# Jump-Diffusion Models for Valuing the Future: Discounting under Extreme Situations

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Main Definitions

#### 2.2. Diffusion Process in the Presence of Random Jumps

#### The Ornstein–Uhlenbeck Process and Poissonian Jumps

## 3. Results

#### 3.1. Asymptotic Discount Function

#### Bounded and Symmetric Jump Density

#### 3.2. Some Specific Jump Distributions

#### 3.2.1. Fixed Jump Amplitudes

#### 3.2.2. Laplacian Jump Amplitudes

- 1.
- When $c<1$ (i.e., $\gamma <\alpha \sqrt{2}$) we prove in the Appendix C that the function $\varphi \left(t\right)$ is given by$$\varphi \left(t\right)=-\frac{t}{1-{c}^{2}}-\frac{1}{2\alpha}\left(\right)open="["\; close="]">\frac{1}{1-c}ln\left(\right)open="("\; close=")">1-c(1-{e}^{-\alpha t})$$In this case, the discount function is finite and follows from Equation (65) after substituting Equation (66). Figure 3 illustrate this result considering the OU parameters estimated in Ref. [17] while considering different jumps frequencies $\lambda $ and different jumps amplitudes in terms of c. For large values of t, we have$$\varphi \left(t\right)\simeq -\frac{t}{1-{c}^{2}},\phantom{\rule{2.em}{0ex}}(t\to \infty ).$$Since ${D}^{\left(0\right)}\left(t\right)\simeq {e}^{-{r}_{\infty}^{\left(0\right)}t}$ as $t\to \infty $ (cf. Equations (36) and (43)), we finally obtain$$D\left(t\right)\simeq exp\left(\right)open="\{"\; close="\}">-\left(\right)open="["\; close="]">{r}_{\infty}^{\left(0\right)}-\lambda {c}^{2}/(1-{c}^{2}),\phantom{\rule{2.em}{0ex}}(t\to \infty ),$$$${r}_{\infty}={r}_{\infty}^{\left(0\right)}-\frac{\lambda {c}^{2}}{1-{c}^{2}}<{r}_{\infty}^{\left(0\right)},\phantom{\rule{2.em}{0ex}}(c<1),$$
- 2.
- When $c>1$ (i.e., $\gamma >\alpha \sqrt{2}$), we prove in the Appendix C that the discount becomes infinite for times greater than a critical time,$$D\left(t\right)=\infty ,\phantom{\rule{2.em}{0ex}}(t\ge {t}^{\ast}),$$$${t}^{\ast}=-\frac{1}{\alpha}ln\left(\right)open="("\; close=")">1-\frac{1}{c}$$
- 3.
- For the threshold case $c=1$ (i.e., $\gamma =\alpha \sqrt{2}$), the discount function grows exponentially. Thus, in the Appendix C we show that$$D\left(t\right)\simeq exp\left(\right)open="\{"\; close="\}">\frac{\lambda}{2\alpha}{e}^{\alpha t}$$Note that this behavior is not contradictory with our previous results since, as ${r}_{\infty}^{\left(0\right)}>0$ and$$1<\frac{2{\alpha}^{2}}{{\gamma}^{2}}<1+\frac{\lambda}{{r}_{\infty}^{\left(0\right)}},$$${r}_{\infty}$ becomes negative, and discount turns into an increasing function for t large enough.

#### 3.3. Discount in the Continuous Time Random Walk Formalism

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. The Method of Characteristics

## Appendix B. Long-Run Discount Rate for Asymmetric Jump Distributions

## Appendix C. Discount Function for Laplacian Jumps

## References

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**Figure 1.**The discount rate (54) (in %) as a function of time (in years). We present the effects of the addition of jumps to the Ornstein–Uhlenbeck process by modifying the scaled jump size $\left|\gamma \right|/\alpha $ < 0. Jumps frequency $\lambda $ is 1/50 years. General parameters for the Ornstein–Uhlenbeck process are those estimated by Ref. [17] for the case of the United States of America, whose estimated parameters are $\widehat{m}=0.0319$ year${}^{-1}$, $\widehat{\alpha}=0.0603$ year${}^{-1}$, $\widehat{{k}^{2}}=10.03\times {10}^{-5}$ year${}^{-3}$.

