# Quadratic Spline Wavelets for Sparse Discretization of Jump–Diffusion Models

## Abstract

**:**

## 1. Introduction

## 2. Construction of Wavelets

**Definition**

**1.**

- (i)
- Ψ is a Riesz basis for H, i.e., the span of Ψ is dense in H and there exist constants $c,C\in \left(0,\infty \right)$ such that$$c\u2225\mathbf{b}\u2225\le {\u2225{\displaystyle \sum _{\lambda \in \mathcal{J}}}{b}_{\lambda}{\psi}_{\lambda}\u2225}_{H}\le C\u2225\mathbf{b}\u2225,$$
- (ii)
- The functions are local in the sense that$$diam\phantom{\rule{4pt}{0ex}}supp\phantom{\rule{4pt}{0ex}}{\psi}_{\lambda}\le C{2}^{-\left|\lambda \right|},\phantom{\rule{1.em}{0ex}}\lambda \in \mathcal{J},$$
- (iii)
- The family Ψ has the hierarchical structure$$\mathrm{\Psi}={\mathrm{\Phi}}_{{j}_{0}}\cup {\displaystyle \bigcup _{j={j}_{0}}^{K}}{\mathrm{\Psi}}_{j}$$
- (iv)
- There exists $L\ge 1$ such that all functions ${\psi}_{\lambda}\in {\mathrm{\Psi}}_{j}$, ${j}_{0}\le j\le K$, have L vanishing moments, i.e.,$$\underset{a}{\overset{b}{\int}}}{x}^{k}\phantom{\rule{0.166667em}{0ex}}{\psi}_{\lambda}\left(x\right)=0,\phantom{\rule{1.em}{0ex}}k=0,\dots ,L-1.$$

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 3. Discretization of the Jump–Diffusion Option Pricing Models

**Theorem**

**5.**

**Proof.**

- (1)
- Choose a tolerance $\u03f5$.
- (2)
- Compute all the entries ${\tilde{\mathbf{C}}}_{\lambda ,\mu}^{s}$ for indexes $\lambda =\left(i,k\right)$ and $\mu =\left(j,l\right)$ such that $g\notin {C}^{2L}\left({I}_{i,j,k,l}\right)$.
- (3)
- Based on estimate in Equation (81), set the level $\tilde{L}$ such that ${\tilde{\mathbf{C}}}_{\lambda ,\mu}^{s}<\u03f5$ for any $i+j>\tilde{L}$ and $\lambda ,\mu $ such that $g\in {C}^{2L}\left({I}_{i,j,k,l}\right)$.
- (4)
- If $i+j\le \tilde{L}$, then use a local estimate in Equation (81) to compute only entries for which it is not guaranteed that ${\tilde{\mathbf{C}}}_{\lambda ,\mu}^{s}<\u03f5$.

## 4. Numerical Example

**Example**

**1.**

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Sparse structure of the upper left part of the (infinite) matrix $\u2329{\mathrm{\Psi}}^{\prime},{\mathrm{\Psi}}^{\prime}\u232a$ for the wavelet basis $\mathrm{\Psi}$ from [16] (

**left**); the basis from this paper (

**middle**); and the structure of upper left part of the mass matrix $\u2329\mathrm{\Psi},\mathrm{\Psi}\u232a$ for the basis from this paper (

**right**).

**Figure 2.**Scaling functions ${\varphi}^{b}$ and $\varphi $ (

**left**); wavelets ${\psi}^{b}$, ${\psi}^{1}$ and ${\psi}^{2}$ (

**middle**); and wavelets ${\psi}^{3}$ and ${\psi}^{4}$ (

**right**).

**Figure 3.**The structure of matrix ${\tilde{\mathbf{A}}}^{8}$ truncated using the threshold ${10}^{-7}$.

**Figure 4.**Functions representing the values of a European put (

**left**) and call (

**right**) option for the Merton model.

**Table 1.**The condition numbers of the wavelet bases ${\mathrm{\Psi}}^{s}$ with respect to the ${L}^{2}$-norm and the ${H}^{1}$-seminorm. s is the number of wavelet levels and N is the number of basis functions.

