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Between ℙ and ℚ: The ℙ^{ℚ} Measure for Pricing in Asset Liability Management

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## Abstract

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## 1. Introduction

## 2. Solvency Capital Requirement

#### Guaranteed Minimum Accumulation Benefit

- If ${F}_{T}\ge G$, the policyholder receives ${F}_{T}$ and the insurer has no liabilities.
- If ${F}_{T}<G$, the policyholder receives G and the liabilities of the insurer are equal to $G-{F}_{T}$.

- First, the net policy value ${N}_{0}$ is determined, according to definition Equation (3).
- Thereafter, the fund value ${F}_{t}$ along with other explanatory variables is simulated according to the real-world measure up to time $t=1$.
- Subsequently, the values of $\mathrm{Put}({F}_{1},G)$ and $g({F}_{1},1)$ are evaluated for each trajectory. The value of $g({F}_{1},1)$ can be obtained directly from the trajectory of ${F}_{t}$, however, $\mathrm{Put}({F}_{1},G)$ requires a risk-neutral valuation for which the risk-neutral measure at time $t=1$ is required.
- Finally, the simulated values are combined to construct the loss distribution. The SCR corresponds to the 99.5% quantile of this distribution.

## 3. Dynamic Stochastic Volatility Model

#### 3.1. Heston Model

#### 3.2. VIX Heston Model

- The initial volatility $\sqrt{{v}_{0,t}}$ and the volatility of the volatility ${\gamma}_{t}$ are highly correlated to the VIX index, with correlation coefficients of 0.99 and 0.76, respectively.
- The long term volatility $\sqrt{{\overline{v}}_{t}}$ appears to be correlated to the VIX index trend line (estimated by a Kalman filter) with a correlation coefficient of 0.74.

## 4. Hedge Test

#### 4.1. Hedge Test Experiments

#### 4.1.1. Simulated Market

#### 4.1.2. Empirical Market

- The classical Delta-Vega hedge does not take changes of the parameters into account and appears to be the most unstable method. This strategy has the most and highest error “peaks” and is therefore most unreliable.
- The adjusted Delta-Vega hedge is still not perfect but appears to be more stable than the classical Delta-Vega hedge, the error “peaks” happen less frequently and are less pronounced. The error can be minimized by optimizing the correlation and volatility of ${\overline{v}}_{t}$ and ${\gamma}_{t}$, but the optimization can only be performed afterwards, which is not the objective of this test.
- The full hedge is the most stable out of the three strategies. It does not have any error “peaks” and outperforms the other two strategies in most cases. Moreover, this strategy does not depend on additional parameters which may introduce an error if chosen poorly, such as in the dynamic Heston Delta-Vega hedge.

## 5. VIX Heston Model Results

#### 5.1. Data and Calibration

#### 5.2. SCR Impact Study

#### 5.2.1. Guaranteed Minimum Accumulation Benefit

- A time-dependent risk-neutral measure where all parameters depend on the simulated state of the market.
- A risk-neutral measure where ${F}_{1}$ and ${v}_{1}$ depend on the simulated market and the risk-neutral parameters are equal to the parameters as observed on $t=0$. This measure is equivalent to the original risk-neutral measure that we have previously defined.
- A risk-neutral measure where ${F}_{1}$ and ${v}_{1}$ depend on the simulated market and the risk-neutral parameters are equal to the parameters as observed on $t=1$. We refer to this measure as the future risk-neutral measure.
- A risk-neutral measure where ${F}_{1}$ and ${v}_{1}$ depend on the simulated market and the risk-neutral parameters are equal to the realized regression model predictions at $t=1$ of Figure 3. This measure is different from the time-dependent risk-neutral measure, as it depends on the realized state VIX, instead of the simulated VIX. We refer to this measure as the future VIX risk-neutral measure.

#### 5.2.2. Discussion on Impact

- 2005–2007: The volatility in these years was relatively low and this translates to a somewhat higher SCR under the original measure and an even higher SCR under the time-dependent measure, which is similar to Scenario 2. The SCR under the future measure is rising, due to higher expected liabilities at $t=1$, indicating that more volatile times are coming.
- Early 2008: The market has not crashed yet, but volatility is starting to increase, resulting in a smaller difference between the time-dependent and original measures, which is comparable to Scenario 1. The future measure, however, takes the fact that the market will crash into account. Hence, the SCR is the highest under the future measure.
- Late 2008–2012: During these years, several spikes occurred in the implied volatility surface, which will increase the initial liabilities and therefore the expected losses will decrease, analogously to Scenario 3. This results in lower SCRs during this period. Moreover, we see that the SCRs under the future risk-neutral measure are lowest during these highly volatile periods, since this measure depends on the realized market at $t=1$, which has returned to its less volatile state. Consequently, the expected liabilities at $t=1$ and the SCR are the lowest under the future risk-neutral measure.
- 2013–2017: This period is comparable to 2005–2007, apart from the fact that the volatility is approximately constant throughout these years. This translates into an almost equal SCR prediction under the original and future risk-neutral measures.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Least-Squares Monte Carlo Method

