# Pricing, Risk and Volatility in Subordinated Market Models

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## Abstract

**:**

## 1. Introduction

#### 1.1. Time Subordination in Financial Modelling

#### 1.2. Contributions of the Paper

- (a)
- demonstrate that the recent pricing formulas for the VG, NIG and FD models are precise and fast converging, and can be successfully used for other applications (e.g., calculations of volatility curve);
- (b)
- provides efficient closed-form formulas for first and second-order risk sensitivities (Delta, Gamma) and compare them with numerical techniques; and,
- (c)
- deduce from these formulas several practical features regarding delta-hedging policies and portfolio performance.

#### 1.3. Structure of the Paper

## 2. Exponential Lévy Processes

#### 2.1. Basics of Lévy Processes

#### 2.2. Exponential Lévy Motions

#### 2.3. Option Pricing

## 3. Subordinated Models

#### 3.1. Exponential VG Model

#### 3.1.1. Model Characteristics

**Remark**

**1.**

#### 3.1.2. Financial Applications

#### 3.2. Exponential NIG Model

#### 3.2.1. Model Characteristics

**Remark**

**2.**

#### 3.2.2. Financial Applications

#### 3.3. Fractional Diffusion Model

#### 3.3.1. Lévy-Stable Processes and Fractional Derivatives

**Remark**

**3.**

#### 3.3.2. Model Characteristics

**Remark**

**4.**

#### 3.3.3. Financial Applications

## 4. Pricing and Volatility Modelling

**Formula**

**1**

- -
- (OTM price) If ${k}_{VG}<0$,$$\begin{array}{c}{C}_{VG}^{-}({k}_{VG},{\sigma}_{\nu})\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{F}{2\Gamma \left(\frac{\tau}{\nu}\right)}\sum _{\begin{array}{c}{n}_{1}=0\\ {n}_{2}=1\end{array}}^{\infty}\frac{{(-1)}^{{n}_{1}}}{{n}_{1}!}\left(\right)open="["\; close>\frac{\Gamma (\frac{-{n}_{1}+{n}_{2}+1}{2}+{\tau}_{\nu})}{\Gamma (\frac{-{n}_{1}+{n}_{2}}{2}+1)}{\left(\right)}^{\frac{-{k}_{VG}}{{\sigma}_{\nu}}}{n}_{1}{\sigma}_{\nu}^{{n}_{2}}\hfill \end{array}\hfill \left(\right)open\; close="]">+\phantom{\rule{0.166667em}{0ex}}2\phantom{\rule{0.166667em}{0ex}}\frac{\Gamma (-2{n}_{1}-{n}_{2}-1-2{\tau}_{\nu})}{\Gamma (-{n}_{1}+\frac{1}{2}-{\tau}_{\nu})}{\left(\right)}^{\frac{-{k}_{VG}}{{\sigma}_{\nu}}}2{n}_{1}+1+2{\tau}_{\nu}{(-{k}_{VG})}^{{n}_{2}}& .$$
- -
- (ITM price) If ${k}_{VG}>0$,$${C}_{VG}^{+}({k}_{VG},{\sigma}_{\nu})\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{S}_{t}{e}^{-q\tau}-F-{C}_{VG}^{-}({k}_{VG},-{\sigma}_{\nu}).$$
- -
- (ATM price) If ${k}_{VG}=0$,$${C}_{VG}^{-}({k}_{VG},{\sigma}_{\nu})\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{C}_{VG}^{+}({k}_{VG},{\sigma}_{\nu})\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{F}{2\Gamma \left(\frac{\tau}{\nu}\right)}\sum _{n=1}^{\infty}\frac{\Gamma (\frac{n+1}{2}+{\tau}_{\nu})}{\Gamma (\frac{n}{2}+1)}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{\nu}^{n}.$$

