
7.1 Background7.2 The generic Monte Carlo algorithm7.3 Simulating asset price processes following geometric Brownian motion7.4 Discretising stochastic differential equations7.5 Predictor–Corrector methods7.6 Variance reduction techniques7.6.1 Antithetic sampling7.6.2 Control variates7.6.3 Optimal control variate weights7.7 Pricing instruments with early exercise features7.7.1 The regression–based algorithm7.7.2 Implementation7.8 Quasi-random Monte Carlo7.8.1 Generating quasi–random sequences7.8.2 Sobol’ sequence construction7.8.3 Implementation of quasi–random Monte Carlo7.8.4 Randomised quasi–random Monte CarloFigure 7.1Figure 7.2Figure 7.3Figure 7.4Figure 7.5Figure 7.6Figure 7.7Table 7.1Table 7.2Table 7.3Table 7.4Table 7.5Table 7.6Listing 7.1Listing 7.2Listing 7.3Listing 7.4Listing 7.5Listing 7.6Listing 7.7Listing 7.8Listing 7.9Listing 7.10Listing 7.11Listing 7.12Listing 7.13Listing 7.14Listing 7.15Listing 7.16Listing 7.17Listing 7.18Listing 7.19Listing 7.20Listing 7.21Listing 7.22Listing 7.23Listing 7.24Listing 7.25Listing 7.26Listing 7.27
Chapter 8 The Heath/Jarrow/Morton model
8.1 The model framework8.2 Gauss/Markov HJM8.2.1 Zero coupon bond prices — one –factor case8.2.2 Zero coupon bond prices — multifactor case8.3 Option pricing in the Gaussian HJM framework8.3.1 Pricing a European call option on a zero coupon bond8.3.2 Options on coupon bonds8.3.3 Caps, floors and swaptions8.4 Adding a foreign currency8.4.1 Standard FX options8.4.2 A “Quintessential Formula” in the multicurrency Gauss/Markov HJM model8.5 Implementing closed–form solutions8.6 Monte Carlo simulation in the HJM framework8.6.1 Simulation of interest rate dynamics in a single currency8.6.2 Joint simulation in several currencies8.7 Implementing Monte Carlo simulation8.7.1 Propagating the state variables and calculating the realisations of financial variables8.7.2 Initialising the variance/covariance matrices and state variable drifts8.7.3 Options with early exercise featuresTable 8.1Table 8.2Table 8.3Listing 8.1Listing 8.2Listing 8.3Listing 8.4Listing 8.5Listing 8.6Listing 8.7Listing 8.8Listing 8.9
Appendix A Interfacing between C++ and Microsoft Excel
A.1 The XLW projectA.2 The QuantLib ObjectHandlerTable A.1Listing A.1Listing A.2Listing A.3Listing A.4Listing A.5Listing A.6Listing A.7Listing A.8Listing A.9
Appendix B Automatic generation of documentation using Doxygen
Listing B.1Listing B.2
References

Content preview from Quantitative Finance

Chapter 5

Finite difference methods for partial differential equations

Option pricing problems can typically be represented as a partial differential equation (PDE) subject to boundary conditions, see for example the Black/Scholes PDE in Section 4.2.1 or the option pricing PDE in the presence of stochastic volatility in Section 6.3.. The idea behind finite difference methods is to approximate the partial derivatives in the PDE by a difference quotient, e.g.

$\begin{matrix} \partial_{1} C (t, S (t)) \approx \frac{C (t + Δ t, S (t)) - C (t, S (t))}{Δ t} & (5.1) \end{matrix}$

$\begin{matrix} \partial_{2} C (t, S (t)) \approx \frac{C (t, S (t) + Δ S) - C (t, S (t))}{Δ S} & (5.2) \end{matrix}$

Thus the PDE is discretised in both the time and the space dimension and the resulting solution converges to the continuous (true) solution as the discretisation grid is made finer and finer. Potentially there ...

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Quantitative Finance

by Erik Schlogl

Finite difference methods for partial differential equations

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
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