s | N | ${\mathit{\lambda}}_{\mathit{min}}$ | ${\mathit{\lambda}}_{\mathit{max}}$ | ${\mathit{\kappa}}_{{\mathit{L}}^{2}}$ | ${\mathit{\lambda}}_{\mathit{min}}$ | ${\mathit{\lambda}}_{\mathit{max}}$ | ${\mathit{\kappa}}_{{\mathit{H}}^{1}}$ |
---|---|---|---|---|---|---|---|

1 | 8 | 0.22 | 1.77 | 8.06 | 0.50 | 1.53 | 3.05 |

2 | 16 | 0.18 | 1.87 | 10.23 | 0.50 | 1.67 | 3.34 |

3 | 32 | 0.16 | 1.96 | 12.21 | 0.50 | 1.72 | 3.45 |

4 | 64 | 0.15 | 2.01 | 13.70 | 0.50 | 1.74 | 3.49 |

5 | 128 | 0.14 | 2.04 | 14.86 | 0.50 | 1.75 | 3.51 |

6 | 256 | 0.13 | 2.06 | 15.79 | 0.50 | 1.76 | 3.52 |

7 | 512 | 0.13 | 2.08 | 16.59 | 0.50 | 1.76 | 3.53 |

8 | 1024 | 0.12 | 2.10 | 17.28 | 0.50 | 1.77 | 3.53 |

9 | 2048 | 0.12 | 2.11 | 17.86 | 0.50 | 1.77 | 3.54 |

10 | 4096 | 0.12 | 2.12 | 18.35 | 0.50 | 1.77 | 3.54 |

Crank–Nicolson | Richardson | ||||||
---|---|---|---|---|---|---|---|

S | $\mathit{N}$ | $\mathit{M}$ | Put | Error | $\mathit{M}$ | Put | Error |

90 | 32 | 6 | 9.284895 | 5.23 × ${10}^{-4}$ | 10 | 9.287676 | 2.26 × ${10}^{-3}$ |

64 | 16 | 9.282265 | 3.15 × ${10}^{-3}$ | 16 | 9.282679 | 2.74 × ${10}^{-3}$ | |

128 | 46 | 9.285090 | 3.78 × ${10}^{-4}$ | 27 | 9.285143 | 2.75 × ${10}^{-4}$ | |

256 | 128 | 9.285427 | 8.69 × ${10}^{-6}$ | 46 | 9.285433 | 1.51 × ${10}^{-5}$ | |

512 | 363 | 9.285413 | 4.97 × ${10}^{-6}$ | 77 | 9.285414 | 4.17 × ${10}^{-6}$ | |

1024 | 1024 | 9.285417 | 6.24 × ${10}^{-7}$ | 128 | 9.285418 | 5.39 × ${10}^{-7}$ | |

100 | 32 | 6 | 3.166832 | 1.78 × ${10}^{-2}$ | 10 | 3.165157 | 1.61 × ${10}^{-2}$ |

64 | 16 | 3.148937 | 8.86 × ${10}^{-5}$ | 16 | 3.148590 | 4.36 × ${10}^{-4}$ | |

128 | 46 | 3.149050 | 2.44 × ${10}^{-5}$ | 27 | 3.149012 | 1.30 × ${10}^{-5}$ | |

256 | 128 | 3.149038 | 1.20 × ${10}^{-5}$ | 46 | 3.149032 | 6.56 × ${10}^{-6}$ | |

512 | 363 | 3.149027 | 9.07 × ${10}^{-7}$ | 77 | 3.149026 | 2.20 × ${10}^{-7}$ | |

1024 | 1024 | 3.149026 | 5.64 × ${10}^{-8}$ | 128 | 3.149026 | 1.76 × ${10}^{-7}$ | |

110 | 32 | 6 | 1.389539 | 1.16 × ${10}^{-2}$ | 10 | 1.389646 | 1.15 × ${10}^{-2}$ |

