# Simple Formulas for Pricing and Hedging European Options in the Finite Moment Log-Stable Model

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## Abstract

**:**

## 1. Introduction

## 2. Lévy-Stable Option Pricing

#### 2.1. Model Definition

#### 2.2. Stable Distributions

- −
- If $1<\alpha <2$, then ${g}_{\alpha ,-1}$ decays exponentially on the positive real axis and has a heavy tail on the negative real axis (that is, decays in ${\left|x\right|}^{-\alpha}$);
- −
- If $\alpha =2$, then $\omega (k,\alpha )=0$ and in that case the transform (5) is independent of $\beta $ and resumes to ${e}^{{\left|k\right|}^{2}}$, that is, the (re-scaled) Fourier transform of the heat kernel. Therefore, ${L}_{\alpha ,\beta}\left(t\right)$ degenerates into the usual Brownian motion $W\left(t\right)$ and the process (2) is a geometric Brownian motion.

#### 2.3. Mellin-Barnes Representation of the European Option

**Proposition**

**1.**

#### 2.4. Pricing Formulas

**Theorem**

**1**(Pricing formula)

**.**

**Proof.**

**Corollary**

**1**(At-the-money price)

**.**

## 3. Risk Sensitivities (Greeks)

#### 3.1. Delta

- In left figure, we plot the value of ${\Delta}_{{C}_{\alpha}}$ in function of the market price, for different cases of $\alpha $; in all cases, $0<{\Delta}_{{C}_{\alpha}}<1$ for all S, and ${\Delta}_{{C}_{\alpha}}$ admits an inflection in the “out-of-the-money” region ($S<K$). However, we can observe that in this region, ${\Delta}_{{C}_{\alpha}}$ grows faster when $\alpha $ decays, and the inflection occurs for smaller market prices.
- In the right figure, we choose 3 different values of S corresponding to the in, at or out-of-the-money situation and we plot the evolution of ${\Delta}_{{C}_{\alpha}}$ in function of $\alpha $. We can observe that ${\Delta}_{{C}_{\alpha}}$ is in all cases a decreasing function of $\alpha $ (as could be expected from the overall $\frac{1}{\alpha}$ factor in (25)) meaning that when $\alpha $ becomes smaller, then the options become more sensitive to variations of the underlying price than in the Gaussian ($\alpha =2$) case. This stronger sensitivity can be regarded as a conservative feature of the FMLS model (similar features have also been observed in Robinson (2015)).

#### 3.2. Gamma

#### 3.3. Theta

## 4. Expected P&L

#### 4.1. Long-Call Position

#### 4.2. Delta-Hedged Portfolio

#### 4.3. Gamma-Hedged Portfolio (Synthetic Future)

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Call (

**left graph**) and put (

**right graph**) option prices, as a function of S and for various stability parameters $\alpha $ (parameters: $K=4000$, $r=1\%$ and $\sigma =20\%$). In both the call and put cases, the prices become higher as $\alpha $ decreases.

**Figure 2.**(

**Left graph**): Plot of the call’s Delta, in function of the market price S and for different stability parameters $\alpha $. (

**Right graph**): Plot of the call’s Delta, in function of the stability parameter $\alpha $ and for different market configurations. In both cases, $K=4000$, $r=1\%$ and $\sigma =20\%$.

**Figure 3.**(

**Left graph**): Theta of an at-the-money call ($S=K{e}^{-r\tau})$. (

**Right graph**): Theta of a deeply in the money put ($S=3500$). In both cases $K=4000$, $r=1\%$, $\sigma =20\%$ and $T=2$.

**Figure 4.**Expected daily P&L of a long-call position, for various values of the stability parameter; one observes that daily losses or gains are accentuated when $\alpha $ departs from 2.

**Figure 5.**Expected daily P&L of a delta-hedged portfolio; as before, the loss is accentuated when departing from the Gaussian case.

**Table 1.**Numerical values for the $(n,m)$-term in the series (16) for the option price ($S=3800$, $K=4000,\phantom{\rule{0.166667em}{0ex}}r=1\%,\phantom{\rule{0.166667em}{0ex}}\sigma =20\%,\phantom{\rule{0.166667em}{0ex}}\tau =1Y,\phantom{\rule{0.166667em}{0ex}}\alpha =1.7$). The call price converges to a precision of ${10}^{-3}$ after summing only very few terms of the series.

n /m | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

0 | 395.167 | 49.052 | 4.962 | 0.431 | 0.033 | 0.002 | 0.000 |

1 | −190.223 | −32.268 | −4.005 | −0.405 | −0.035 | −0.003 | −0.000 |

2 | 23.829 | 7.767 | 1.317 | 0.164 | 0.017 | 0.001 | 0.000 |

3 | 1.430 | −0.649 | −0.211 | −0.036 | −0.004 | −0.000 | −0.000 |

4 | −0.246 | −0.029 | 0.013 | 0.001 | 0.000 | 0.000 | 0.000 |

5 | −0.046 | 0.004 | 0.000 | −0.000 | −0.000 | −0.000 | −0.000 |

6 | 0.001 | 0.000 | −0.000 | −0.000 | 0.000 | 0.000 | 0.000 |

7 | 0.001 | −0.000 | −0.000 | 0.000 | 0.000 | −0.000 | −0.000 |

8 | 0.000 | −0.000 | 0.000 | 0.000 | −0.000 | −0.000 | 0.000 |

Call | 229.914 | 253.790 | 255.866 | 256.024 | 256.035 | 256.035 | 256.035 |

**Table 2.**Numerical values for the $(n,m)$-term in the series (25) for the call option’s Delta ($S=3800$, $K=4000,\phantom{\rule{0.166667em}{0ex}}r=1\%,\phantom{\rule{0.166667em}{0ex}}\sigma =20\%,\phantom{\rule{0.166667em}{0ex}}\tau =1Y,\phantom{\rule{0.166667em}{0ex}}\alpha =1.7$).

