# Sparse Modeling Approach to the Arbitrage-Free Interpolation of Plain-Vanilla Option Prices and Implied Volatilities

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Relation between Terminal Density and Option Price

#### 2.2. Matrix Representation of the Relation between Option Price and Terminal Density

#### 2.3. The Difficulty in Implying the Terminal Density from Option Prices

- application of a basis transformation from $\varphi $ to ${\varphi}^{\prime}$ via ${\varphi}^{\prime}={V}^{T}\varphi $
- weighting the elements of ${\varphi}^{\prime}$ with the singular values S to get ${\mathrm{Pr}}^{\prime}$ via ${\mathrm{Pr}}^{\prime}=S{\varphi}^{\prime}$
- application of a basis transformation from ${\mathrm{Pr}}^{\prime}$ to Pr via $\mathrm{Pr}=U{\mathrm{Pr}}^{\prime}$

#### 2.4. Rapid Decay of the Kernel Matrix Singular Values

#### 2.5. Optimization Problem for Finding the Density

#### 2.6. Finding a Solution to the Optimization Problem

`CVXPY`(Agrawal et al. 2018; Diamond and Boyd 2016), available as a package for the Python programming language.

`ECOS`(Domahidi et al. 2013) and

`SCS`(O’Donoghue et al. 2016) both deliver excellent performances, especially with systems, such as the system we attempt to solve, which usually have only a few degrees of freedom (remember that ${\varphi}^{\prime}$ has only Q entries). Therefore, we refer readers who are interested in details of the implementation of these solvers to the respective papers. In our case,

`ECOS`seemed to be a good choice for the numerical solver.

#### 2.7. A Measure for the Similarity of Probability Distributions

## 3. Examples

#### 3.1. Normal Density

#### 3.2. Log-Normal Density

`ECOS`solver, probably because we retained a large number of singular values Q, which allowed for many similarly good solutions. This could probably be resolved by either fine-tuning of numerical parameters in

`ECOS`or by reducing the number of singular values.

#### 3.3. Multimodal Density

#### 3.4. Density Implied from Prices with Arbitrage

#### 3.5. Density Implied from S&P 500 Option Prices

`ECOS`solver, so we divided all strikes in the inputs by 1000. The same transformation was also applied to the forward price. This simple rescaling of the problem solved the numerical issues we encountered with the original inputs.

## 4. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

SVD | Singular Value Decomposition |

LV | Local Volatility |

LSV | Local Stochastic Volatility |

SABR | Stochastic Alpha, Beta, Rho |

SPX500 | Standard & Poor’s 500 Stock Index |

ITM | In-The-Money |

OTM | Out-Of-The-Money |

## Appendix A. Treatment of In-the-Money Options in the Error Function Calculation

## Appendix B. Ideas for Performance Optimization

## Note

1 | https://github.com/danielguterding/svdensity (accessed on 21 April 2023). |

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**Figure 1.**Log–log plot of the condition number of the kernel matrix G defined in Equation (6) as a function of the number of strikes. The fit with $f\left(x\right)=a\xb7{x}^{k}$ clearly shows that the growth of the condition number follows such a power law with exponent $k\approx 2$.

**Figure 2.**Logarithmic plot of the normalized singular values ${s}_{i}/{s}_{1}$ for various numbers of discretization points N. The number of strikes is fixed to $M=25$. The normalized singular values decay with an inverse power law of the form $f\left(x\right)={x}^{-k}$ with $k\approx 2.7$.

**Figure 3.**Comparison of the exact (bold line) and implied (dashed line) densities $\varphi \left(x\right)$ for two different values of the regularization parameter $\lambda $. The exact density was normal with $\sigma =0.1$ and shifted to $\mu ={S}_{0}exp\left(r\tau \right)\approx 0.105$. The top panel shows an under-regularized implied density ($\lambda ={10}^{-12}$), while the bottom panel shows a close to optimal implied density with $\lambda ={10}^{-7.5}$.

