# Quantifying the Model Risk Inherent in the Calibration and Recalibration of Option Pricing Models

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## Abstract

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## 1. Introduction

- Parameter uncertainty (and sensitivity to parameters)—let us call this “Type 0” model risk for short. If model parameters need to be statistically estimated, then they will only be known up to some level of statistical confidence, and this parameter uncertainty induces uncertainty regarding the correctness of the model outputs. Examples of where this type of risk is considered explicitly in the literature include Löffler (2003); Bannör and Scherer (2013) and Kerkhof et al. (2010).
- Inability to fit a model to a full set of simultaneous market observations —this is “calibration error”, let’s call this “Type 1” model risk for short. “Calibration” in this context refers to choosing model parameters in such a way that prices for derivative financial instruments that are implied by the model match market prices at one given point in time. Thus, calibration error on single-time-point (a.k.a. “cross-sectional”) market data means that these data already contradict the model assumptions. The classical example of this is the Black/Scholes implied volatility smile.
- Change in parameters due to recalibration—let us call this “Type 2” model risk for short. Once one moves from one day to the next, this aspect of model risk becomes apparent: in order to again fit the market as closely as possible, it is common practice in the industry to recalibrate models. This recalibration results in model parameters (which the models assume to be fixed) changing from day-to-day, contradicting the model assumptions.
- The “true” dynamics of state variables do not match model dynamics—let us call this violation of model assumptions “Type 3” model risk. The classical example of this is the econometric rejection of the hypothesis that asset prices follow geometric Brownian motion, thus invalidating the key assumption in the seminal model of Black and Scholes (1973). This type of model risk is considered, for example, in Kerkhof et al. (2010), who also relate this to identification risk, which they define as risk that “arises when observationally indistinguishable models have different consequences for capital reserves”. Boucher et al. (2014) present a method for making value-at-risk more robust with respect to this source of model risk by “learning” from the results of model backtesting. Type 3 model risk would impact, in particular, the effectiveness of hedging strategies based on a model, for example Detering and Packham (2016) take the approach of measuring model risk that is based on the residual profit/loss from hedging in a misspecified model.

- Less stringent requirements of an exact fit to market observations (Type 1) allows for less frequent recalibration (Type 2).
- Instead of different model dynamics (Type 3), one could consider a parameterised family of models (Type 2).
- Regime-switching models “legalise” changes in parameters, so Type 2 becomes more like Type 3.
- Adding parameters shifts model risk from Type 1 to Type 2 (or, to a certain extent, to Type 0).
- Adding state variables shifts model risk from Type 2 to Type 3.

## 2. Quantifying Model Risk by Relative Entropy

#### 2.1. Quantifying Calibration Error

#### 2.2. Including Model Risk Due to Recalibration

#### 2.3. The Treatment of Latent State Variables

## 3. Numerical Implementation

- (1)
- Produce ${P}^{\left(0\right)}$ from a parametric model that is based on an initial guess of the model parameters (and latent state variables, where required).
- (2)
- Solve, for ${Q}^{\left(0\right)}$, via Lagrange multipliers for the constrained problem that minimises $D\left({Q}^{\left(0\right)}\right|\left|{P}^{\left(0\right)}\right)$.
- (3)
- Solve, for ${P}^{\left(1\right)}$, to obtain model parameters for the ${P}^{\left(1\right)}$ that minimises $D\left({Q}^{\left(0\right)}\right|\left|{P}^{\left(1\right)}\right)$.
- (4)
- Iterate steps 2 and 3: ${P}^{\left(n\right)}\to {Q}^{\left(n\right)}\to {P}^{(n+1)}$ until convergence.

#### 3.1. Step 1

#### 3.2. Step 2

#### 3.3. Steps 3 and 4

## 4. Examining the Trade-Off between Calibration Error and Model Risk Due to Recalibration

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation of the Recalibration Model Risk Quantity in the Black/Scholes Model

## References

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**Figure 1.**Graphic illustration of the mathematical definitions of model risks for (

**a**) observable state variable, (

**b**) latent state variable.

**Figure 2.**Decomposition of model risks of Apple (AAPL) options while using the Black/Scholes model as the nominal model, for a selection of option maturity dates (given as the title of each graph). The vertical axis denotes the numerical value of the relative entropy.

**Figure 3.**Decomposition of model risks of Google (GOOG) options while using the Heston model as the nominal model, for a selection of option maturity dates (given as the title of each graph). The vertical axis denotes the numerical value of the relative entropy.