64 | 16 | 1.401664 | 4.78 × ${10}^{-4}$ | 16 | 1.401750 | 5.65 × ${10}^{-4}$ | |

128 | 46 | 1.401350 | 1.64 × ${10}^{-4}$ | 27 | 1.401362 | 1.76 × ${10}^{-4}$ | |

256 | 128 | 1.401196 | 1.06 × ${10}^{-5}$ | 46 | 1.401198 | 1.20 × ${10}^{-5}$ | |

512 | 363 | 1.401183 | 3.22 × ${10}^{-6}$ | 77 | 1.401183 | 3.04 × ${10}^{-6}$ | |

1024 | 1024 | 1.401186 | 3.22 × ${10}^{-7}$ | 128 | 1.401186 | 3.00 × ${10}^{-7}$ |

**Table 3.**Errors in the ${L}^{\infty}\left(0,2K\right)$-norm and in the ${L}^{2}\left(0,2K\right)$-norm and the corresponding experimental rates of convergence.

Crank–Nicolson | Richardson | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

$\mathit{N}$ | $\mathit{M}$ | ${\mathit{L}}^{\infty}$ | Rate | ${\mathit{L}}^{\mathbf{2}}$ | Rate | $\mathit{M}$ | ${\mathit{L}}^{\infty}$ | Rate | ${\mathit{L}}^{\mathbf{2}}$ | Rate |

32 | 6 | 3.48 × ${10}^{-2}$ | - | 9.91 × ${10}^{-2}$ | - | 10 | 3.26 × ${10}^{-2}$ | - | 9.63 × ${10}^{-2}$ | - |

64 | 16 | 3.31 × ${10}^{-3}$ | 3.39 | 8.00 × ${10}^{-3}$ | 3.63 | 16 | 2.89 × ${10}^{-3}$ | 3.49 | 7.67 × ${10}^{-3}$ | 3.65 |

128 | 46 | 3.59 × ${10}^{-4}$ | 3.20 | 8.80 × ${10}^{-4}$ | 3.18 | 27 | 3.07 × ${10}^{-4}$ | 3.23 | 8.55 × ${10}^{-4}$ | 3.17 |

256 | 128 | 4.24 × ${10}^{-5}$ | 3.08 | 1.07 × ${10}^{-4}$ | 3.04 | 46 | 3.78 × ${10}^{-5}$ | 3.02 | 1.05 × ${10}^{-4}$ | 3.03 |

512 | 363 | 5.28 × ${10}^{-6}$ | 3.01 | 1.33 × ${10}^{-5}$ | 3.01 | 77 | 4.66 × ${10}^{-6}$ | 3.02 | 1.30 × ${10}^{-5}$ | 3.01 |

1024 | 1024 | 7.35 × ${10}^{-7}$ | 2.85 | 1.91 × ${10}^{-6}$ | 2.80 | 128 | 6.52 × ${10}^{-7}$ | 2.84 | 1.87 × ${10}^{-6}$ | 2.80 |

**Table 4.**The condition numbers (cond) of discretization matrices, and numbers of GMRES iterations (it).

N | M | Cond | It |
---|---|---|---|

32 | 6 | 11.5 | 5(1) |

64 | 16 | 12.9 | 5(3) |

128 | 46 | 14.0 | 5(7) |

256 | 128 | 14.9 | 5(10) |

512 | 363 | 15.7 | 6(2) |

1024 | 1024 | 16.3 | 6(3) |

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**MDPI and ACS Style**

Černá, D. Quadratic Spline Wavelets for Sparse Discretization of Jump–Diffusion Models. *Symmetry* **2019**, *11*, 999.
https://doi.org/10.3390/sym11080999

**AMA Style**

Černá D. Quadratic Spline Wavelets for Sparse Discretization of Jump–Diffusion Models. *Symmetry*. 2019; 11(8):999.
https://doi.org/10.3390/sym11080999

**Chicago/Turabian Style**

Černá, Dana. 2019. "Quadratic Spline Wavelets for Sparse Discretization of Jump–Diffusion Models" *Symmetry* 11, no. 8: 999.
https://doi.org/10.3390/sym11080999