**Proof.**

#### 4.1. At-the-Money Forward Approximations

#### 4.2. Implied Volatility

## 5. First-Order Sensitivities

**Formula**

**2**

- -
- (OTM sensitivity) If ${k}_{VG}<0$,$$\begin{array}{c}{\Delta}_{VG}^{-}({k}_{VG},{\sigma}_{\nu})\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{F}{2{S}_{t}\Gamma \left(\frac{\tau}{\nu}\right)}\sum _{\begin{array}{c}{n}_{1}=0\\ {n}_{2}=1\end{array}}^{\infty}\frac{{(-1)}^{{n}_{1}}}{{n}_{1}!}\left(\right)open="["\; close>-\frac{{n}_{1}\Gamma (\frac{-{n}_{1}+{n}_{2}+1}{2}+{\tau}_{\nu})}{\Gamma (\frac{-{n}_{1}+{n}_{2}}{2}+1)}{\left(\right)}^{\frac{-{k}_{VG}}{{\sigma}_{\nu}}}{n}_{1}-1{\sigma}_{\nu}^{{n}_{2}-1}\hfill \end{array}\hfill \left(\right)open\; close="]">+\phantom{\rule{0.166667em}{0ex}}2\phantom{\rule{0.166667em}{0ex}}\frac{\Gamma (-2{n}_{1}-{n}_{2}-2{\tau}_{\nu})}{\Gamma (-{n}_{1}+\frac{1}{2}-{\tau}_{\nu})}{\left(\right)}^{\frac{-{k}_{VG}}{{\sigma}_{\nu}}}2{n}_{1}+1+2{\tau}_{\nu}{(-{k}_{VG})}^{{n}_{2}-1}& .$$
- -
- (ITM sensitivity) If ${k}_{VG}>0$,$${\Delta}_{VG}^{+}({k}_{VG},{\sigma}_{\nu})\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{e}^{-q\tau}-{\Delta}_{VG}^{-}({k}_{VG},-{\sigma}_{\nu}).$$
- -
- (ATM sensitivity) If ${k}_{VG}=0$,$${\Delta}_{VG}^{-}({k}_{VG},{\sigma}_{\nu})\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\Delta}_{VG}^{+}({k}_{VG},{\sigma}_{\nu})\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{F}{2{S}_{t}\Gamma \left(\frac{\tau}{\nu}\right)}\sum _{n=1}^{\infty}\frac{\Gamma (\frac{n}{2}+{\tau}_{\nu})}{\Gamma \left(\frac{n+1}{2}\right)}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{\nu}^{n-1}.$$

#### 5.1. Delta Hedging

#### 5.2. Comparisons with Numerical Techniques

## 6. Second-Order Sensitivities and Portfolio Performance

#### 6.1. Gamma, Dollar Gamma

**Formula**

**3**

- -
- (OTM sensitivity) If ${k}_{VG}<0$,$$\begin{array}{c}{\Gamma}_{VG}^{-}({k}_{VG},{\sigma}_{\nu})\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{F}{2{S}_{t}^{2}{\sigma}_{\nu}\Gamma \left(\frac{\tau}{\nu}\right)}\sum _{n=0}^{\infty}\frac{{(-1)}^{n}}{n!}\left(\right)open="["\; close>\frac{\Gamma (-\frac{n}{2}+{\tau}_{\nu})}{\Gamma \left(\frac{-n+1}{2}\right)}{\left(\right)}^{\frac{-{k}_{VG}}{{\sigma}_{\nu}}}n\hfill \end{array}\hfill \left(\right)open\; close="]">+\phantom{\rule{0.166667em}{0ex}}2\phantom{\rule{0.166667em}{0ex}}\frac{\Gamma (-2n-2{\tau}_{\nu})}{\Gamma (-n+\frac{1}{2}-{\tau}_{\nu})}{\left(\right)}^{\frac{-{k}_{VG}}{{\sigma}_{\nu}}}2n+2{\tau}_{\nu}& .$$
- -
- (ITM sensitivity) If ${k}_{VG}>0$,$${\Gamma}_{VG}^{+}({k}_{VG},{\sigma}_{\nu})\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}-{\Gamma}_{VG}^{-}({k}_{VG},-{\sigma}_{\nu}).$$
- -
- (ATM sensitivity) If ${k}_{VG}=0$,$${\Gamma}_{VG}^{-}({k}_{VG},{\sigma}_{\nu})\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\Gamma}_{VG}^{+}({k}_{VG},{\sigma}_{\nu})\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{F}{2\sqrt{\pi}{S}_{t}^{2}{\sigma}_{\nu}\Gamma \left(\frac{\tau}{\nu}\right)}\frac{\Gamma ({\tau}_{\nu}-\frac{1}{2})}{\Gamma \left(\frac{\tau}{\nu}\right)}.$$