**Table 1.**Model risks (in terms of relative entropy) under different models and recalibration frequencies for AAPL.

Risk Measure | Black-Scholes | Heston | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

1 Day | 3 Days | 1 Week | 2 Weeks | 1 Quarter | 1 Day | 3 Days | 1 Week | 2 Weeks | 1 Quarter | ||

Aggregate Model Risk | |||||||||||

mean | 0.070 | 0.071 | 0.073 | 0.073 | 0.085 | 0.037 | 0.035 | 0.038 | 0.039 | 0.046 | |

median | 0.045 | 0.047 | 0.047 | 0.051 | 0.057 | 0.005 | 0.005 | 0.005 | 0.006 | 0.010 | |

Quantile | 99% | 0.474 | 0.427 | 0.471 | 0.455 | 0.462 | 0.508 | 0.462 | 0.512 | 0.503 | 0.649 |

95% | 0.212 | 0.221 | 0.221 | 0.212 | 0.251 | 0.173 | 0.169 | 0.177 | 0.177 | 0.185 | |

90% | 0.158 | 0.160 | 0.165 | 0.160 | 0.192 | 0.096 | 0.097 | 0.098 | 0.105 | 0.121 | |

75% | 0.092 | 0.094 | 0.095 | 0.099 | 0.112 | 0.027 | 0.027 | 0.029 | 0.031 | 0.034 | |

Calibration Error | |||||||||||

mean | 0.055 | 0.066 | 0.069 | 0.072 | 0.084 | 0.008 | 0.026 | 0.032 | 0.036 | 0.045 | |

median | 0.038 | 0.043 | 0.045 | 0.049 | 0.057 | 0.001 | 0.003 | 0.004 | 0.005 | 0.009 | |

Quantile | 99% | 0.239 | 0.397 | 0.433 | 0.443 | 0.455 | 0.163 | 0.416 | 0.496 | 0.495 | 0.648 |

95% | 0.171 | 0.207 | 0.212 | 0.210 | 0.249 | 0.015 | 0.130 | 0.158 | 0.168 | 0.182 | |

90% | 0.135 | 0.151 | 0.158 | 0.158 | 0.191 | 0.008 | 0.065 | 0.083 | 0.097 | 0.119 | |

75% | 0.075 | 0.087 | 0.091 | 0.096 | 0.111 | 0.003 | 0.012 | 0.019 | 0.026 | 0.034 | |

Model Risk due to Recalibration | |||||||||||

mean | 0.024 | 0.009 | 0.005 | 0.002 | 0.001 | 0.057 | 0.019 | 0.011 | 0.006 | 0.001 | |

median | 0.002 | 0.001 | 0.001 | 0.000 | 0.000 | 0.004 | 0.001 | 0.001 | 0.001 | 0.000 | |

Quantile | 99% | 0.458 | 0.196 | 0.057 | 0.030 | 0.012 | 0.630 | 0.226 | 0.117 | 0.067 | 0.011 |

95% | 0.103 | 0.034 | 0.021 | 0.010 | 0.005 | 0.315 | 0.108 | 0.062 | 0.032 | 0.006 | |

90% | 0.056 | 0.020 | 0.011 | 0.006 | 0.002 | 0.164 | 0.052 | 0.031 | 0.016 | 0.004 | |

75% | 0.013 | 0.006 | 0.004 | 0.002 | 0.001 | 0.055 | 0.016 | 0.011 | 0.005 | 0.001 |

**Table 2.**Model risks (in terms of relative entropy) under different models and recalibration frequencies for GOOG.

Risk Measure | Black-Scholes | Heston | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

1 Day | 3 Days | 1 Week | 2 Weeks | 1 Quarter | 1 Day | 3 Days | 1 Week | 2 Weeks | 1 Quarter | ||

Aggregate Model Risk | |||||||||||

mean | 0.165 | 0.168 | 0.165 | 0.165 | 0.188 | 0.115 | 0.105 | 0.113 | 0.114 | 0.127 | |

median | 0.109 | 0.111 | 0.112 | 0.115 | 0.138 | 0.053 | 0.050 | 0.055 | 0.053 | 0.059 | |

Quantile | 99% | 0.705 | 0.722 | 0.722 | 0.699 | 0.728 | 0.726 | 0.668 | 0.711 | 0.699 | 0.744 |

95% | 0.519 | 0.530 | 0.521 | 0.496 | 0.549 | 0.475 | 0.438 | 0.455 | 0.467 | 0.560 | |

90% | 0.393 | 0.391 | 0.387 | 0.385 | 0.408 | 0.330 | 0.287 | 0.320 | 0.324 | 0.380 | |

