Market calibration under a long memory stochastic volatility model

ABSTRACT

In this article, we study a long memory stochastic volatility model (LSV), under which stock prices follow a jump-diffusion stochastic process and its stochastic volatility is driven by a continuous-time fractional process that attains a long memory. LSV model should take into account most of the observed market aspects and unlike many other approaches, the volatility clustering phenomenon is captured explicitly by the long memory parameter. Moreover, this property has been reported in realized volatility time-series across different asset classes and time periods. In the first part of the article, we derive an alternative formula for pricing European securities. The formula enables us to effectively price European options and to calibrate the model to a given option market. In the second part of the article, we provide an empirical review of the model calibration. For this purpose, a set of traded FTSE 100 index call options is used and the long memory volatility model is compared to a popular pricing approach – the Heston model. To test stability of calibrated parameters and to verify calibration results from previous data set, we utilize multiple data sets from NYSE option market on Apple Inc. stock.

KEYWORDS:

• European call option
• stochastic volatility
• long memory
• fractional process
• market calibration

Acknowledgements

This work was supported by the GACR Grant 14-11559S Analysis of Fractional Stochastic Volatility Models and their Grid Implementation. Computational resources were provided by the CESNET LM2015042 and the CERIT Scientific Cloud LM2015085, provided under the programme “Projects of Large Research, Development, and Innovations Infrastructures”.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. A risk-neutral probability measure for this model is not uniquely defined due to the incompleteness of the market, purely for derivatives pricing we do not need to specify it. Comments on the equivalent martingale measures for classical stochastic volatility models are available, for instance, in Sircar and Papanicolaou (1999) and references therein.

2. This assumption is taken into consideration in many jump-diffusion stock models, e.g. Bates (1996).

3. This is possible due to the stochastic independence with diffusion processes and log-normal distribution of the jumps, see Gatheral (2006).

4. In case of the presented approach, ${\mathrm{\Theta }}^{\star }$ takes form: ${\mathrm{\Theta }}^{\star }:=\left\{{\kappa }^{\star },v_0^{\star },{\bar{v}}^{\star },{\xi }^{\star },{\rho }^{\star },{\lambda }_J^{\star },{\alpha }_J^{\star },{\gamma }_J^{\star },H^{\star }\right\}$

5. Data set obtained from OMON Screen, Bloomberg L.P. 2014.

6. Other data sets possessed slightly narrower moneyness range.

7. In fact, $G({\mathrm{\Theta }}^{\star })$ represents weighted least squares of the market fit.

8. One can calibrate the model using heuristic algorithms that evaluate model prices very frequently.

9. Under Heston model, $\xi$ represents volatility of volatility and thus, one would intuitively expect that the increased upper bound would not affect the solution. Under the LSV approach, however, vol. of vol. takes the following form, $\xi {\epsilon }^{H-1/2}$ and thus, $\xi$ might take greater values.

Additional information

Funding

This work was supported by the GACR Grant 14-11559S Analysis of Fractional Stochastic Volatility Models and their Grid Implementation.