**Proof.**

#### 6.2. Properties and Particular Cases

## 7. Concluding Remarks

- (a)
- The pricing formulas are smooth and fast converging, and provide excellent agreement with efficient numerical techniques (such as the PROJ method). Moreover, these formulas can provide useful approximations for at-the-money options, and allow for the construction of volatility curves.
- (b)
- We have derived several analytical formulas for risk sensitivities and shown that they also provide excellent agreement with standard numerical (Fourier) evaluations.
- (c)
- Thanks to these formulas, we were able to show that the presence of a time subordination in the VG, NIG, and FD models has a minimal impact on the delta hedging policy of an at-the-money option, but, on the contrary, has a direct impact on the P&L of the delta hedged portfolio.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Black–Scholes (circles) as the limit of VG$(\sigma ,\nu ,0)\stackrel{\nu \to 0}{\u27f6}$ BS$\left(\sigma \right)$.

**Figure 2.**Log-return densities for several maturities $\tau $, for the VG$(0.3,0.5,0)$ model (

**Left**) and the NIG$(9.0,\phantom{\rule{3.33333pt}{0ex}}0,\phantom{\rule{3.33333pt}{0ex}}1.2)$ model (

**Right**).

**Figure 3.**Exponential convergence of pricing Formula (1) for a call option: $\tau =1,{S}_{t}=K=4000,r\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.01,q=0$. (

**Left**) VG$(\sigma ,\nu ,0)$ with $\sigma =0.3$. (

**Right**) NIG$(\alpha ,0,\delta )$ with $\delta =1.2$. Here $N={n}_{1}={n}_{2}$ is the number of terms in the truncated series.

**Figure 4.**Implied volatility smiles of VG$(\sigma ,\nu ,0)$ obtained by Formula (1). Params: $\sigma =0.3$, $\tau =0.2$, $r=1\%,q=0\%$, ${S}_{t}=4000$. The moneyness is determined by $F:={S}_{t}exp\left((r-q)\tau \right)$.

**Figure 5.**Delta (

**Left**) and Dollar Gamma (

**Right**) of a call option under VG$(\sigma ,\nu ,0)$ using Formula (2) and Formula (3). Greeks of BS$\left(\sigma \right)$ are provided for reference (dash lines). Params: $\sigma =0.3$, $\tau =1$, $r=1\%,q=0\%$, $K=4000$.

**Figure 6.**Delta (

**Left**) and Dollar Gamma (

**Right**) of a call option under NIG$(\alpha ,0,\delta )$ using Formulas (2) and (3). Greeks of BS$(\sigma =0.3)$ are provided for reference (dash lines). Params: $\delta =1.2$, $\tau =1$, $r=1\%,q=0\%$, $K=4000$.

**Table 1.**Some subordinated market models, and their limiting cases. Time changed models (exponential VG and NIG), FD, and sub-BS models recover the Black–Scholes (BS) model for specific values of their subordination parameters.