**Figure 2.**The long-run discount rate ${r}_{\infty}$ as a function of the the scaled jump size ($\left|\gamma \right|/\alpha $) for jumps with fixed and bounded amplitudes ($\gamma >0$ and $\gamma <0$, cf. Equation (57)), with two fixed and bounded symmetric jumps $\pm \gamma $ (cf. Equation (58)) and for Laplacian jumps with average absolute value equals to $\gamma $ (cf. Equation (68)). In all cases, jumps frequency $\lambda $ is 1/50 years and ${r}_{\infty}^{\left(0\right)}=1.81\%$.

**Figure 3.**Discount function for Laplacian jump amplitudes with $c<1$ (cf. Equations (65) and (66)) as a function of time (in years) and for different jumps time frequency $\lambda $. We take the Ornstein–Unlenbeck parameters estimated somewhere else with initial interest rate ${r}_{0}=1\%$ [17] with United States of America (USA) and Sweden (SWE) dates, which are considered to be stable countries. (

**a**,

**b**) we explore the effect of increasing jumps time frequency (thinner lines, $1/\lambda $ = {500 and 50 years}) while fixing jumps amplitude ($c=0.5$, cf. Equation (64)). The panel figures (

**a**,

**b**) show that the higher the frequency, the lower the discount function curve. The panel figures (

**c**,

**d**) explore the effect of increasing jumps amplitude (thinner lines, c = {0.2, 0.9}) while fixing jumps frequency ($1/\lambda $ = 50 years). The higher the c (jump size), the lower the discount. The case of the United States of America (USA), whose Ornstein–Uhlenbeck estimated parameters are $\widehat{m}=0.0319$ year${}^{-1}$, $\widehat{\alpha}=0.0603$ year${}^{-1}$, $\widehat{{k}^{2}}=10.03\times {10}^{-5}$ year${}^{-3}$. The case of Sweden (SWE), whose Ornstein–Uhlenbeck estimated parameters are $\widehat{m}=0.0279$ year${}^{-1}$, $\widehat{\alpha}=0.0676$ year${}^{-1}$, $\widehat{{k}^{2}}=16.9\times {10}^{-5}$ year${}^{-3}$.

**Figure 4.**Critical time as a function of the Laplacian jumps amplitude $c=\gamma /\left(\sqrt{2}\widehat{\alpha}\right)>1$ (cf. Equation (70)) and for several values of $\widehat{\alpha}$ attributed to different countries. We take the Ornstein–Uhlenbeck $\widehat{\alpha}$ parameter estimated somewhere else [17]: $1/\left(6.068years\right)$ for Netherlands (NED), $1/\left(16.58years\right)$ for United States of America (USA), $1/\left(70.42years\right)$ for Canada (CAN), and $1/\left(188.7years\right)$ for Japan (JAP). The inset shows in further detail the cases with shorter critical times.

**Table 1.**Summary of the results proved in Section 3. The Ornstein–Uhlenbeck diffusion process already shows that the presence of noise ($k\ne 0$) reduces the long-run discount rate ${r}_{\infty}^{\left(0\right)}$. The inclusion of Poissonian jumps with specific scenarios leads to several discount functions $D\left(t\right)$ and several long-run discount rate ${r}_{\infty}$.