## Appendix B. Dynamic Heston Model

## Appendix C. VIX Heston: UK and Europe

#### Appendix C.1. Calibrated Parameters

#### Appendix C.1.1. Parameters obtained from UK data

#### Appendix C.1.2. Parameters obtained from Europe data

#### Appendix C.2. Predicted Parameters

#### Appendix C.2.1. Parameters predicted for UK

#### Appendix C.2.2. Parameters predicted for Europe

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**Figure 2.**Mean error and mean squared error for different hedging strategies performed on monthly historical data.

**Figure 4.**Probability density functions of the one-year loss distribution for a variable annuity with the GMAB rider, under the original and time-dependent risk-neutral measure. Scenario 1 = average initial volatility; Scenario 2 = low initial volatility; and Scenario 3 = high initial volatility.

**Table 1.**Hedge errors of the different hedging strategies. The standard errors of the estimates are given in parentheses.

Frequency | ${\overline{\mathit{E}}}_{\mathbf{Mean}}$ | ${\overline{\mathit{E}}}_{\mathbf{Std}}$ |
---|---|---|

classical Delta-Vega: | ||

Once per week | −0.339 (0.0207) | 0.228 (0.0084) |

Once per day | −0.354 (0.0227) | 0.210 (0.0080) |

Adjusted Delta-Vega: | ||

Once per week | −0.204 (0.0097) | 0.137 (0.0072) |

Once per day | −0.220 (0.0071) | 0.114 (0.0047) |

Full hedge: | ||

Once per week | −0.044 (0.0040) | 0.068 (0.0042) |

Once per day | −0.051 (0.0030) | 0.045 (0.0022) |

**Table 2.**Out-of-sample accuracy of the regression models according to the error measures defined in Equation (40).

$\mathbf{SSE}$ | $\mathbf{MAE}$ | ${\mathit{R}}^{2}$ | ${\mathit{R}}_{\mathbf{Min}}^{2}$ | |
---|---|---|---|---|

US | ||||

VIX Heston | 0.2286 | 0.0124 | 0.8948 | 0.8159 |

Unrestricted | 0.0185 | 0.0034 | 0.9915 | 0.9798 |

UK | ||||

VIX Heston | 0.0577 | 0.0093 | 0.9340 | 0.7401 |

Unrestricted | 0.0126 | 0.0045 | 0.9855 | 0.9690 |

Europe | ||||

VIX Heston | 0.0433 | 0.0078 | 0.9389 | 0.7693 |

Unrestricted | 0.0154 | 0.0051 | 0.9783 | 0.9190 |

Parameter | Value |
---|---|

${F}_{0}$ | 1000 |

G | 1000 |

${T}_{\mathrm{GMAB}}$ | 10 |

r | 0.04 |

Parameter | Scenario 1 | Scenario 2 | Scenario 3 |
---|---|---|---|

${v}_{0}$ | 0.04 | 0.01 | 0.27 |

${\overline{v}}_{0}$ | 0.08 | 0.025 | 0.24 |

${\gamma}_{0}$ | 0.55 | 0.05 | 1.4 |

${\alpha}_{\mathrm{GMAB}}$ | 0.0174 | 0.0057 | 0.0345 |

**Table 5.**Solvency Capital Requirement of the scenarios for the original and time-dependent risk-neutral measure.

Original | Time-Dependent | |
---|---|---|

Scenario 1 | 170.1 | 195.5 |

Scenario 2 | 166.9 | 237.5 |

Scenario 3 | 178.8 | 150.8 |

Value | % | |
---|---|---|

Mean absolute difference | 41.1 | 28.7% |

Maximum absolute difference | 85.3 | 52.0% |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Van Dijk, M.T.P.; De Graaf, C.S.L.; Oosterlee, C.W. Between ℙ and ℚ: The ℙ^{ℚ} Measure for Pricing in Asset Liability Management. *J. Risk Financial Manag.* **2018**, *11*, 67.
https://doi.org/10.3390/jrfm11040067

**AMA Style**

Van Dijk MTP, De Graaf CSL, Oosterlee CW. Between ℙ and ℚ: The ℙ^{ℚ} Measure for Pricing in Asset Liability Management. *Journal of Risk and Financial Management*. 2018; 11(4):67.
https://doi.org/10.3390/jrfm11040067

**Chicago/Turabian Style**

Van Dijk, Marcel T. P., Cornelis S. L. De Graaf, and Cornelis W. Oosterlee. 2018. "Between ℙ and ℚ: The ℙ^{ℚ} Measure for Pricing in Asset Liability Management" *Journal of Risk and Financial Management* 11, no. 4: 67.
https://doi.org/10.3390/jrfm11040067