Subordinated Model | Limiting Regimes |
---|---|

VG$(\sigma ,\nu ,\theta )$ | VG$(\sigma ,\nu ,0)\stackrel{\nu \to 0}{\u27f6}$ BS$\left(\sigma \right)$ |

NIG$(\alpha ,\beta ,\delta )$ | NIG$(\alpha ,0,\delta )\stackrel{\alpha \to \infty}{\u27f6}$ BS$\left(\sqrt{\delta /\alpha}\right)$ |

FD$(\sigma ,\alpha ,\gamma )$ | FD$(\sigma ,\alpha ,\gamma )\stackrel{\gamma \to 1}{\u27f6}$ FMLS$(\sigma ,\alpha )\stackrel{\alpha \to 2}{\u27f6}$ BS$\left(\sigma \right)$ |

sub-BS($\sigma $,$\gamma $) $:=$ FD$(\sigma ,2,\gamma )$ | sub-BS$(\sigma ,\gamma )\stackrel{\gamma \to 1}{\u27f6}$ BS$\left(\sigma \right)$ |

**Table 2.**Volatility modelling for ATMF options in various subordinated models, and their limiting cases.

ATMF Implied Volatility (European Options) | |
---|---|

Exponential VG | ${\sigma}_{VG}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\sqrt{\frac{2\pi}{\nu}}\frac{\Gamma \left(\frac{\tau}{\nu}\right)}{\Gamma (\frac{1}{2}+\frac{\tau}{\nu})}\phantom{\rule{0.166667em}{0ex}}\frac{{C}_{t}}{{S}_{t}}$ |

Low variance regime ($\nu \to 0$): | |

${\sigma}_{VG}\to \sqrt{\frac{2\pi}{\tau}}\frac{{C}_{t}}{{S}_{t}}$ | |

Exponential NIG | Solve $\frac{{S}_{t}\delta \tau {e}^{\alpha \delta \tau}}{\pi}{K}_{0}\left(\alpha \delta \tau \right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{C}_{t}$ |

Large steepness regime ($\alpha \to \infty $): | |

$\delta \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{2\pi \alpha}{\tau}\frac{{C}_{t}^{2}}{{S}_{t}^{2}}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}\frac{1}{4\alpha \tau}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}O\left(\right)open="("\; close=")">\frac{1}{{\alpha}^{2}}$ | |

At order ${\alpha}^{0}$: | |

${\sigma}_{NIG}:=\sqrt{\frac{\delta}{\alpha}}=\sqrt{\frac{2\pi}{\tau}}\frac{{C}_{t}}{{S}_{t}}$ | |

sub-BS | ${\sigma}_{FD}=\frac{2\sqrt{\Gamma (1+2\gamma )}\Gamma (1+\frac{\gamma}{2})}{{\tau}^{\frac{\gamma}{2}}}\phantom{\rule{0.166667em}{0ex}}\frac{{C}_{t}}{{S}_{t}}$ |

Non-fractional regime ($\gamma \to 1$): | |

${\sigma}_{FD}\to \sqrt{\frac{2\pi}{\tau}}\frac{{C}_{t}}{{S}_{t}}$ |

**Table 3.**First order sensitivity (Delta) of European call options in the exponential VG and NIG models, obtained by truncations of Formula (2), and by a numerical evaluation of (74). Here, $N={n}_{1}={n}_{2}$ is the number of terms in the truncated series. Parameters: $K=4000$, $r=1\%$, $q=0\%$, $\tau =1$.

Exponential VG Model [$\mathit{\sigma}=0.2$, $\mathit{\nu}=0.85$] | |||||
---|---|---|---|---|---|

Formula (2) | Lewis (74) | ||||

$\mathit{N}=\mathbf{3}$ | $\mathit{N}=\mathbf{5}$ | $\mathit{N}=\mathbf{10}$ | $\mathit{N}=\mathbf{15}$ | ||

Deep OTM (${S}_{t}=3000$) | 2.1823 | 0.6347 | 0.0941 | 0.0940 | 0.0940 |

OTM (${S}_{t}=3500$) | 0.4113 | 0.2567 | 0.2455 | 0.2455 | 0.2455 |

ATM (${S}_{t}=4040.90$) | 0.5703 | 0.5718 | 0.5719 | 0.5719 | 0.5719 |

ITM (${S}_{t}=4500$) | 0.7569 | 0.8113 | 0.8134 | 0.8134 | 0.8134 |

Deep ITM (${S}_{t}=5000$) | 0.4729 | 0.8589 | 0.9206 | 0.9206 | 0.9206 |

Exponential NIG Model [$\mathbf{\alpha}=\mathbf{9}$, $\mathbf{\delta}=\mathbf{1}.\mathbf{2}$] | |||||