**Figure 4.**Log–log plot of the squared error in prices ${\chi}^{2}$ (top panel) and the Bhattacharyya distance ${d}_{B}$ (bottom panel). Both measures were calculated based on a comparison between the exact input data and our implied output data for prices and densities, respectively. The input option prices were based on a Bachelier model. The related input density was a normal distribution with $\sigma =0.2$ and shifted to $\mu ={S}_{0}exp\left(r\tau \right)\approx 0.105$. The vertical lines mark the positions of $\lambda ={10}^{-12}$ and $\lambda ={10}^{-7.5}$, for which we show the implied densities in Figure 3.

**Figure 5.**Visualization of the effect of regularization on the number of relevant parameters for the implied transformed density ${\varphi}^{\prime}$. A larger regularization parameter $\lambda $ led to fewer entries with significant magnitude in ${\varphi}^{\prime}$, i.e., regularization turned ${\varphi}^{\prime}$ into a sparse representation of the true density $\varphi $. The number of entries in ${\varphi}^{\prime}$ with a magnitude above the positive threshold a is denoted as $n(\left|{\varphi}_{i}^{\prime}\right|>a)$ and shown as a function of the regularization parameter $\lambda $. For $\lambda $ we chose to show the axis in logarithmic scale. The figure is based on the same Bachelier model as that in Figure 3 and Figure 4. The effect of regularization is clearly visible around $\lambda \approx {10}^{-7.5}$, where there was a sharp decrease in the number of parameters with significant magnitude.

**Figure 6.**Comparison of the exact (bold line) and implied (dashed line) density $\varphi \left(x\right)$ for two different values of the regularization parameter $\lambda $. The exact density was log-normal with $\sigma =0.2$ and shifted to $\mu ={S}_{0}=0.5$. The top panel shows a close to optimal regularized implied density ($\lambda ={10}^{-7.5}$), while the bottom panel shows an over-regularized implied density with $\lambda ={10}^{-3}$.

**Figure 7.**Log–log plot of the squared error in prices ${\chi}^{2}$ (top panel) and the Bhattacharyya distance ${d}_{B}$ (bottom panel). Both measures were calculated based on a comparison between exact input data and our implied output data for prices and densities, respectively. The input option prices were based on the Black–Scholes model. The related input density was a log-normal distribution with $\sigma =0.2$ and shifted to $\mu ={S}_{0}=0.5$. The vertical lines mark the positions of $\lambda ={10}^{-7.5}$ and $\lambda ={10}^{-3}$, for which we show the implied densities in Figure 6.

**Figure 8.**Comparison of the exact (bold line) and implied (dashed line) density $\varphi \left(x\right)$ for two different values of the regularization parameter $\lambda $. The exact density is a linear combination of normal distributions, according to Equation (25), with parameters taken from Table 1. The top panel shows an optimal regularized implied density ($\lambda ={10}^{-8.5}$), while the bottom panel shows an over-regularized implied density with $\lambda ={10}^{-4}$.

**Figure 9.**Log–log plot of the squared error in prices ${\chi}^{2}$ (top panel) and the Bhattacharyya distance ${d}_{B}$ (bottom panel). Both measures were calculated based on a comparison between exact input data and our implied output data for prices and density, respectively. The input density was given by Equation (25), with parameters taken from Table 1. The input option prices were calculated from an equivalent linear combination of Bachelier models with the same parameters, as in Table 1. The vertical lines mark the positions of $\lambda ={10}^{-8.5}$ and $\lambda ={10}^{-4}$, for which we show the implied densities in Figure 8.

**Figure 10.**Log–log plot of the squared error in prices ${\chi}^{2}$ (top panel) and log plot of the Bhattacharyya distance ${d}_{B}$ (bottom panel). Both measures were calculated based on a comparison between exact input data and our implied output data for prices and densities, respectively. The input density was given by Equation (25) with parameters taken from Table 2. The input option prices were calculated from an equivalent linear combination of Bachelier models with the same parameters as those in Table 2. The vertical lines mark the positions of $\lambda ={10}^{-7}$ and $\lambda ={10}^{-5}$, for which we show the implied densities in Figure 11.