75% | 0.219 | 0.223 | 0.214 | 0.215 | 0.256 | 0.145 | 0.137 | 0.142 | 0.153 | 0.157 | |

Calibration Error | |||||||||||

mean | 0.125 | 0.155 | 0.158 | 0.162 | 0.187 | 0.055 | 0.083 | 0.103 | 0.107 | 0.126 | |

median | 0.082 | 0.102 | 0.106 | 0.111 | 0.137 | 0.006 | 0.022 | 0.043 | 0.046 | 0.058 | |

Quantile | 99% | 0.655 | 0.701 | 0.709 | 0.696 | 0.728 | 0.626 | 0.651 | 0.706 | 0.688 | 0.744 |

95% | 0.400 | 0.482 | 0.507 | 0.494 | 0.548 | 0.329 | 0.381 | 0.438 | 0.445 | 0.557 | |

90% | 0.295 | 0.359 | 0.362 | 0.379 | 0.408 | 0.172 | 0.250 | 0.298 | 0.306 | 0.375 | |

75% | 0.171 | 0.208 | 0.207 | 0.212 | 0.255 | 0.034 | 0.103 | 0.127 | 0.141 | 0.155 | |

Model Risk due to Recalibration | |||||||||||

mean | 0.036 | 0.014 | 0.008 | 0.004 | 0.001 | 0.119 | 0.044 | 0.012 | 0.012 | 0.003 | |

median | 0.004 | 0.002 | 0.001 | 0.001 | 0.000 | 0.043 | 0.016 | 0.003 | 0.003 | 0.001 | |

Quantile | 99% | 0.405 | 0.172 | 0.091 | 0.050 | 0.010 | 0.757 | 0.255 | 0.087 | 0.077 | 0.012 |

95% | 0.173 | 0.066 | 0.035 | 0.018 | 0.004 | 0.543 | 0.187 | 0.056 | 0.064 | 0.011 | |

90% | 0.109 | 0.038 | 0.022 | 0.012 | 0.003 | 0.411 | 0.151 | 0.034 | 0.045 | 0.009 | |

75% | 0.036 | 0.013 | 0.008 | 0.004 | 0.001 | 0.142 | 0.052 | 0.016 | 0.013 | 0.003 |

**Table 3.**Model risks (in terms of relative entropy) under different models, by maturity buckets, for AAPL (daily recalibration frequency).

Risk Measure | Black-Scholes | Heston | |||||||
---|---|---|---|---|---|---|---|---|---|

All | 0–0.2 Year | 0.2–0.7 Year | >0.7 Year | All | 0–0.2 Year | 0.2–0.7 Year | >0.7 Year | ||

