# Real-Option Valuation in a Finite-Time, Incomplete Market with Jump Diffusion and Investor-Utility Inflation

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## Abstract

**:**

## 1. Introduction

## 2. Model Description

#### 2.1. Fundamentals of the Model

#### 2.2. Time-Dependence of the Exponential Utility Function

## 3. Jump-Diffusion Model

#### 3.1. Distribution of the Jump Process

#### 3.2. Setup of the Multinomial Tree

#### 3.3. Discretisation of the Random Variables

^{rd}percentile and above the 66.7

^{th}percentile of the respective continuous distributions. For Figure 1c,d, the percentiles are the 20

^{th}, 80

^{th}respectively. Thus, the most extreme values for the $M=5$ case are more extreme than those for the $M=3$ case (consistent with the earlier statements about $\mathrm{max}\left\{\right|{x}_{i}\left|\right\}$), but they have lower weight.

## 4. Numerical Calculation

## 5. Simulation Results

#### 5.1. Time-Dependent Utility Function

#### 5.2. Inflation Effects

#### 5.3. Incorporation of Jump Diffusion into the Project Value

#### 5.3.1. Effect of Varying the Parameter ${\rho}_{X}$

#### 5.3.2. Effect of Varying Jump Rate $\lambda $

#### 5.3.3. Effect of Varying Skewness ${\mathrm{Sk}}_{\mathrm{X}}$

#### 5.3.4. Further Investigation of ${\rho}_{X}$, $\lambda $, ${\mathrm{Sk}}_{X}$

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Comparison between the double-exponential density function and its discrete approximations for the cases: (

**a**) ${p}_{X}=1/2,M=3,N=2$; (

**b**) ${p}_{X}=1/3,M=3,N=2$; (

**c**) ${p}_{X}=1/2,M=5,N=2$; (

**d**) ${p}_{X}=1/3,M=5,N=2$.

**Figure 2.**(

**a**) Case $T=10$, $\alpha =i$. Colour scale represents the undiscounted time value of the real option. The vertical axis (undiscounted project value) is plotted with a logarithmic scale. Time is plotted on the horizontal axis. Blue line: at the money; Red line: exercise threshold; Green line: Infinite-maturity exercise threshold; (

**b**) Case $T=30$, $\alpha =i$; (

**c**) Case $T=10$, $r=i$; (

**d**) Case $T=30$, $r=i$. Exercise threshold at $t=0$: (

**a**) 1.4731; (

**b**) 1.5226; (

**c**) 1.4816; (

**d**) 1.5425. Option value at the money ($t=0$): (

**a**) 0.1531; (

**b**) 0.1797; (

**c**) 0.1564; (

**d**) 0.1881. Calculation time (backwards induction step): (

**a**) 1.37 s; (

**b**) 5.37 s; (

**c**) 1.38 s; (

**d**) 5.31 s.

**Figure 3.**(

**a**) Case $T=20$, $\alpha =i=0.01$. Colour scale represents the undiscounted time value of the real option. The vertical axis (undiscounted project value) is plotted with a logarithmic scale, and the horizontal axis represents time. Blue line: at the money; Red line: exercise threshold; Green line: Infinite-maturity exercise threshold. (

**b**) Case $T=20$, $\alpha =i=0$. (

**c**) Case $T=20$, $\alpha =i=-0.01$.; (

**d**) Plot of undiscounted option value at the money, versus time to expiry, for the three cases represented in (

**a**–

**c**). Exercise threshold at $t=0$: (

**a**) 1.4593; (

**b**) 1.5102; (

**c**) 1.5652. Option value at the money ($t=0$): (

**a**) 0.1570; (

**b**) 0.1727; (

**c**) 0.1892. Calculation times (backwards induction step): (

**a**) 3.26 s; (

**b**) 3.19 s; (

**c**) 3.61 s.

**Figure 4.**Process as in Figure 3b, but incorporating jump diffusion with parameters $\lambda =0.1,\chi =1$. The parameter ${\rho}_{X}$ is, sequentially: (

**a**) 0.5; (

**b**) 0.6; (

**c**) 0.7; (

**d**) 0.8. Sub-figure (

**e**) Plot of undiscounted option value at the money, versus time to expiry, for the four cases represented in (

**a**–

**d**). Sub-figure (

**f**) Continuous and discrete (incorporating rounding of the ${x}_{i}$ values) approximations of the random variable ${\sigma}_{2}{\rho}_{X}\Delta X$, for the case ${\rho}_{X}=0.6$. Exercise threshold at $t=0$: (