n /m | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|

0 | 0.613034 | 0.103991 | 0.0129082 | 0.001306 | 0.000113 | 8.8 × ${10}^{-6}$ |

1 | −0.153590 | −0.050059 | −0.008492 | −0.001054 | −0.000107 | −9.3 × ${10}^{-6}$ |

2 | −0.013825 | 0.006271 | 0.002044 | 0.000347 | 0.000043 | 4.4 × ${10}^{-6}$ |

3 | 0.003174 | 0.000376 | −0.000171 | −0.000056 | −9.4 × ${10}^{-6}$ | −1.2 × ${10}^{-6}$ |

4 | 0.000743 | −0.000065 | −7.6 × ${10}^{-6}$ | 3.5 × ${10}^{-6}$ | 1.1 × ${10}^{-6}$ | 1.9 × ${10}^{-7}$ |

5 | −0.000026 | −0.000012 | 1.1 × ${10}^{-6}$ | 1.2 × ${10}^{-7}$ | −5.7 × ${10}^{-8}$ | −1.9 × ${10}^{-8}$ |

6 | −0.000023 | 3.5 × ${10}^{-7}$ | 1.7 × ${10}^{-7}$ | −1.4 × ${10}^{-8}$ | −1.7 × ${10}^{-9}$ | 7.7 × ${10}^{-10}$ |

7 | −1.3 × ${10}^{-6}$ | 2.7 × ${10}^{-7}$ | −4.1 × ${10}^{-9}$ | −1.9 × ${10}^{-9}$ | 1.7 × ${10}^{-10}$ | 2.0 × ${10}^{-11}$ |

Delta | 0.449486 | 0.509990 | 0.516273 | 0.516819 | 0.516861 | 0.516864 |

**Table 3.**Monthly P&L explain for various portfolios on the S&P 500 and different stability parameters.

Long-Call Position | ||||
---|---|---|---|---|

Stability | Total Expected P&L | Time Effect | Spot Price Effect | Gamma Effect |

$\alpha $ = 2 (Black-Scholes) | 36.2548 | −9.6994 | 45.9514 | 0.0027 |

$\alpha $ = 1.8 | 38.6730 | −10.9094 | 49.5790 | 0.0034 |

$\alpha $ = 1.6 | 42.5902 | −12.8088 | 55.3948 | 0.0042 |

$\alpha $ = 1.4 | 48.4861 | −15.6892 | 64.1705 | 0.0049 |

Long Put Position | ||||

Stability | Total Expected P&L | Time Effect | Spot Price Effect | Gamma Effect |

$\alpha $ = 2 (Black-Scholes) | −127.1943 | −6.8985 | −120.2986 | 0.0027 |

$\alpha $ = 1.8 | −124.7761 | −8.1085 | −116.6710 | 0.0034 |

$\alpha $ = 1.6 | −120.8589 | −10.0079 | −110.8552 | 0.0042 |

$\alpha $ = 1.4 | −114.9630 | −12.8883 | −102.0795 | 0.0049 |

Delta-Hedged Portfolio | ||||

Stability | Total Expected P&L | Time Effect | Spot Price Effect | Gamma Effect |

$\alpha $ = 2 (Black-Scholes) | −9.6966 | −9.6994 | - | 0.0027 |

$\alpha $ = 1.8 | −10.9060 | −10.9094 | - | 0.0034 |

$\alpha $ = 1.6 | −12.8045 | −12.8088 | - | 0.0042 |

$\alpha $ = 1.4 | −15.6843 | −15.6892 | - | 0.0049 |

Gamma-Hedged Portfolio | ||||

Stability | Total Expected P&L | Time Effect | Spot Price Effect | Gamma Effect |

All $\alpha $ | 163.449 | −2.8009 | 166.25 | - |

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## Share and Cite

**MDPI and ACS Style**

Aguilar, J.-P.; Korbel, J.
Simple Formulas for Pricing and Hedging European Options in the Finite Moment Log-Stable Model. *Risks* **2019**, *7*, 36.
https://doi.org/10.3390/risks7020036

**AMA Style**

Aguilar J-P, Korbel J.
Simple Formulas for Pricing and Hedging European Options in the Finite Moment Log-Stable Model. *Risks*. 2019; 7(2):36.
https://doi.org/10.3390/risks7020036

**Chicago/Turabian Style**

Aguilar, Jean-Philippe, and Jan Korbel.
2019. "Simple Formulas for Pricing and Hedging European Options in the Finite Moment Log-Stable Model" *Risks* 7, no. 2: 36.
https://doi.org/10.3390/risks7020036