# Local Refinement and Adaptive Strategy for a System of Free Boundary Power Options with High Order Compact Differencing

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## Abstract

**:**

## 1. Mathematical Model

## 2. Numerical Methods

#### 2.1. Fourth-Order Non-Equidistant Hermitian Differencing on a Locally Refined Grid

**Lemma 1.**

**Proof.**

**Lemma 2.**

**Proof.**

**Remark 1.**

#### 2.2. 5(4) Dormand–Prince, Runge–Kutta Embedded Time Integration Method

**Preliminary stage**:$${s}_{f}^{\prime}\left({\tau}_{n}\right)=\frac{{\mathcal{F}}^{n}}{{\mathcal{G}}^{n}}+O\left({h}^{4}\right),\phantom{\rule{2.em}{0ex}}{\xi}_{m\left(n\right)}=mr+\frac{{s}_{f}^{\prime}\left({\tau}_{n}\right)}{{s}_{f}\left({\tau}_{n}\right)}-m\frac{{\sigma}^{2}}{2}.$$

**First stage**:$${\mathbf{u}}_{\tau}^{n}=\frac{{\overline{\sigma}}^{2}}{2}{A}_{u}^{-1}\left({B}_{u}{\mathbf{u}}^{n}+{\mathbf{b}}_{u}^{n}\right)+{\xi}_{m\left(n\right)}{\mathbf{w}}^{n}-r{\mathbf{u}}^{n}+O\left({h}^{4}\right)$$$${\mathbf{u}}^{n+1/7}={\mathbf{u}}^{n}+\frac{k}{5}{\mathbf{u}}_{\tau}^{n}$$$${\mathbf{w}}_{\tau}^{n}=\frac{{\overline{\sigma}}^{2}}{2}{A}_{w,y}^{-1}\left({B}_{w,y}{\mathbf{y}}^{n}+{\mathbf{b}}_{w,y}^{n}\right)+{\xi}_{m\left(n\right)}{A}_{u}^{-1}\left({B}_{u}{\mathbf{u}}^{n}+{\mathbf{b}}_{u}^{n}\right)-r{\mathbf{w}}^{n}+O\left({h}^{4}\right)$$$${\mathbf{w}}^{n+1/7}={\mathbf{w}}^{n}+\frac{k}{5}{\mathbf{w}}_{\tau}^{n}$$$${\mathbf{y}}_{\tau}^{n}=\frac{{\overline{\sigma}}^{2}}{2}{A}_{w,y}^{-1}\left({B}_{w,y}{\mathbf{w}}^{n}+{\mathbf{b}}_{w,y}^{n}\right)+{\xi}_{m\left(n\right)}{A}_{w,y}^{-1}\left({B}_{w,y}{\mathbf{w}}^{n}+{\mathbf{b}}_{w,y}^{n}\right)-r{\mathbf{y}}^{n}+O\left({h}^{4}\right)$$$${\mathbf{y}}^{n+1/7}={\mathbf{y}}^{n}+\frac{k}{5}{\mathbf{y}}_{\tau}^{n}$$$${s}_{f}\left({\tau}_{n+1/7}\right)=-\frac{\mathcal{M}\left({\tau}_{n+1/7}\right)-{a}_{1}K}{{a}_{2}}+O\left({h}^{5}\right),\phantom{\rule{1.em}{0ex}}{s}_{f}^{\prime}\left({\tau}_{n+1/7}\right)=-\frac{{\mathcal{M}}^{\prime}\left({\tau}_{n+1/7}\right)}{{a}_{2}}+O\left({h}^{5}\right)$$$$u({\tau}_{n+1/7},{x}_{0})=K-{s}_{f}\left({\tau}_{n+1/7}\right),\phantom{\rule{1.em}{0ex}}w({\tau}_{n+1/7},{x}_{0})=-{s}_{f}\left({\tau}_{n+1/7}\right)$$$$\mathcal{M}\left({\tau}_{n+1/7}\right)=6u({\tau}_{n+1/7},{x}_{1})-\frac{3}{2}u({\tau}_{n+1/7},{x}_{2})+\frac{2}{9}u({\tau}_{n+1/7},{x}_{3})$$$${\xi}_{m(n+1/7)}=mr+\frac{{s}_{f}^{\prime}\left({\tau}_{n+1/7}\right)}{{s}_{f}\left({\tau}_{n+1/7}\right)}-m\frac{{\sigma}^{2}}{2}$$