Aggregate Model Risk | |||||||||

mean | 0.070 | 0.041 | 0.066 | 0.109 | 0.037 | 0.039 | 0.024 | 0.046 | |

median | 0.045 | 0.021 | 0.052 | 0.081 | 0.005 | 0.004 | 0.004 | 0.008 | |

Quantile | 99% | 0.474 | 0.471 | 0.266 | 0.567 | 0.508 | 0.590 | 0.302 | 0.472 |

95% | 0.212 | 0.120 | 0.185 | 0.282 | 0.173 | 0.243 | 0.091 | 0.191 | |

90% | 0.158 | 0.081 | 0.143 | 0.213 | 0.096 | 0.074 | 0.066 | 0.142 | |

75% | 0.092 | 0.047 | 0.090 | 0.141 | 0.027 | 0.021 | 0.015 | 0.054 | |

Calibration Error | |||||||||

mean | 0.055 | 0.026 | 0.057 | 0.087 | 0.008 | 0.012 | 0.004 | 0.007 | |

median | 0.038 | 0.014 | 0.048 | 0.068 | 0.001 | 0.001 | 0.001 | 0.001 | |

Quantile | 99% | 0.239 | 0.157 | 0.230 | 0.289 | 0.163 | 0.072 | 0.052 | 0.116 |

95% | 0.171 | 0.091 | 0.157 | 0.212 | 0.015 | 0.021 | 0.011 | 0.017 | |

90% | 0.135 | 0.062 | 0.126 | 0.181 | 0.008 | 0.006 | 0.007 | 0.010 | |

75% | 0.075 | 0.036 | 0.077 | 0.128 | 0.003 | 0.002 | 0.003 | 0.003 | |

Model Risk due to Recalibration | |||||||||

mean | 0.024 | 0.020 | 0.013 | 0.038 | 0.057 | 0.062 | 0.048 | 0.060 | |

median | 0.002 | 0.002 | 0.001 | 0.003 | 0.004 | 0.004 | 0.003 | 0.009 | |

Quantile | 99% | 0.458 | 0.506 | 0.116 | 0.662 | 0.630 | 0.702 | 0.536 | 0.512 |

95% | 0.103 | 0.066 | 0.066 | 0.169 | 0.315 | 0.443 | 0.274 | 0.247 | |

90% | 0.056 | 0.030 | 0.042 | 0.102 | 0.164 | 0.223 | 0.127 | 0.160 | |

75% | 0.013 | 0.009 | 0.010 | 0.031 | 0.055 | 0.030 | 0.048 | 0.091 |

**Table 4.**Model risks (in terms of relative entropy) under different models, by maturity buckets, for GOOG (daily recalibration frequency).

Risk Measure | Black-Scholes | Heston | |||||||
---|---|---|---|---|---|---|---|---|---|

All | 0–0.2 Year | 0.2–0.7 Year | >0.7 Year | All | 0–0.2 Year | 0.2–0.7 Year | >0.7 Year | ||

Aggregate Model Risk | |||||||||

mean | 0.165 | 0.167 | 0.162 | 0.164 | 0.115 | 0.115 | 0.120 | 0.109 | |

median | 0.109 | 0.104 | 0.099 | 0.120 | 0.053 | 0.052 | 0.057 | 0.048 | |

Quantile | 99% | 0.705 | 0.736 | 0.709 | 0.661 | 0.726 | 0.714 | 0.707 | 0.736 |

95% | 0.519 | 0.527 | 0.551 | 0.459 | 0.475 | 0.468 | 0.500 | 0.476 | |

90% | 0.393 | 0.417 | 0.382 | 0.371 | 0.330 | 0.351 | 0.322 | 0.304 | |

75% | 0.219 | 0.221 | 0.216 | 0.218 | 0.145 | 0.149 | 0.155 | 0.136 | |

Calibration Error | |||||||||

mean | 0.125 | 0.121 | 0.123 | 0.132 | 0.055 | 0.055 | 0.052 | 0.057 | |

median | 0.082 | 0.074 | 0.075 | 0.099 | 0.006 | 0.005 | 0.006 | 0.006 | |

Quantile | 99% | 0.655 | 0.662 | 0.650 | 0.639 | 0.626 | 0.658 | 0.531 | 0.665 |

95% | 0.400 | 0.403 | 0.423 | 0.391 | 0.329 | 0.349 | 0.311 | 0.330 | |

90% | 0.295 | 0.298 | 0.301 | 0.285 | 0.172 | 0.155 | 0.185 | 0.171 | |

75% | 0.171 | 0.162 | 0.160 | 0.176 | 0.034 | 0.031 | 0.034 | 0.034 | |

Model Risk due to Recalibration | |||||||||

mean | 0.036 | 0.038 | 0.039 | 0.033 | 0.119 | 0.117 | 0.117 | 0.124 | |

median | 0.004 | 0.005 | 0.003 | 0.005 | 0.043 | 0.043 | 0.037 | 0.049 | |

Quantile | 99% | 0.405 | 0.419 | 0.479 | 0.284 | 0.757 | 0.746 | 0.767 | 0.754 |

95% | 0.173 | 0.176 | 0.188 | 0.155 | 0.543 | 0.510 | 0.558 | 0.554 | |

90% | 0.109 | 0.111 | 0.115 | 0.098 | 0.411 | 0.390 | 0.418 | 0.418 | |

75% | 0.036 | 0.038 | 0.032 | 0.036 | 0.142 | 0.144 | 0.137 | 0.146 |

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**MDPI and ACS Style**

Feng, Y.; Rudd, R.; Baker, C.; Mashalaba, Q.; Mavuso, M.; Schlögl, E.
Quantifying the Model Risk Inherent in the Calibration and Recalibration of Option Pricing Models. *Risks* **2021**, *9*, 13.
https://doi.org/10.3390/risks9010013

**AMA Style**

Feng Y, Rudd R, Baker C, Mashalaba Q, Mavuso M, Schlögl E.
Quantifying the Model Risk Inherent in the Calibration and Recalibration of Option Pricing Models. *Risks*. 2021; 9(1):13.
https://doi.org/10.3390/risks9010013

**Chicago/Turabian Style**

Feng, Yu, Ralph Rudd, Christopher Baker, Qaphela Mashalaba, Melusi Mavuso, and Erik Schlögl.
2021. "Quantifying the Model Risk Inherent in the Calibration and Recalibration of Option Pricing Models" *Risks* 9, no. 1: 13.
https://doi.org/10.3390/risks9010013