**a**) 1.4736; (

**b**) 1.4483; (

**c**) 1.4105; (

**d**) 1.3563. Option value at the money ($t=0$): (

**a**) 0.1653; (

**b**) 0.1588 (

**c**) 0.1490; (

**d**) 0.1348. Calculation times (backwards induction step): (

**a**) 4.07 s (1.3×); (

**b**) 4.48 s (1.4×) ; (

**c**) 4.98 s (1.6×); (

**d**) 6.14 s (1.9×), where the figures in parentheses represent a multiple of the ${\rho}_{X}=0$ case of Figure 3b.

**Figure 5.**As in Figure 4d, but with jump rates $\lambda $ equal to: (

**a**) 2; (

**b**) 1; (

**c**) 0.5; (

**d**) 0.3; (

**e**) 0.2. (

**f**) Plot of undiscounted option valuen at the money, versus time to expiry, for the five cases represented in (

**a**–

**e**). Exercise threshold at $t=0$: (

**a**) 1.4664; (

**b**) 1.4481; (

**c**) 1.4272; (

**d**) 1.4075; (

**e**) 1.3902. Option value at the money ($t=0$): (

**a**) 0.1700; (

**b**) 0.1668 (

**c**) 0.1613; (

**d**) 0.1548; (

**e**) 0.1484. Calculation times (backwards induction step): (

**a**) 4.37 s (1.4×); (

**b**) 4.52 s (1.4×) ; (

**c**) 4.84 s (1.5×); (

**d**) 5.06 s (1.6×); (

**e**) 5.46 s (1.7×), where the figures in parentheses represent a multiple of the ${\rho}_{X}=0$ case of Figure 3b.

**Figure 6.**As in Figure 4b, but with skewness ${\mathrm{Sk}}_{\mathrm{X}}$ equal to: (

**a**) $-3$; (

**b**) 3; (

**c**) Plot of undiscounted option value at the money, versus time to expiry, for the three cases in ${\mathrm{Sk}}_{\mathrm{X}}=-3,0,3$. Exercise threshold at $t=0$: (

**a**) 1.4767; (

**b**) 1.3762. Option value at the money ($t=0$): (

**a**) 0.1614; (

**b**) 0.1359. Calculation times (backwards induction step): (

**a**) 4.12 s; (

**b**) 5.04 s.

**Figure 7.**Plots of exercise threshold (

**a**) and option value (

**b**) at time $t=0$, vs. jump rate $\lambda $, for different values of skewness (−3, 0, 3). Other parameters are given in the text. In particular, ${\rho}_{X}=0.6$. The comparison level when ${\rho}_{X}=0$ is indicated using a black dotted line.

**Figure 8.**Plots of exercise threshold (

**a**) and option value (

**b**) at time $t=0$, vs. skewness ${\mathrm{Sk}}_{\mathrm{X}}$, for different values of ${\rho}_{X}$ (0, 0.5, 0.6, 0.7, 0.8). Other parameters are given in the text. In particular, $\lambda =0.1$.

**Table 1.**Exercise thresholds (ET), option values at the money (OVATM), calculation times (backwards induction step) (CT), for multiple simulations of the process. The centre columns correspond to Figure 3b (${\rho}_{X}=0$), and Figure 4a–d (${\rho}_{X}=0.5,0.6,0.7,0.8$, respectively). The other columns were generated through changing only the values of M, N as indicated. Figures in parentheses represent calculation times as a multiple of the ${\rho}_{X}=0$ case.

${\mathit{\rho}}_{\mathit{X}}$ | M = 6, N = 3 | M = 5, N = 2 | M = 3, N = 2 | ||||||
---|---|---|---|---|---|---|---|---|---|

ET | OVATM | CT (s) | ET | OVATM | CT (s) | ET | OVATM | CT (s) | |

0 | 1.510 | 0.173 | 3.19 | 1.510 | 0.173 | 3.19 | 1.510 | 0.173 | 3.19 |

0.5 | 1.474 | 0.166 | 4.53 (1.4×) | 1.474 | 0.165 | 4.07 (1.3×) | 1.473 | 0.165 | 3.98 (1.2×) |

0.6 | 1.449 | 0.159 | 5.21 (1.6×) | 1.448 | 0.159 | 4.48 (1.4×) | 1.447 | 0.158 | 4.46 (1.4×) |

0.7 | 1.411 | 0.150 | 5.94 (1.9×) | 1.411 | 0.149 | 4.98 (1.6×) | 1.409 | 0.148 | 4.83 (1.5×) |

0.8 | 1.357 | 0.136 | 7.34 (2.3×) | 1.356 | 0.135 | 6.1 (1.9×) | 1.354 | 0.133 | 5.88 (1.8×) |

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**MDPI and ACS Style**

Hillman, T.; Zhang, N.; Jin, Z.
Real-Option Valuation in a Finite-Time, Incomplete Market with Jump Diffusion and Investor-Utility Inflation. *Risks* **2018**, *6*, 51.
https://doi.org/10.3390/risks6020051

**AMA Style**

Hillman T, Zhang N, Jin Z.
Real-Option Valuation in a Finite-Time, Incomplete Market with Jump Diffusion and Investor-Utility Inflation. *Risks*. 2018; 6(2):51.
https://doi.org/10.3390/risks6020051

**Chicago/Turabian Style**

Hillman, Timothy, Nan Zhang, and Zhuo Jin.
2018. "Real-Option Valuation in a Finite-Time, Incomplete Market with Jump Diffusion and Investor-Utility Inflation" *Risks* 6, no. 2: 51.
https://doi.org/10.3390/risks6020051