**Second stage**:$${\mathbf{u}}_{\tau}^{n+1/7}=\frac{{\overline{\sigma}}^{2}}{2}{A}_{u}^{-1}\left({B}_{u}{\mathbf{u}}^{n+1/7}+{\mathbf{b}}_{u}^{n+1/7}\right)+{\xi}_{m(n+1/7)}{\mathbf{w}}^{n+1/7}-r{\mathbf{u}}^{n+1/7}+O\left({h}^{4}\right)$$$${\mathbf{u}}^{n+2/7}={\mathbf{u}}^{n}+\frac{3k}{40}{\mathbf{u}}_{\tau}^{n}+\frac{9k}{40}{\mathbf{u}}_{\tau}^{n+1/7}$$$$\begin{array}{cc}\hfill {\mathbf{w}}_{\tau}^{n+1/7}=\frac{{\overline{\sigma}}^{2}}{2}{A}_{w,y}^{-1}\left({B}_{w,y}{\mathbf{w}}^{n+1/7}+{\mathbf{b}}_{w}^{n+1/7}\right)+{\xi}_{n+1/7}& {A}_{u}^{-1}\left({B}_{u}{\mathbf{u}}^{n+1/7}+{\mathbf{b}}_{u}^{n+1/7}\right)\hfill \\ & \hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& -r{\mathbf{w}}^{n+1/7}+O\left({h}^{4}\right)\hfill \end{array}$$$${\mathbf{w}}^{n+2/7}={\mathbf{w}}^{n}+\frac{3k}{40}{\mathbf{w}}_{\tau}^{n}+\frac{9k}{40}{\mathbf{w}}_{\tau}^{n+1/7}$$$$\begin{array}{cc}\hfill {\mathbf{y}}_{\tau}^{n+1/7}=\frac{{\overline{\sigma}}^{2}}{2}{A}_{w,y}^{-1}\left({B}_{w,y}{\mathbf{y}}^{n+1/7}+{\mathbf{b}}_{y}^{n+1/7}\right)+{\xi}_{m(n+1/7)}{A}_{w,y}^{-1}& \left({B}_{w,y}{\mathbf{w}}^{n+1/7}+{\mathbf{b}}_{w}^{n+1/7}\right)\hfill \\ & \hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& -r{\mathbf{y}}^{n+1/7}+O\left({h}^{4}\right)\hfill \end{array}$$$${\mathbf{y}}^{n+2/7}={\mathbf{y}}^{n}+\frac{3k}{40}{\mathbf{y}}_{\tau}^{n}+\frac{9k}{40}{\mathbf{y}}_{\tau}^{n+1/7}$$$${s}_{f}\left({\tau}_{n+2/7}\right)=-\frac{\mathcal{M}\left({\tau}_{n+2/7}\right)-{a}_{1}K}{{a}_{2}}+O\left({h}^{5}\right),\phantom{\rule{1.em}{0ex}}{s}_{f}^{\prime}\left({\tau}_{n+2/7}\right)=-\frac{{\mathcal{M}}^{\prime}\left({\tau}_{n+2/7}\right)}{{a}_{2}}+O\left({h}^{5}\right)$$$$u({\tau}_{n+2/7},{x}_{0})=K-{s}_{f}\left({\tau}_{n+2/7}\right),\phantom{\rule{1.em}{0ex}}w({\tau}_{n+2/7},{x}_{0})=-{s}_{f}\left({\tau}_{n+2/7}\right)$$$$\mathcal{M}\left({\tau}_{n+2/7}\right)=6u({\tau}_{n+2/7},{x}_{1})-\frac{3}{2}u({\tau}_{n+2/7},{x}_{2})+\frac{2}{9}u({\tau}_{n+2/7},{x}_{3})$$$${\xi}_{m(n+2/7)}=mr+\frac{{s}_{f}^{\prime}\left({\tau}_{n+2/7}\right)}{{s}_{f}\left({\tau}_{n+2/7}\right)}-m\frac{{\sigma}^{2}}{2}$$

**Third stage**:

**Fourth stage**:

**Fifth stage**:

**Sixth stage**:

**Seventh stage:**

## 3. Numerical Experiment

- DPC-Uniform: Fourth-order compact scheme with 5(4) Dormand–Prince embedded pairs on an equidistant space grid;
- DPC-Loc1: Fourth-order compact scheme with 5(4) Dormand–Prince embedded pairs on a locally refined space grid (${h}_{a}=0.5\mathrm{h}$).
- DPC-Loc2: Fourth-order compact scheme with 5(4) Dormand–Prince embedded pairs on a locally refined space grid (${h}_{a}=0.25\mathrm{h}$).
- DPC-Loc3: Fourth-order compact scheme with 5(4) Dormand–Prince embedded pairs on a locally refined space grid (${h}_{a}=0.125\mathrm{h}$).