Formula (2) | Lewis (74) | ||||

$\mathit{N}=\mathbf{3}$ | $\mathit{N}=\mathbf{5}$ | $\mathit{N}=\mathbf{10}$ | $\mathit{N}=\mathbf{15}$ | ||

Deep OTM (${S}_{t}=3000$) | 0.2921 | 0.2722 | 0.2747 | 0.2748 | 0.2748 |

OTM (${S}_{t}=3500$) | 0.4289 | 0.4309 | 0.4311 | 0.4311 | 0.4311 |

ATM (${S}_{t}=4234.09$) | 0.6336 | 0.6410 | 0.6412 | 0.6412 | 0.6412 |

ITM (${S}_{t}=4500$) | 0.6936 | 0.7030 | 0.7033 | 0.7033 | 0.7033 |

Deep ITM (${S}_{t}=5000$) | 0.7827 | 0.7966 | 0.7971 | 0.7971 | 0.7971 |

**Table 4.**First and second order market sensitivities (ATMF situation) for European call options in various subordinated models, and their limiting cases. Time subordination does not affect the Delta, but it directly impacts the Gamma of options.

1st Order ($\mathbf{\Delta}$) | 2nd Order ($\mathbf{\Gamma}$) | |
---|---|---|

Exponential VG | $\frac{1}{2}$ | $\frac{1}{2\sqrt{\pi}{S}_{t}{\sigma}_{\nu}}\frac{\Gamma (\frac{\tau}{\nu}-\frac{1}{2})}{\Gamma \left(\frac{\tau}{\nu}\right)}$ |

Low variance regime ($\nu \to 0$): | ||

$\frac{1}{{S}_{t}\sigma \sqrt{2\pi \tau}}$ | ||

Exponential NIG | $\frac{1}{2}$ | $\frac{\alpha {e}^{\alpha \delta \tau}}{\pi {S}_{t}}{K}_{1}\left(\alpha \delta \tau \right)$ |

Large steepness regime ($\alpha \to \infty $): | ||

$\frac{1}{{S}_{t}\sigma \sqrt{2\pi \tau}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\sigma}^{2}:=\frac{\delta}{\alpha}$ | ||

FD | $\frac{1}{\alpha}$ | $\frac{1}{\alpha {S}_{t}}\phantom{\rule{0.166667em}{0ex}}\frac{{(-{\omega}_{FD}{\tau}^{\gamma})}^{-\frac{1}{\alpha}}}{\Gamma (1-\frac{\gamma}{\alpha})}$ |

sub-BS | $\frac{1}{2}$ | $\frac{1}{2{S}_{t}}\frac{\sqrt{\Gamma (1+2\gamma )}}{\Gamma (1-\frac{\gamma}{2})\sigma {\tau}^{\frac{\gamma}{2}}}$ |

Non fractional regime ($\gamma \to 1$): | ||

$\frac{1}{{S}_{t}\sigma \sqrt{2\pi \tau}}$ |

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**MDPI and ACS Style**

Aguilar, J.-P.; Kirkby, J.L.; Korbel, J.
Pricing, Risk and Volatility in Subordinated Market Models. *Risks* **2020**, *8*, 124.
https://doi.org/10.3390/risks8040124

**AMA Style**

Aguilar J-P, Kirkby JL, Korbel J.
Pricing, Risk and Volatility in Subordinated Market Models. *Risks*. 2020; 8(4):124.
https://doi.org/10.3390/risks8040124

**Chicago/Turabian Style**

Aguilar, Jean-Philippe, Justin Lars Kirkby, and Jan Korbel.
2020. "Pricing, Risk and Volatility in Subordinated Market Models" *Risks* 8, no. 4: 124.
https://doi.org/10.3390/risks8040124