**Figure 11.**Comparison of the exact (bold line) and implied (dashed line) density $\varphi \left(x\right)$ for two different values of the regularization parameter $\lambda $. The exact “density” is a linear combination of normal distributions according to Equation (25), using parameters from Table 2, which contains a region with negative probability. The top panel shows an under-regularized implied density ($\lambda ={10}^{-7}$), while the bottom panel shows a well-regularized implied density with $\lambda ={10}^{-5}$.

**Figure 12.**Comparison of the log-normal implied volatilities $\sigma $ (as a function of the option strike K) calculated from the input prices containing arbitrage (bold line), which were based on Equation (25) and parameters from Table 2, and the de-arbitraged prices calculated from our method (dashed line) at $\lambda ={10}^{-5}$. The calculation of de-arbitraged implied volatilities is based on the density shown in Figure 11 (bottom panel).

**Figure 13.**Log–log plot of the squared error in prices ${\chi}^{2}$ as a function of the regularization parameter $\lambda $ for SPX500 1M options as of 5 February 2018. The squared error was calculated from a comparison between exact input data and our implied output prices. The input option prices were calculated from the Black (1976) model with market data of Table 11 in Le Floc’h and Osterlee (2019a). The vertical lines mark the positions of $\lambda ={10}^{-7}$ and $\lambda ={10}^{-4}$, for which we show the implied densities in Figure 14.

**Figure 14.**Comparison of implied densities $\varphi \left(x\right)$ for SPX500 1M options as of 5 February 2018. A good compromise between accuracy and smoothness was achieved for $\lambda ={10}^{-7}$ (bold line), while $\lambda ={10}^{-4}$ yielded a density that contained fewer features (dashed line) and was potentially over-regularized. As explained in the main text, we rescaled both the price x of the underlying asset and the density $\varphi \left(x\right)$, for numerical reasons, by a factor of 1/1000 and 1000, respectively.

**Figure 15.**Comparison of input implied volatilities $\sigma $ (open circles) and the volatility smile provided by our method at $\lambda ={10}^{-7}$ (bold line) for SPX500 1M options as of 5 February 2018. The volatilities $\sigma $ are shown as a function of the option strike K in units of thousands. Clearly, our method reproduced the inputs with a high degree of accuracy and, additionally, provided a sensible extrapolation of the available data.

**Table 1.**Parameters for a multimodal density. These parameters are used in Equation (25) to generate a density which is a superposition of multiple normally distributed components.

i | ${\mathit{c}}_{\mathit{i}}$ | ${\mathit{\mu}}_{\mathit{i}}$ | ${\mathit{\sigma}}_{\mathit{i}}$ |
---|---|---|---|

1 | 0.50 | −0.20 | 0.10 |

2 | 0.45 | 0.15 | 0.15 |

3 | 0.05 | 0.55 | 0.05 |

**Table 2.**Parameters for a multimodal density. These parameters were used in Equation (25) to generate a density which was a superposition of multiple normally distributed components. This parameter set contained arbitrage, i.e., the resulting “density” contained negative “probabilities”. This was due to the negative pre-factor.

i | ${\mathit{c}}_{\mathit{i}}$ | ${\mathit{\mu}}_{\mathit{i}}$ | ${\mathit{\sigma}}_{\mathit{i}}$ |
---|---|---|---|

1 | 0.55 | 0.80 | 0.10 |

2 | −0.20 | 1.15 | 0.07 |

3 | 0.65 | 1.35 | 0.20 |

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## Share and Cite

**MDPI and ACS Style**

Guterding, D.
Sparse Modeling Approach to the Arbitrage-Free Interpolation of Plain-Vanilla Option Prices and Implied Volatilities. *Risks* **2023**, *11*, 83.
https://doi.org/10.3390/risks11050083

**AMA Style**

Guterding D.
Sparse Modeling Approach to the Arbitrage-Free Interpolation of Plain-Vanilla Option Prices and Implied Volatilities. *Risks*. 2023; 11(5):83.
https://doi.org/10.3390/risks11050083

**Chicago/Turabian Style**

Guterding, Daniel.
2023. "Sparse Modeling Approach to the Arbitrage-Free Interpolation of Plain-Vanilla Option Prices and Implied Volatilities" *Risks* 11, no. 5: 83.
https://doi.org/10.3390/risks11050083