#### 3.1. Investigating Solution Accuracy on Both Uniform and Non-Equidistant Space Grids with Unit Power Terms

**Remark 2.**

#### 3.2. Investigating Solution Accuracy on Both Uniform and Non-Equidistant Space Grids with an Arbitrary Power Term

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Code Availability

## References

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**Figure 2.**Plot of the optimal exercise boundary with uniform and non-uniform Hermitian scheme $\u03f5={10}^{-3}$.

**Figure 5.**Optimal time step for each time level with varying power m ($h=0.025$, ${h}_{a}=0.25\mathrm{h}$).

K | T | r | $\mathit{\sigma}$ | ${\mathit{x}}_{\mathbf{max}}$ |
---|---|---|---|---|

100 | 6/12 | 5% | 20% | 3.00 |

**Table 2.**Comparing the optimal exercise boundary and runtime in seconds ($\u03f5={10}^{-3}$). ${s}_{f}\left(T\right)=83.91996$ is the benchmark value.

h | DPC-Loc1 | DPC-Loc2 | DPC-Loc3 | DPC-Uniform |
---|---|---|---|---|

0.05 | 83.92016 | 83.92055 | 83.92042 | 83.92703 |

0.025 | 83.92008 | 83.91988 | 83.91985 | 83.91961 |

h | DPC-Loc1 | DPC-Loc2 | DPC-Loc3 | DPC-Uniform |
---|---|---|---|---|

0.05 | 0.186678 | 0.355253 | 1.161236 | 0.279023 |

0.025 | 1.059206 | 2.429464 | 8.883537 | 0.628332 |

**Table 4.**Initializing with varying initial time step ${k}_{0}$ and fixed step size ($h=0.025$ and $\u03f5={10}^{-3}$).

${\mathit{k}}_{0}$ | DPC-Loc1 | DPC-Loc2 | DPC-Loc3 | DPC-Uniform |
---|---|---|---|---|

6.25 | 83.92008 | 83.91988 | 83.91985 | 83.91961 |

0.000625 | 83.92008 | 83.91988 | 83.91985 | 83.91961 |

K | T | r | $\mathit{\sigma}$ | ${\mathit{x}}_{\mathbf{max}}$ | $\mathit{\u03f5}$ |
---|---|---|---|---|---|

35 | 4/12 | 4.88% | 30% | 3.00 | ${10}^{-4}$ |

K | DPC-Uniform | DPC Loc1 | DPC Loc2 | DPC Loc3 | Binomial Method |
---|---|---|---|---|---|

35 | 0.6944 | 0.6967 | 0.6974 | 0.6976 | 0.6975 |

40 | 2.4782 | 2.4815 | 2.4823 | 2.4825 | 2.4825 |

45 | 5.7028 | 5.7050 | 5.7055 | 5.7057 | 5.7056 |

K | DPC-Uniform | DPC Loc1 | DPC Loc2 | DPC Loc3 | Binomial Method |
---|---|---|---|---|---|

35 | −0.1738 | −0.1740 | −0.1741 | −0.1741 | −0.1741 |

40 | −0.4422 | −0.4420 | −0.4420 | −0.4420 | −0.4420 |

45 | −0.7272 | −0.7268 | −0.7267 | −0.7266 | −0.7266 |

K | T | r | $\mathit{\sigma}$ | ${\mathit{x}}_{\mathbf{max}}$ |
---|---|---|---|---|

100 | 3 | 5% | 20% | 3.00 |

S | $\mathit{h}=0.2$ | $\mathit{h}=0.1$ | $\mathit{h}=0.05$ | Binomial Method |
---|---|---|---|---|

80 | 0.0138 | 0.0340 | 0.0361 | 0.0361 |

90 | 0.0257 | 0.0236 | 0.0234 | 0.0234 |

100 | 0.0147 | 0.0153 | 0.0154 | 0.0155 |

110 | 0.0104 | 0.0103 | 0.0102 | 0.0103 |

120 | 0.0064 | 0.0067 | 0.0068 | 0.0068 |

Runtime (s) | 0.2651 | 0.5294 | 1.2769 | None |

S | $\mathit{h}=0.2$ | $\mathit{h}=0.1$ | $\mathit{h}=0.05$ | Binomial Method |
---|---|---|---|---|

80 | 0.0340 | 0.0361 | 0.0361 | 0.0361 |

90 | 0.0236 | 0.0234 | 0.0234 | 0.0234 |

100 | 0.0153 | 0.0154 | 0.0154 | 0.0155 |

110 | 0.0103 | 0.0103 | 0.0103 | 0.0103 |

120 | 0.0067 | 0.0068 | 0.0068 | 0.0068 |

Runtime (s) | 0.4810 | 0.7951 | 3.4038 | None |

S | $\mathit{h}=0.2$ | $\mathit{h}=0.1$ | $\mathit{h}=0.05$ | Binomial Method |
---|---|---|---|---|