Model | Discount |
---|---|

Main definitions | Discount function: |

$dx/dt=r$ | $D\left(t\right)=\mathbb{E}\left(\right)open="["\; close="]">exp\left(\right)open="("\; close=")">-{\int}_{0}^{t}r\left({t}^{\prime}\right)d{t}^{\prime}$ |

$dr/dt=f\left(r\right)+g\left(t\right)\xi \left(t\right)+n\left(t\right)$ | Discount rate: $lnD\left(t\right)/t$ |

Long-run discount rate: | |

${r}_{\infty}={lim}_{t\to \infty}lnD\left(t\right)/t$ | |

Ornstein–Uhlenbeck (OU) | |

$dr/dt=-\alpha (r-m)dt+k\xi \left(t\right)$ | ${r}_{\infty}^{\left(0\right)}=m-{k}^{2}/\left(2{\alpha}^{2}\right)$ |

OU and Poissonian jumps | |

$dr/dt=-\alpha (r-m)dt+k\xi \left(t\right)+n\left(t\right)$ | |

$n\left(t\right)={\sum}_{j}{\gamma}_{j}\delta (t-{t}_{j})$ | |

Jumps size PDF $h\left({\gamma}_{j}\right)$ | |

Poissonian $\tau ={t}_{i+1}-{t}_{i}$ time interval PDF | |

$\psi \left(\tau \right)=\lambda {e}^{-\lambda \tau}$ | |

If $\tilde{h}(-i/\alpha )$ is finite | ${r}_{\infty}={r}_{\infty}^{\left(0\right)}+\lambda \left(\right)open="["\; close="]">1-\tilde{h}(-i/\alpha )$ |

If $\tilde{h}(-i/\alpha )$ is finite and $\tilde{h}(-i/\alpha )>1$ | ${r}_{\infty}<{r}_{\infty}^{\left(0\right)}$ |

If $\tilde{h}(-i/\alpha )$ is finite and jumps are symmetric | ${r}_{\infty}<{r}_{\infty}^{\left(0\right)}$ |

If jumps have two fixed amplitudes $\pm \gamma $ | ${r}_{\infty}={r}_{\infty}^{\left(0\right)}+\lambda [1-cosh(\gamma /\alpha )]<{r}_{\infty}^{\left(0\right)}$ |

Laplacian jumps with absolute jump average $\gamma $ | |

If $0<\gamma <\alpha \sqrt{2}$ | ${r}_{\infty}={r}_{\infty}^{\left(0\right)}-\frac{\lambda}{2{\alpha}^{2}/{\overline{\gamma}}^{2}-1}<{r}_{\infty}^{\left(0\right)}$ |

If $\gamma >\alpha \sqrt{2}$ | Not defined |

Critical explosive time | |

If jumps have one-fixed increasing amplitude | ${r}_{\infty}={r}_{\infty}^{\left(0\right)}+\lambda \left(\right)open="("\; close=")">1-{e}^{-\left|\gamma \right|/\alpha}$ |

If jumps have one-fixed decreasing amplitude | ${r}_{\infty}={r}_{\infty}^{\left(0\right)}+\lambda \left(\right)open="("\; close=")">1-{e}^{\left|\gamma \right|/\alpha}$ |

If jumps have two-fixed amplitudes $\pm \gamma $ | ${r}_{\infty}={r}_{\infty}^{\left(0\right)}+\lambda [1-cosh(\gamma /\alpha )]<{r}_{\infty}^{\left(0\right)}$ |

Continuous Time Random Walk | |

$dr/dt=n\left(t\right)$ | |

Laplacian jumps with absolute jump average $\gamma $ | Not defined |

Critical explosive time ${t}^{\ast}=\sqrt{2}/\gamma $ |

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**MDPI and ACS Style**

Masoliver, J.; Montero, M.; Perelló, J.
Jump-Diffusion Models for Valuing the Future: Discounting under Extreme Situations. *Mathematics* **2021**, *9*, 1589.
https://doi.org/10.3390/math9141589

**AMA Style**

Masoliver J, Montero M, Perelló J.
Jump-Diffusion Models for Valuing the Future: Discounting under Extreme Situations. *Mathematics*. 2021; 9(14):1589.
https://doi.org/10.3390/math9141589

**Chicago/Turabian Style**

Masoliver, Jaume, Miquel Montero, and Josep Perelló.
2021. "Jump-Diffusion Models for Valuing the Future: Discounting under Extreme Situations" *Mathematics* 9, no. 14: 1589.
https://doi.org/10.3390/math9141589