80 | 0.0361 | 0.0361 | 0.0361 | 0.0361 |

90 | 0.0234 | 0.0234 | 0.0234 | 0.0234 |

100 | 0.0154 | 0.0154 | 0.0155 | 0.0155 |

110 | 0.0102 | 0.0103 | 0.0103 | 0.0103 |

120 | 0.0067 | 0.0068 | 0.0068 | 0.0068 |

Runtime (s) | 0.5354 | 1.7565 | 9.5990 | None |

S | Scenario 1 (with ${\mathit{e}}_{\mathit{y}}$) | Scenario 2 (without ${\mathit{e}}_{\mathit{y}}$) | Binomial Method |
---|---|---|---|

80 | 0.0361 | −2.548 $\times {10}^{40}$ | 0.0361 |

90 | 0.0234 | −9.047 $\times {10}^{39}$ | 0.0234 |

100 | 0.0154 | −4.343 $\times {10}^{39}$ | 0.0155 |

110 | 0.0103 | −1.555 $\times {10}^{39}$ | 0.0103 |

120 | 0.0068 | −6.230 $\times {10}^{38}$ | 0.0068 |

S | Scenario 1 (with ${\mathit{e}}_{\mathit{y}}$) | Scenario 2 (without ${\mathit{e}}_{\mathit{y}}$) | Binomial Method |
---|---|---|---|

80 | 0.0361 | 0.0361 | 0.0361 |

90 | 0.0234 | 0.0234 | 0.0234 |

100 | 0.0154 | 0.0154 | 0.0155 |

110 | 0.0103 | 0.0103 | 0.0103 |

120 | 0.0068 | 0.0068 | 0.0068 |

K | T | r | $\mathit{\sigma}$ | ${\mathit{x}}_{\mathbf{max}}$ |
---|---|---|---|---|

100 | 6/12 | 8% | 10% | 3.00 |

**Table 15.**Optimal exercise boundary with varying power term m ($h=0.025$, ${h}_{a}=0.25\mathrm{h}$).

Parameter | $\mathit{m}=1$ | $\mathit{m}=2$ | $\mathit{m}=3$ | $\mathit{m}=4$ | $\mathit{m}=5$ |
---|---|---|---|---|---|

${s}_{f}\left(T\right)$ | 95.17035 | 90.80024 | 86.82542 | 83.19370 | 79.86161 |

S | $\mathit{m}=1$ | $\mathit{m}=2$ | $\mathit{m}=3$ | $\mathit{m}=4$ | $\mathit{m}=5$ |
---|---|---|---|---|---|

80 | 20.00000 | 20.00000 | 20.00000 | 20.00000 | 20.00051 |

100 | 1.63347 | 3.19848 | 4.70042 | 6.14327 | 7.53094 |

120 | 0.00185 | 0.20491 | 0.85746 | 1.79875 | 2.88475 |

S | $\mathit{m}=1$ | $\mathit{m}=2$ | $\mathit{m}=3$ | $\mathit{m}=4$ | $\mathit{m}=5$ |
---|---|---|---|---|---|

80 | −1.00000 | −1.00000 | −1.00000 | −1.00000 | −0.99282 |

100 | −0.39823 | −0.38896 | −0.38013 | −0.37174 | −0.36375 |

120 | −0.00080 | −0.03126 | −0.07634 | −0.11197 | −0.13754 |

S | $\mathit{m}=1$ | $\mathit{m}=2$ | $\mathit{m}=3$ | $\mathit{m}=4$ | $\mathit{m}=5$ |
---|---|---|---|---|---|

80 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00947 |

100 | 0.08271 | 0.04223 | 0.02870 | 0.02192 | 0.01784 |

120 | 0.00033 | 0.00446 | 0.00647 | 0.00683 | 0.00664 |

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**MDPI and ACS Style**

Nwankwo, C.; Dai, W.
Local Refinement and Adaptive Strategy for a System of Free Boundary Power Options with High Order Compact Differencing. *Axioms* **2023**, *12*, 602.
https://doi.org/10.3390/axioms12060602

**AMA Style**

Nwankwo C, Dai W.
Local Refinement and Adaptive Strategy for a System of Free Boundary Power Options with High Order Compact Differencing. *Axioms*. 2023; 12(6):602.
https://doi.org/10.3390/axioms12060602

**Chicago/Turabian Style**

Nwankwo, Chinonso, and Weizhong Dai.
2023. "Local Refinement and Adaptive Strategy for a System of Free Boundary Power Options with High Order Compact Differencing" *Axioms* 12, no. 6: 602.
https://doi.org/10.3390/axioms12060602